Quantum Mechanics in NonLinear Systems

QUANTUMMECHANICS IN

NONLINEAR SYSTEMS

QUANTUMMECHANICS IN

NONLINEAR SYSTEMS

Pang Xiao-FengUniversity of Electronic Science and Technology of China, China

Feng Yuan-PingNational University of Singapore, Singapore

\[p World ScientificNEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Co vent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication DataPang, Xiao-Feng, 1945-

Quantum mechanics in nonlinear systems / Pang Xiao-Feng, Feng Yuan-Ping,p. cm.

Includes bibliographical references and index.ISBN 9812561161 (alk. paper) ISBN 9812562990 (pbk)

1. Nonlinear theories. 2. Quantum theory. I. Feng, Yuang-Ping. II. Title.

QC20.7.N6P36 2005530.15'5252-dc22

2004060119

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright ClearanceCenter, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopyis not required from the publisher.

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface

This book discusses the properties of microscopic particles in nonlinear systems,principles of the nonlinear quantum mechanical theory, and its applications in con-densed matter, polymers, and biological systems. It is intended for researchers,graduate students, and upper level undergraduate students.

About the Book

Some materials in the book are based on the lecture notes for a graduate course"Problems in nonlinear quantum theory" given by one of the authors (X. F. Pang)in his university in the 1980s, and a book entitled "Theory of Nonlinear QuantumMechanics" (in Chinese) by the same author in 1994. However, the contents werecompletely rewritten in this English edition, and in the process, we incorporatedrecent results related to the nonlinear Schrodinger equations and the nonlinearKlein-Gordon equations based on research of the authors as well as other scientistsin the field.

The following topics are covered in 10 chapters in this book, the necessity forconstructing a nonlinear quantum mechanical theory; the theoretical and experi-mental foundations on which the nonlinear quantum mechanical theory is based; theelementary principles and the theory of nonlinear quantum mechanics; the wave-corpuscle duality of particles in the theory; nonlinear interaction and localization ofparticles; the relations between nonlinear and linear quantum theories; the proper-ties of nonlinear quantum mechanics, including simultaneous determination of po-sition and momentum of particles, self-consistence and completeness of the theory;methods of solving nonlinear quantum mechanical problems; properties of particlesin various nonlinear systems and applications to exciton, phonon, polaron, electron,magnon and proton in physical, biological and polymeric systems. In particular,an in-depth discussion on the wave-corpuscle duality of microscopic particles innonlinear systems is given in this book.

The book is organized as follows. We start with a brief review on the postu-lates of linear quantum mechanics, its successes and problems encountered by thelinear quantum mechanics in Chapter 1. In Chapter 2, we discuss some macro-

V

vi Quantum Mechanics in Nonlinear Systems

scopic quantum effects which form the experimental foundation for a new nonlinearquantum theory, and the properties of microscopic particles in the macroscopicquantum systems which provide a theoretical base for the establishment of the non-linear quantum theory. The fundamental principles on which the new theory isbased and the theory of nonlinear quantum mechanics as proposed by Pang et al.are given in Chapter 3. The close relations among the properties of macroscopicquantum effects; nonlinear interactions and soliton motions of microscopic particlesin macroscopic quantum systems play an essential role in the establishment of thistheory. In Chapter 4, we examine in details the wave-corpuscle duality of parti-cles in nonlinear systems. In Chapter 5, we look into the mechanisms of nonlinearinteractions and their relations to localization of particles. In the next chapter,features of the nonlinear and linear quantum mechanical theories are compared; theself-consistence and completeness of the theory were examined; and finally solutionsand properties of the time-independent nonlinear quantum mechanical equations,and their relations to the original quantum mechanics are discussed. We will showthat problems existed in the original quantum mechanics can be explained by thenew nonlinear quantum mechanical theory. Chapter 7 shows the methods of solvingvarious kinds of nonlinear quantum mechanical problems. The dynamic propertiesof microscopic particles in different nonlinear systems are discussed in Chapter 8.Finally in Chapters 9 and 10, applications of the theory to exciton, phonon, elec-tron, polaron, proton and magnon in various physical systems, such as condensedmatter, polymers, molecules and living systems, are explored.

The book is essentially composed of three parts. The first part consists of Chap-ters 1 and 2, gives a review on the linear quantum mechanics, and the importantexperimental and theoretical studies that lead to the establishment of the nonlinearquantum-mechanical theory. The nonlinear theory of quantum mechanics itself aswell as its essential features are described in second part (Chapters 3-8). In thethird part (Chapters 9 and 10), we look into applications of this theory in physics,biology and polymer, etc.

An Overview

Nonlinear quantum mechanics (NLQM) is a theory for studying properties andmotion of microscopic particles (MIPs) in nonlinear systems which exhibit quantumfeatures. It was named so in relation to the quantum mechanics established by Bohr,Heisenberg, Schrodinger, and many others. The latter deals with only propertiesand motion of microscopic particles in linear systems, and will be referred to as thelinear quantum mechanics (LQM) in this book.

The concept of nonlinearity in quantum mechanics was first proposed by deBroglie in the 1950s in his book, "Nonlinear wave mechanics". LQM had difficultiesexplaining certain problems right from the start, de Broglie attempted to clarifyand solve these problems of LQM using the concept of nonlinearity. Even thougha great idea, de Broglie did not succeed because his approach was confined to the

Preface vii

framework of the original LQM.Looking back to the modern history of physics and science, we know that quan-

tum mechanics is really the foundation of modern science. It had great successesin solving many important physical problems, such as the light spectra of hydro-gen and hydrogen-like atoms, the Lamb shift in these atoms, and so on. Jargonssuch as "quantum jump" have their scientific origins and become ever fashion-able in our normal life. In this particular case, the phrase "quantum jump" givesa vivid description for major qualitative changes and is almost universally used.However, it was also known that LQM has its problems and difficulties relatedto the fundamental postulates of the theory, for example, the implications of theuncertainty principle between conjugate dynamical variables, such as position andmomentum. Different opinions on how to resolve such issues and further developquantum mechanics lead to intense arguments and debates which lasted almost acentury. The long-time controversy showed that these problems cannot be solvedwithin the framework of LQM. It was also through such debates that the directionto take for improving and further developing quantum mechanics became clear,which was to extend the theory from the linear to the nonlinear regime. Certainfundamental assumptions such as the principle of linear superposition, linearity ofthe dynamical equation and the independence of the Hamiltonian of a system onits wave function must be abandoned because they are the roots of the problems ofLQM. In other words, a new nonlinear quantum theory should be developed.

A series of nonlinear quantum phenomena including the macroscopic quantumeffects and motion of soli tons or solitary waves have, in recent decades, been dis-covered one after another from experiments in superconductors, superfluid, fer-romagnetic, antiferromagnetic, organic molecular crystals, optical fiber materialsand polymer and biological systems, etc. These phenomena did underlie nonlin-ear quantum mechanics because they could not be explained by LQM. Meanwhile,the theories of nonlinear partial differential equations and of solitary wave havebeen very well established which build the mathematical foundation of nonlinearquantum mechanics. Due to these developments of nonlinear science, a lot of newbranches of science, for example, nonlinear vibrational theory, nonlinear Newtonmechanics, nonlinear fluid mechanics, nonlinear optics, chaos, synergetics and frac-tals, have been established or being developed. In such a case, it is necessary tobuild the nonlinear quantum mechanics described the law of motion of microscopicparticles in nonlinear systems.

However, how do we establish such a theory? Experiences in the study of quan-tum mechanics for several decades tell us that it is impossible to establish such atheory if we followed the direction of de Broglie et al. A completely new way ofthinking, a new idea and method must be adopted and developed.

According to this idea we will, first of all, study the properties of macroscopicquantum effects, which is a nonlinear quantum effect on macroscopic scale occurredin some matters, for example, superconductors and superfluid. To be more precise,

viii Quantum Mechanics in Nonlinear Systems

these effects occur in systems with ordered states over a long-range, or, coherentstates, or, Bose-like condensed states, which are formed through phase transitionsafter a spontaneous symmetry breakdown in the systems by means of nonlinearinteractions. These results show that the properties of microscopic particles in themacroscopic quantum systems cannot be well represented by LQM. In these systemsthe microscopic particles are self-localized to become soliton with wave-corpuscleduality. The observed macroscopic quantum effects are just a result produced bysoliton motions of the particles in these systems. Therefore, the macroscopic quan-tum effect is closely related to the nonlinear interaction and to solitary motion ofthe particles. The close relations among them prompt us to propose and establishthe fundamental principles and the theory of NLQM which describes the propertiesof microscopic particles in the nonlinear systems. We then demonstrate that theNLQM is truely a self-consistent and complete theory. It has so far enjoyed greatsuccesses in a wide range of applications in condensed matter, polymers and biolog-ical systems. In exploring these applications, we also obtain many important resultswhich are consistent with experimental data. These results confirm the correctnessof the NLQM on one hand, and provide further theoretical understanding to manyphenomena occurred in these systems on the other hand.

Therefore, we can say that the experimental foundation of the nonlinear quan-tum mechanics established is the macroscopic quantum effects, and the coherentphenomena. Its theoretical basis is superconducting and superfluidic theories. Itsmathematical framework is the theories of nonlinear partial differential equationsand of solitary waves. The elementary principles and theory of the NLQM proposedhere are established on the basis of results of research on properties of microscopicparticles in nonlinear systems and the close relations among the macroscopic quan-tum effects, nonlinear interactions and soliton motions. The linearity in the LQMis removed and dependence of Hamiltonian of systems on the state wave functionof particles is assumed in this theory. Through careful investigations and extensiveapplications, we demonstrate that this new theory is correct, self-consistent andcomplete. The new theory solves the problems and difficulties in the LQM.

One of the authors (X. F. Pang) has been studying the NLQM for about 25 yearsand has published about 100 papers related to this topic. The newly establishednonlinear quantum theory has been reported and discussed in many internationalconferences, for example, International Conference of Nonlinear Physics (ICNP), In-ternational Conference of Material Physics (ICMP), Asia Pacific Physics Conference(APPC), International Workshop of Nonlinear Problems in Science and Engineering(IWNPSE), National Quantum Mechanical Conference of China (NQMCC), etc..Pang also published a monograph entitled "The problems for nonlinear quantumtheory" in 1985 and a book entitled "The theory of nonlinear quantum mechanics"in 1994 in Chinese. Pang has also lectured in many Universities and Institutes onthis subject. Certain materials in this book are based on the above lecture materialsand book. It also incorporates many recent results published by Pang and other

Preface ix

scientists related to nonlinear Schrodinger equation and nonlinear Klein-Gordonequations.

Finally, we should point out that the NLQM presented here is completely dif-ferent from the LQM. It is intended for studying properties and motion of micro-scopic particles in nonlinear systems, in which the microscopic particles becomeself-localized particles, or solitons, under the nonlinear interaction. Sources ofsuch nonlinear interation can be intrinsic nonlinearity or persistent self-interactionsthrough mechanisms such as self-trapping, self-condensation, self-focusing and self-coherence by means of phase transitions, sudden changes and spontaneous break-down of symmetry of the systems, and so on. In such cases, the particles haveexactly wave-corpuscle duality, and obey simultaneously the classical and quantumlaws of motion, i. e., the nature and properties of the microscopic particle are es-sentially changed from that in LQM. For example, the position and momentum ofa particle can be determined to a certain degree. Thus, the linear feature of theoryand the principles for independences of the Hamiltonian of the systems on the state-wave function of particle are completely removed. However, this is not to deny thevalidity of LQM. Rather we believe that it is an approximate theory which is onlysuitable for systems with linear interactions and the nonlinear interaction is smalland can be neglected. In other words, LQM is a special case of the NLQM. Thisrelation between the LQM and the NLQM is similar to that between the relativityand Newtonian mechanics. The NLQM established here is a necessary result ofdevelopment of quantum mechanics in nonlinear systems.

The establishment of the NLQM can certainly advance and facilitate furtherdevelopments of natural sciences including physics, biology and astronomy. Mean-while, it is also useful in understanding the properties and limitations of the LQM,and in solving problems and difficulties encountered by the LQM. Therefore, wehope that by publishing this book on quantum mechanics in the nonlinear systemswould add some value to science and would contribute to our understanding of thewonderful nature.

X. F. Pang and Y. P. Feng2004

Contents

Preface v

1. Linear Quantum Mechanics: Its Successes and Problems 1

1.1 The Fundamental Hypotheses of the Linear Quantum Mechanics . 11.2 Successes and Problems of the Linear Quantum Mechanics 51.3 Dispute between Bohr and Einstein 101.4 Analysis on the Roots of Problems of Linear Quantum Mechanics

and Review on Recent Developments 15Bibliography 21

2. Macroscopic Quantum Effects and Motions of Quasi-Particles 23

2.1 Macroscopic Quantum Effects 232.1.1 Macroscopic quantum effect in superconductors 23

2.1.1.1 Quantization of magnetic flux 242.1.1.2 Structure of vortex lines in type-II superconductors . 252.1.1.3 Josephson effect 26

2.1.2 Macroscopic quantum effect in liquid helium 282.1.3 Other macroscopic quantum effects 31

2.1.3.1 Quantum Hall effect 312.1.3.2 Spin polarized atomic hydrogen system 332.1.3.3 Bose-Einstein condensation of excitons 33

2.2 Analysis on the Nature of Macroscopic Quantum Effect 342.3 Motion of Superconducting Electrons 47

2.3.1 Motion of electrons in the absence of external fields 492.3.2 Motion of electrons in the presence of an electromagnetic field 50

2.4 Analysis of Macroscopic Quantum Effects in Inhomogeneous Super-conductive Systems 542.4.1 Proximity effect 542.4.2 Josephson current in S-I-S and S-N-S junctions 56

xi

xii Quantum Mechanics in Nonlinear Systems

2.4.3 Josephson effect in SNIS junction 592.5 Josephson Effect and Transmission of Vortex Lines Along the Su-

perconductive Junctions 602.6 Motion of Electrons in Non-Equilibrium Superconductive Systems . 662.7 Motion of Helium Atoms in Quantum Superfluid 72Bibliography 77

3. The Fundamental Principles and Theories of Nonlinear QuantumMechanics 81

3.1 Lessons Learnt from the Macroscopic Quantum Effects 813.2 Fundamental Principles of Nonlinear Quantum Mechanics 843.3 The Fundamental Theory of Nonlinear Quantum Mechanics . . . . 89

3.3.1 Principle of nonlinear superposition and Backlund transfor-mation 89

3.3.2 Nonlinear Fourier transformation 943.3.3 Method of quantization 953.3.4 Nonlinear perturbation theory 100

3.4 Properties of Nonlinear Quantum-Mechanical Systems 101Bibliography 106

4. Wave-Corpuscle Duality of Microscopic Particles in NonlinearQuantum Mechanics 109

4.1 Invariance and Conservation Laws, Mass, Momentum and Energyof Microscopic Particles in the Nonlinear Quantum Mechanics . . . 110

4.2 Position of Microscopic Particles and Law of Motion 1174.3 Collision between Microscopic Particles 126

4.3.1 Attractive interaction (b > 0) 1264.3.2 Repulsive interaction (b < 0) 1364.3.3 Numerical simulation 139

4.4 Properties of Elastic Interaction between Microscopic Particles . . 1434.5 Mechanism and Rules of Collision between Microscopic Particles . 1494.6 Collisions of Quantum Microscopic Particles 1544.7 Stability of Microscopic Particles in Nonlinear Quantum Mechanics 161

4.7.1 "Initial" stability 1624.7.2 Structural stability 164

4.8 Demonstration on Stability of Microscopic Particles 1694.9 Multi-Particle Collision and Stability in Nonlinear Quantum Me-

chanics 1734.10 Transport Properties and Diffusion of Microscopic Particles in Vis-

cous Environment 1784.11 Microscopic Particles in Nonlinear Quantum Mechanics versus

Macroscopic Point Particles 188

Contents xiii

4.12 Reflection and Transmission of Microscopic Particles at Interfaces . 1934.13 Scattering of Microscopic Particles by Impurities 2004.14 Tunneling and Praunhofer Diffraction 2094.15 Squeezing Effects of Microscopic Particles Propagating in Nonlinear

Media 2184.16 Wave-corpuscle Duality of Microscopic Particles in a Quasiperiodic

Perturbation Potential 221Bibliography 228

5. Nonlinear Interaction and Localization of Particles 233

5.1 Dispersion Effect and Nonlinear Interaction 2335.2 Effects of Nonlinear Interactions on Behaviors of Microscopic

Particles 2385.3 Self-Interaction and Intrinsic Nonlinearity 2435.4 Self-localization of Microscopic Particle by Inertialess

Self-interaction 2505.5 Nonlinear Effect of Media and Self-focusing Mechanism 2525.6 Localization of Exciton and Self-trapping Mechanism 2585.7 Initial Condition for Localization of Microscopic Particle 2635.8 Experimental Verification of Localization of Microscopic Particle . 267

5.8.1 Observation of nonpropagating surface water soliton in watertroughs 269

5.8.2 Experiment on optical solitons in fibers 272Bibliography 274

6. Nonlinear versus Linear Quantum Mechanics 277

6.1 Nonlinear Quantum Mechanics: An Inevitable Result of Develop-ment of Quantum Mechanics 277

6.2 Relativistic Theory and Self-consistency of Nonlinear QuantumMechanics 2816.2.1 Bound state and Lorentz relations 2836.2.2 Interaction between microscopic particles in relativistic

theory 2866.2.3 Relativistic dynamic equations in the nonrelativistic limit . 2886.2.4 Nonlinear Dirac equation 291

6.3 The Uncertainty Relation in Linear and Nonlinear QuantumMechanics 2926.3.1 The uncertainty relation in linear quantum mechanics . . . . 2926.3.2 The uncertainty relation in nonlinear quantum mechanics . 293

6.4 Energy Spectrum of Hamiltonian and Vector Form of the NonlinearSchrodinger Equation 3036.4.1 General approach 304

xiv Quantum Mechanics in Nonlinear Systems

6.4.2 System with two degrees of freedom 3066.4.3 Perturbative method 3096.4.4 Vector nonlinear Schrodinger equation 313

6.5 Eigenvalue Problem of the Nonlinear Schrodinger Equation . . . . 3156.6 Microscopic Causality in Linear and Nonlinear Quantum Mechanics 321Bibliography 326

7. Problem Solving in Nonlinear Quantum Mechanics 329

7.1 Overview of Methods for Solving Nonlinear Quantum MechanicsProblems 3297.1.1 Inverse scattering method 3307.1.2 Backlund transformation 3307.1.3 Hirota method 3317.1.4 Function and variable transformations 331

7.1.4.1 Function transformation 3317.1.4.2 Variable transformation and characteristic line . . . 3327.1.4.3 Other variable transformations 3327.1 A A Self-similarity transformation 3337.1.4.5 Galilei transformation 3347.1.4.6 Traveling-wave method 3357.1.4.7 Perturbation method 3357.1.4.8 Variational method 3357.1.4.9 Numerical method 3357.1.4.10 Experimental simulation 335

7.2 Traveling-Wave Methods 3367.2.1 Nonlinear Schrodinger equation 3367.2.2 Sine-Gordon equation 337

7.3 Inverse Scattering Method 3407.4 Perturbation Theory Based on the Inverse Scattering Transforma-

tion for the Nonlinear Schrodinger Equation 3457.5 Direct Perturbation Theory in Nonlinear Quantum Mechanics . . . 352

7.5.1 Method of Gorshkov and Ostrovsky 3527.5.2 Perturbation technique of Bishop 356

7.6 Linear Perturbation Theory in Nonlinear Quantum Mechanics . . . 3587.6.1 Nonlinear Schrodinger equation 3597.6.2 Sine-Gordon equation 364

7.7 Nonlinearly Variational Method for the Nonlinear SchrodingerEquation 366

7.8 D Operator and Hirota Method 3757.9 Backlund Transformation Method 379

7.9.1 Auto-Backlund transformation method 3797.9.2 Backlund transform of Hirota 382

Contents xv

7.10 Method of Separation of Variables 3847.11 Solving Higher-Dimensional Equations by Reduction 387Bibliography 394

8. Microscopic Particles in Different Nonlinear Systems 397

8.1 Charged Microscopic Particles in an Electromagnetic Field 3978.2 Microscopic Particles Interacting with the Field of an External

Traveling Wave 4018.3 Microscopic Particle in Time-dependent Quadratic Potential . . . . 4048.4 2D Time-dependent Parabolic Potential-field 4118.5 Microscopic Particle Subject to a Monochromatic Acoustic Wave . 4158.6 Effect of Energy Dissipation on Microscopic Particles 4198.7 Motion of Microscopic Particles in Disordered Systems 4238.8 Dynamics of Microscopic Particles in Inhomogeneous Systems . . . 4268.9 Dynamic Properties of Microscopic Particles in a Random Inhomo-

geneous Media 4318.9.1 Mean field method 4318.9.2 Statistical adiabatic approximation 4338.9.3 Inverse-scattering transformation based statistical perturba-

tion theory 4368.10 Microscopic Particles in Interacting Many-particle Systems 4388.11 Effects of High-order Dispersion on Microscopic Particles 4448.12 Interaction of Microscopic Particles and Its Radiation Effect in Per-

turbed Systems with Different Dispersions 4538.13 Microscopic Particles in Three and Two Dimensional Nonlinear Me-

dia with Impurities 459Bibliography 467

9. Nonlinear Quantum-Mechanical Properties of Excitons and Phonons 471

9.1 Excitons in Molecular Crystals 4719.2 Raman Scattering from Nonlinear Motion of Excitons 4809.3 Infrared Absorption of Exciton-Solitons in Molecular Crystals . . . 4879.4 Finite Temperature Excitonic Mossbauer Effect 4939.5 Nonlinear Excitation of Excitons in Protein 5019.6 Thermal Stability and Lifetime of Exciton-Soliton at Biological

Temperature 5109.7 Effects of Structural Disorder and Heart Bath on Exciton

Localization 5209.7.1 Effects of structural disorder 5219.7.2 Influence of heat bath 526

9.8 Eigenenergy Spectra of Nonlinear Excitations of Excitons 529

xvi Contents

9.9 Experimental Evidences of Exciton-Soliton State in MolecularCrystals and Protein Molecules 5369.9.1 Experimental data in acetanilide 536

9.9.1.1 Infrared absorption and Raman spectra 5379.9.1.2 Dynamic test of soliton excitation in acetanilide . . . 538

9.9.2 Infrared and Raman spectra of collagen, E. coli. and humantissue 5419.9.2.1 Infrared spectra of collagen proteins 5419.9.2.2 Raman spectrum of collagen 544

9.9.3 Infrared radiation spectrum of human tissue and Ramanspectrum of E. col 545

9.9.4 Specific heat of ACN and protein 5479.10 Properties of Nonlinear Excitations of Phonons 549Bibliography 551

10. Properties of Nonlinear Excitations and Motions of Protons,Polarons and Magnons in Different Systems 557

10.1 Model of Excitation and Proton Transfer in Hydrogen-bondedSystems 557

10.2 Theory of Proton Transferring in Hydrogen Bonded Systems . . . . 56410.3 Thermodynamic Properties and Conductivity of Proton Transfer . 57210.4 Properties of Proton Collective Excitation in Liquid Water 577

10.4.1 States and properties of molecules in liquid water 57810.4.2 Properties of hydrogen-bonded closed chains in liquid water 57910.4.3 Ring electric current and mechanism of magnetization

of water 58110.5 Nonlinear Excitation of Polarons and its Properties 58610.6 Nonlinear Localization of Small Polarons 59310.7 Nonlinear Excitation of Electrons in Coupled Electron-Electron and

Electron-Phonon Systems 59610.8 Nonlinear Excitation of Magnon in Ferromagnetic Systems 60110.9 Collective Excitations of Magnons in Antiferromagnetic Systems . . 607Bibliography 613

Index 619

Chapter 1

Linear Quantum Mechanics: Its Successesand Problems

The quantum mechanics established by Bohr, de Broglie, Schrodinger, Heisenbergand Bohn in 1920s is often referred to as the linear quantum mechanics (LQM). Inthis chapter, the hypotheses of linear quantum mechanics, the successes of and prob-lems encountered by the linear quantum mechanics are reviewed. The directionsfor further development of the quantum theory are also discussed.

1.1 The Fundamental Hypotheses of the Linear Quantum Mechan-ics

At the end of the 19th century, classical mechanics encountered major difficultiesin describing motions of microscopic particles (MIPs) with extremely light masses(~ 10~23 - 10~26 g) and extremely high velocities, and the physical phenomenarelated to such motions. This forced scientists to rethink the applicability of classicalmechanics and lead to fundamental changes in their traditional understanding of thenature of motions of microscopic objects. The wave-corpuscle duality of microscopicparticles was boldly proposed by Bohr, de Broglie and others. On the basis of thisrevolutionary idea and some fundamental hypotheses, Schrodinger, Heisenberg, etc.established the linear quantum mechanics which provided a unique way of describingquantum systems. In this theory, the states of microscopic particles are described bya wave function which is interpreted based on statistics, and physical quantities arerepresented by operators and are given in terms of the possible expectation values(or eigenvalues) of these operators in the states (or eigenstates). The time evolutionof quantum states are governed by the Schrodinger equation. The hypotheses ofthe linear quantum mechanics are summarized in the following.

(1) A state of a microscopic particle is represented by a vector in the Hilbertspace, \ip), or a wave function ip{r,t) in coordinate space. The wave functionuniquely describes the motion of the microscopic particle and reflects the wavenature of microscopic particles. Furthermore, if /? is a constant, then both \ip) and/3\ip) describe the same state. Thus, the normalized wave function, which satisfiesthe condition (ipl'tp) = 1, is often used to describe the state of the particle.

l

2 Quantum Mechanics in Nonlinear Systems

(2) A physical quantity, such as the coordinate X, the momentum P and theenergy E of a particle, is represented by a linear operator in the Hilbert space, andthe eigenvectors of the operator form a basis of the Hilbert space. An observablemechanical quantity is represented by a Hermitian operator whose eigenvalues arereal. Therefore, the values a physical quantity can have are the eigenvalues of thecorresponding linear operator. The eigenvectors corresponding to different eigenval-ues are orthogonal to each other. All eigenstates of a Hermitian operator span anorthogonal and complete set, {IPL}- Any vector of state, ip(f,t), can be expandedin terms of the eigenvectors:

^(r,t) = 2cL^L(r,t), or \ij)(r,t)) = J^^M^PL) (1.1)L L

where Ci = (tpL\ip) is the wave function in representation L. If the spectrum ofL is continuous, then the summation in (1.1) should be replaced by an integral:JdL---. Equation (1.1) can be regarded as a projection of the wave functionip(f, t) of a microscopic particle system on to those of its subsystems and it isthe foundation of transformation between different representations in the linearquantum mechanics. In the quantum state described by tjj(f,t), the probability ofgetting the value L' in a measurement of L is \CL'\2 = KV'L'IV')!2 m t n e c a s e °fdiscrete spectrum, or \(ipLi\ip)\2dL if the spectrum of the system is continuous. Ina single measurement of any mechanical quantity, only one of the eigenvalues of thecorresponding linear operator can be obtained, and the system is then said to bein the eigenstate belonging to this eigenvalue. This is a fundamental assumption oflinear quantum mechanics concerning measurements of physical quantities.

(3) The average (A) of a physical quantity A in an arbitrary state \ip) is given

or

(A) = (v|i|V>),if tp is normalized. Possible values of A can be obtained through the determinationof the above average. In order to obtain these possible values, we must find a wavefunction in which A has a precise value. In other words, we must find a state suchthat (AA)2 = 0, where (AA)2 = (A2) - (A)2. This leads to the following eigenvalueproblem for the operator A,

AtpL = AipL. (1.3)

From the above equation we can determine the spectrum of eigenvalues of the oper-ator A and the corresponding eigenfunctions ipL- The eigenvalues of A are possiblevalues observed from a measurement of the physical quantity. All possible valuesof A in any other state are nothing but its eigenvalues in its own eigenstates. This

(1.2)

Linear Quantum Mechanics: Successes and Problems 3

hypothesis reflects the statistical nature in the description of motion of microscopicparticles in the linear quantum mechanics.

(4) The Hilbert space in which the linear quantum mechanics is defined is a linearspace. The operator of a mechanical quantity is a linear operator in this space. Theeigenvectors of a linear operator satisfy the linear superposition principle. That is,if two states, |T/>I) and l^ ) are both eigenfunctions of a given linear operator, thentheir linear combination

\1>) = Cl\ih) + C2\rh), (1.4)

where C\ and C2 are constants, also describes a state of the same particle. The linearsuperposition principle of quantum states is determined by the linear characteristicsof the operators and this is why the quantum theory is referred to as linear quantummechanics. It is noteworthy to point out that such a superposition is different fromthat of classical waves, it does not result in changes in probability and intensity.

(5) The correspondence principle: If two classical mechanical quantities, A andB, satisfy the Poisson brackets,

{ ' ! ^[dqndpn dPndqn)

where qn and pn are generalized coordinate and momentum in the classical system,respectively, then the corresponding operators A and B in quantum mechanicssatisfy the following commutation relation:

[A, B] = (AB - BA) = -ih{A, B) (1.5)

where i = y/—T and h is the Planck's constant. If A and B are substituted by qn

and pn respectively, we have:

\Pn,qm] = -ihSnm, \pn,Pm] = 0,

This reflects the fact that values allowed for a physical quantity in a microscopicsystem are quantized, and thus the name "quantum mechanics". Based on this fun-damental principle, the Heisenberg uncertainty relation can be obtained as follows,

\ri\2

(A4)2 (AB)2 > J^L (1.6)

where iC = [A,B] and AA = {A - {A}). For the coordinate and momentumoperators, the Heisenberg uncertainty relation takes the usual form

|Az||Ap>!

(6) The time dependence of a quantum state \ip) of a microscopic particle isdetermined by the following Schrodinger equation:

- ~ W = *W- (1.7)


This is a fundamental dynamic equation for microscopic particle in space-time. His the Hamiltonian operator of the system and is given by,

H = f + V = -^—W2 + V,

where T is the kinetic energy operator and V the potential energy operator. Thus,the state of a quantum system at any time is determined by the Hamiltonian of thesystem. As a fundamental equation of linear quantum mechanics, equation (1.7) isa linear equation of the wave function ip which is another reason why the theory isreferred as a linear quantum mechanics.

If the quantum state of a system at time io is \ip(t0)), then the wave functionand mechanical quantities at time t are associated with those at time to by a unitaryoperator U(t,to), i.e.

\m) = U(t,to)\iP(to)), (1.8)

where U(to,to) = 1 and U+U = UU+ = I. If we let U(t,0) = U(t), then theequation of motion becomes

-~U(t) = HU(t) (1.9)

when H does not depend explicitly on time t and U(t) = e-l(H/h)t_ jf jj j g a n

explicit function of time t, we then have

U(t) = 1 + i / dhH{h) + — ^ f dhHih) f ' dt2H(t2) + •••. (1 .10)™ Jo \lh) Jo Jo

Obviously, there is an important assumption here: the Hamiltonian operator ofthe system is independent of its state, or its wave function. This is a fundamentalassumption in the linear quantum mechanics.

(7) Identical particles: No new physical state should occur when a pair of iden-tical particles is exchanged in a system. In other words, the wave function satisfiesPkj\ip) = A|"0)> where Pkj is an exchange operator and A = ±1. Therefore, the wavefunction of a system consisting of identical particles must be either symmetric, ips,(A = +1), or antisymmetric, ipa, (A = —1), and this property remains invariantwith time and is determined only by the nature of the particle. The wave functionof a boson particle is symmetric and that of a fermion is antisymmetric.

(8) Measurements of physical quantities: There was no assumption made aboutmeasurements of physical quantities at the beginning of the linear quantum me-chanics. It was introduced later to make the linear quantum mechanics complete.However, this is a nontrivial and contraversal topic which has been a focus of sci-entific debate. This problem will not be discussed here. Interested reader can referto texts and references given at the end of this chapter.


1.2 Successes and Problems of the Linear Quantum Mechanics

On the basis of the fundamental hypotheses mentioned above, Heisenberg,Schrodinger, Bohn, Dirac, and others established the theory of linear quantum me-chanics which describes the properties and motions of microscopic particle systems.This theory states that once the externally applied potential fields and initial statesof the particles are given, the states of the particles at any time later and any posi-tion can be determined by the linear Schrodinger equation, equations (1.7) and (1.8)in the case of nonrelativistic motion, or equivalently, the Dirac equation and theKlein-Gordon equation in the case of relativistic motion. The quantum states andtheir occupations of electronic systems, atoms, molecules, and the band structure ofsolid state matter, and any given atomic configuration are completely determinedby the above equations. Macroscopic behaviors of systems such as mechanical,electrical and optical properties may also be determined by these equations. Thistheory also describes the properties of microscopic particle systems in the presenceof external electromagnetic field, optical and acoustic waves, and thermal radiation.Therefore, to a certain degree, the linear quantum mechanics describes the law ofmotion of microscopic particles of which all physical systems are composed. It isthe foundation and pillar of modern physics.

The linear quantum mechanics had great successes in descriptions of motions ofmicroscopic particles, such as electron, phonon, photon, exciton, atom, molecule,atomic nucleus and elementary particles, and in predictions of properties of matterbased on the motions of these quasi-particles. For example, energy spectra of atoms(such as hydrogen atom, helium atom), molecules (such as hydrogen molecule) andcompounds, electrical, optical and magnetic properties of atoms and condensedmatters can be calculated based on linear quantum mechanics and the calculatedresults are in good agreement with experimental measurements. Being the founda-tion of modern science, the establishment of the theory of quantum mechanics hasrevolutionized not only physics, but many other science branches such as chemistry,astronomy, biology, etc., and at the same time created many new branches of sci-ence, for example, quantum statistics, quantum field theory, quantum electronics,quantum chemistry, quantum biology, quantum optics, etc. One of the great suc-cesses of the linear quantum mechanics is the explanation of the fine energy spectraof hydrogen atom, helium atom and hydrogen molecule. The energy spectra pre-dicted by linear quantum mechanics for these atoms and molecules are completely inagreement with experimental data. Furthermore, modern experiments have demon-strated that the results of the Lamb shift and superfine structure of hydrogen atomand the anomalous magnetic moment of the electron predicted by the theory ofquantum electrodynamics are in agreement with experimental data within an orderof magnitude of 10~5. It is therefore believed that the quantum electrodynamics isone of most successful theories in modern physics.

Despite the great successes of linear quantum mechanics, it nevertheless en-


countered some problems and difficulties. In order to overcome these difficulties,Einstein had disputed with Bohr and others for the whole of his life and the difficul-ties still remained up to now. Some of the difficulties will be discussed in the nextsection. These difficulties of the linear quantum mechanics are well known and havebeen reviewed by many scientists. When one of the founders of the linear quantummechanics, Dirac, visited Australia in 1975, he gave a speech on the developmentof quantum mechanics in New South Wales University. During his talk, Dirac men-tioned that at the time, great difficulties existed in the quantum mechanical theory.One of the difficulties referred to by Dirac was about an accurate theory for inter-action between charged particles and an electromagnetic field. If the charge of aparticle is considered as concentrated at one point, we shall find that the energyof the point charge is infinite. This problem had puzzled physicists for more than40 years. Even after the establishment of the renormalization theory, no actualprogress had been made. Such a situation was similar to the unified field theoryfor which Einstein had struggled for his whole life. Therefore, Dirac concluded histalk by making the following statements: It is because of these difficulties, I believethat the foundation for the quantum mechanics has not been correctly laid down.As part of the current research based on the existing theory, a great deal of workhas been done in the applications of the theory. In this respect, some rules for get-ting around the infinity were established. Even though results obtained based onsuch rules agree with experimental measurements, they are artificial rules after all.Therefore, I cannot accept that the present foundation of the quantum mechanicsis completely correct.

However, what are the roots of the difficulties of the linear quantum mechanicsthat evoked these contentions and raised doubts about the theory among physicists?Actually, if we take a closer look at the history of physics, one would know thatnot so many fundamental assumptions were required for all physical theories butthe linear quantum mechanics. Obviously, these assumptions of linear quantummechanics caused its incompleteness and limited its applicability.

It was generally accepted that the fundamentals of the linear quantum mechan-ics consist of the Heisenberg matrix mechanics, the Schrodinger wave mechanics,Born's statistical interpretation of the wave function and the Heisenberg uncer-tainty principle, etc. These were also the focal points of debate and controversy. Inother words, the debate was about how to interpret quantum mechanics. Some ofthe questions being debated concern the interpretation of the wave-particle duality,probability explanation of the wave function, the difficulty in controlling interactionbetween measuring instruments and objects being measured, the Heisenberg un-certainty principle, Bohr's complementary (corresponding) principle, single particleversus many particle systems, the problems of microscopic causality and probability,process of measuring quantum states, etc. Meanwhile, the linear quantum mechan-ics in principle can describe physical systems with many particles, but it is not easyto solve such a system and approximations must be used to obtain approximate


solutions. In doing this, certain features of the system which could be importanthave to be neglected. Therefore, while many enjoyed the successes of the linearquantum mechanics, others were wondering whether the linear quantum mechanicsis the right theory of the real microscopic physical world, because of the problemsand difficulties it encountered. Modern quantum mechanics was born in 1920s, butthese problems were always the topics of heated debates among different views tillnow. It was quite exceptional in the history of physics that so many prominentphysicists from different institutions were involved and the scope of the debate wasso wide. The group in Copenhagen School headed by Bohr represented the view ofthe main stream in these discussions. In as early as 1920s, heated disputes on thestatistical explanation and completeness of wave function arose between Bohr andother physicists, including Einstein, de Broglie, Schrodinger, Lorentz, etc.

The following is a brief summary of issues being debated and problems encoun-tered by the linear quantum mechanics.

(1) First, the correctness and completeness of the linear quantum mechanics werechallenged. Is linear quantum mechanics correct? Is it complete and self-consistent?Can the properties of microscopic particle systems be completely described by thelinear quantum mechanics? Do the fundamental hypotheses contradict each other?

(2) Is the linear quantum mechanics a dynamic or a statistical theory? Doesit describe the motion of a single particle or a system of particles? The dynamicequation seems an equation for a single particle, but its mechanical quantities aredetermined based on the concepts of probability and statistical average. This causedconfusion about the nature of the theory itself.

(3) How to describe the wave-particle duality of microscopic particles? Whatis the nature of a particle defined based on the hypotheses of the linear quantummechanics? The wave-particle duality is established by the de Broglie relations. Canthe statistical interpretation of wave function correctly describe such a property?There are also difficulties in using wave package to represent the particle natureof microscopic particles. Thus describing the wave-corpuscle duality was a majorchallenge to the linear quantum mechanics.

(4) Was the uncertainty principle due to the intrinsic properties of microscopicparticles or a result of uncontrollable interaction between the measuring instrumentsand the system being measured?

(5) A particle appears in space in the form of a wave, and it has certain probabil-ity to be at a certain location. However, it is always a whole particle, rather than afraction of it, being detected in a measurement. How can this be interpreted? Is theexplanation of this problem based on wave package contraction in the measurementcorrect?

Since these are important issues concerning the fundamental hypotheses of thelinear quantum mechanics, many scientists were involved in the debate. Unfortu-nately, after being debated for almost a century, there are still no definite answersto most of these questions. We will introduce and survey some main views of this


debate in the following.As far as the completeness of the linear quantum mechanics was concerned, Von

Neumann provided a proof in 1932. According to Von Neumann, if O is a set ofobservable quantities in the Hilbert space Q of dimension greater than one, thenthe self-adjoint of any operator in this set represents an observable quantity in thesame set, and its state can be determined by the average (A) for the operator A.If this average value satisfies (1) = 1, we have (rA) = r(A) for any real constantr. If A is non-negative, then {A) > 0. If A,B,C,--- are arbitrary observablequantities, then, there always exists an observable A + B + C + • • • such that(A + B + C H ) = (A) + (B) + (C) H . Von Neumann proved that there exists aself-adjoint operator A in Q such that { 4°) ^ {A)a. This implies that there alwaysexists an observable quantity A which is indefinite or does not have an accuratevalue. In other words, the states as defined by the average value are dispersiveand cannot be determined accurately, which further implies that states in which allobservable quantities have accurate values simultaneously do not exist. To be moreconcrete, not all properties of a physical system can possess accurate values. Atthis stage, this was the best the theory can do. Whether it can be accepted as acomplete theory is subjective. It seemed that any further discussion would lead tonowhere.

It was realized later that Von Neumann's theorem was mathematically flawlessbut ambiguous and vague in physics. In 1957, Gleason made two modificationsto Von Neumann's assumptions: Q should be the Hilbert space of more than twodimensions rather than one; and A, B,C, ••• should be limited to commutable self-adjoint operators in Q. He verified that Von Neumann's theorem is still valid withthese assumptions. Because the operators are commutable, the linear superpositionproperty of average values is, in general, independent of the order in which exper-iments are performed. Hence, these assumptions seem to be physically acceptable.Furthermore, Von Neumann's conclusion ruled out some nontrivial hidden variabletheories in the Hilbert space with dimensions of more than two.

However, in 1966, Bell indicated that Gleason's theorem can essentially onlyremove the hidden variable theories which are independent of environment andarrangements before and after a measurement. It would be possible to establishhidden variable theories which are dependent on environment and arrangementsbefore and after a measurement. At the same time, Bell argued that since thereare more input hidden variables in the hidden variable theory than in quantummechanics, there should be new results that may be compared with experiments,thus to verify whether the quantum mechanics is complete.

Starting from an ideal experiment based on the localized hidden variables theoryand the average value q(a, b) = J Aa(X)Bb(X)d\, Bohm believed that some featuresof a particle could be obtained once those of another particle which is remotelyseparated from the first are measured. This indicates that correlation betweenparticles exists which could be described in terms of "hidden parameters". Based


on this idea, Bell proposed an inequality which is applicable to any "localized"hidden variables theory. Thus, the natures of correlation in a system of particlespredicted by the Bell's inequality and quantum mechanics would differ appreciablywhich can be used to verify which of the two is correct.

To this end, we discuss a system of spin correlation. We shall first discuss spincorrelation from the point of view of quantum mechanics. Assume that there existsa system which consists of two particles A and B, both of spin 1/2, but the totalspin of the system is zero. Let Aa be the spin component measured along a directionspecified by a unit vector a, and similarly B/, the spin component measured alonga direction specified by a unit vector b. According to linear quantum mechanics,it is easy to write down the components of the spin operators along directions aand b. They are {a A • S)/2 and (&B • b)/2, respectively, where <TA/2 and <TB/2 arethe spin operators of particles A and B in terms of the Pauli matrices, respectively.{aA • S)/2 and (<3\B • b)/2 can be regarded as projections of the spin operators onthe unit vectors a and b, respectively. The spin correlation function, q(a,b), maybe defined as the average of the product of Aa and Bb, i.e. q{a,b) = 4AO • B\>,where the factor of 4 is due to "normalization", the horizontal line above Aa • Bb

denotes the statistical average of the product of Aa and Bb over all possible resultsof measurements. According to linear quantum mechanics, we have

A~Wb = ±(0+\(&A-a)(cTB-b)\0+)

where |0+) represents the spin wave function with zero total spin, of the systemconsisting of particles A and B of spin 1/2, and can be expressed as

|0+) = ±= [v+i(A)V_i(£) - V_i(^+i(-B)] .

(0+| in the above equations is the Hermitian conjugate of |0+). Using the aboveexpression and the rules of Pauli matrix, we can obtain

q(a, b) = AAa -Bb = -a-b.

According to this equation, q(a,b) = - 1 if a = b, which results in "negative"correlation for spin projections measured in the same direction.

On the other hand, if we start from Bell's localized hidden variable theory, weobtain the following Bell's inequality:

\q{a,b)-q{a,c)\ < l + q(a,c).

This involves measurements of the spin components in three directions, specifiedby unit vectors a, b, and c, respectively, in contrast to the previous case whichinvolves only two directions. If we let a = b = c — n, then Bell's inequality becomesq(h • h) > — 1, which is the same as that given by quantum mechanics. Differentresults can be expected if three directions are really involved in the measurements.For example, if the angles between a and b and between b and c are 60° and that


between d and c is 120°, then we have g(o, b) = q(b,c) = 1/2, and q{a,c) = -1 /2according to quantum mechanics. Substituting these into the Bell's inequality, it isevident that

which results in 1 < 1/2 that does not make any sense.It is clearly seen that spin correlation described in linear quantum mechanics

contradicts the Bell's inequality. That is to say that all statistical predictions oflinear quantum mechanics cannot be obtained from the localized hidden variabletheory. In some special cases, if statistical predictions based on linear quantummechanics are correct, then the localized hidden variable theory does not hold, andvice versa. However, whether the Bell's inequality is correct remained a question.

Since then many physicists, for example Wigner in 1970, had also derived theBell's inequality using analytical methods which were quite different from Bell'sapproach. Unfortunately, only single state of particles with zero spin was discussedin an ideal experiment setting. This is equivalent to assume that two particles ofspin 1/2 always reach the instrument and therefore the instrument always measuresa definite spin along a given axis. Such a measurement is very hard to realize inactual experiments.

This prompted Clayser et al. to generalize Bell's inequality by removing the re-strictions of single state and spin 1/2, in 1969. The Clayser's generalized inequality

\q(a, b) - q(a, b')\ < 2 ± [q(a', &) + q(a', b)]

is based on some more common and realistic experimental conditions. If q(a',b) =— 1, the Clayser's inequality reduces to the Bell's inequality. Bell himself also ob-tained the same result in 1971. Since 1972, many experiments, as shown in Table 1.1,have been carried out and results have been reported to verify which theory, theBell's inequality of localized hidden variable or the linear quantum mechanics, cor-rectly describes the motion of the microscopic particle.

Among the nine experiments listed in Table 1.1, seven of them gave supportsto linear quantum mechanics and only two experimental findings are in agreementwith the Bell's inequality. It seems that the experimental results are in favor of thelinear quantum mechanics than Bell's localized hidden variable theory. This showsthat linear quantum mechanics does not satisfy the requirement of localization. Theresults, however, cannot exclusively confirm its validity either.

1.3 Dispute between Bohr and Einstein

While the view on linear quantum mechanics and its interpretation by Bohr andothers in the Copenhagen school dominated the debate, many prominent physicistsrespected Einstein as the authority who had doubted and continuously criticized


Table 1.1 List of experiments to verify Bell's inequality.

No. Author(s) Date Experiment Results1 S. T. Freedman 1972 Low-energy photon radiation in Supports linear quantum mechanic!

J. F. Clauser transitional process of a calciumatom

2 R. A. Holt 1973 Low-energy photon radiation in Supports Bell'sF. M. Pipkin transitional process of mercury- inequality

198 atoms3 J. F. Clauser 1976 Low-energy photon radiation in Supports linear quantum mechanic!

transitional process of mercury-202 atoms

4 E. S. Firg 1976 Low-energy photon radiation in Supports linear quantum mechanic!R. C. Thomson transitional process of mercury-

202 atom5 G. Fioraci 1975 High-energy photon annihilation Supports Bell's

S. Gutkowski of electron - positron pair (7 ray) inequalityS. NatarrigoR. Pennisi

6 J. Kasday 1975 High-energy photon annihilation Supports linear quantum mechanic!J. Ulman of electron - positron pair (7 ray)Wu Jianxiong Supports linear quantum mechanic:

7 M. Lamchi-Rachti 1976 Atomic pair in single state Supports linear quantum mechanic!W. Mitting

8 Aspect 1981 Cascade photon radiation in Supports linear quantum mechanicP. Grangier transitional process of atomsG. Roger

9 P. Grangier 1982 Cascade photon radiation in Supports linear quantum mechanicP. Grangier transitional process of 46CaG. Roger

Bohr's interpretation. This resulted in a life-long dispute between Bohr and Ein-stein, which was unprecedented and went through three stages.

The first stage was during the period from 1924 to 1927 when the theory ofquantum mechanics had just been established. Einstein proceeded from his ownphilosophical belief and his scientific goal for an exact description of causality inthe physical world, and expressed his extreme unhappiness with the probabilityinterpretation of linear quantum mechanics. In a letter to Born on December 4,1926, Einstein said that "Quantum mechanics is certainly imposing. But an innervoice tells me that it is not the real thing (der Wahre Jakob). The theory says alot, but it does not bring us any closer to the secret of the "Old One." I, at anyrate, am convinced that He is not playing at dice."

The second stage was from 1927 to 1930. After Bohr had put forward hiscomplementary principle and had established his interpretation as the main streaminterpretation, Einstein was extremely unhappy. His main criticism was directed atthe uncertainty relation on which Bohr's complementary principle was based. At the5th (1927) and the 6th (1930) International Meetings of Physics at Solway, Einsteinproposed two ideal experiments (double slit diffraction and photon box) to provethat the uncertainty relation and formalism of the quantum mechanics contradict


each other, and thus to disprove Bohr's complementary principle. But Einstein'sidea was demolished each time by Bohr through resourceful analysis. Since then,Einstein had to accept the logical consistency of quantum mechanics and turned hiscriticism to the completeness of the linear quantum mechanics theory.

The third stage was from 1930 until the death of Einstein. The dispute duringthis period is reflected in the debate between Einstein and Bohr over the EPRparadox proposed by Einstein together with Podolsky and Rosen. This paradoxconcerned the fundamental problem of the linear quantum mechanics, i.e., whetherit satisfied the deterministic localized theory and the microscopic causality. Sincesome of the subsequent experiments seem to support the linear quantum mechanics,instead of the Bell inequality, it is necessary to understand the nature of the EPRparadox and results it brought about.

The EPR paradox will be briefly introduced below.Consider a system consisting of two particles which move in opposite directions.

For simplicity but without losing its generality, we assume that the initial relativisticmomentum of the pair of particles is p = 0. Then there must be p\ = —p2 afterthe two particles interact and depart. However, the magnitude and direction ofthe momentum of each particle are not known. Assume that the momentum ofparticle 1 is measured, by a detector, and the value p\ = +a is obtained, then themomentum of the particle 2 is determined and it can only be pi = —a accordingto conservation of momentum in the linear quantum mechanics. However, in thelight of the hypothesis of contraction of wave packet in the measuring process, theplane wave with momentum pi = a\ is "selected" out by the detector from thewave packet î(Xi) describing particle 1. In accordance with the traditional linearquantum mechanics, this process of "spectrum resolution" is due to some kind of"uncontrollable interaction" between the instrument and the wave packet. Underthe influence of such an "uncontrollable interaction", the momentum of particle1 could be pi = a, or pi = 6, • • •. However, what is surprising is that there isalways pi — —a as long as p\ = a is measured by the detector. This means thatthis value should be obtained regardless of the measurement on the wave packet•02(^2) is made or not. In other words, when the wave packet ipi(Xi) is measuredand contracted, the wave packet ^2(^2) for particle 2 will also be automaticallycontracted. A series of questions then arise. For example, what mechanism makesthis possible? Does this occur instantaneously, or is it propagating at speed of lightaccording to the special theory of relativity? How can the wave packet contractioncaused by measurement automatically guarantee the conservation of momentum? Itis very difficult to answer these questions. Only after careful studies by Einstein andothers, the following conclusions were obtained: either the description of the linearquantum mechanics was incomplete, or the linear quantum mechanics didn't satisfythe criterion of "localization". Einstein tended to believe that physical phenomenamust satisfy the criterion of "localization", i.e. physical quantities cannot propagatewith speed greater than the speed of light. Thus, he thought that the linear quantum


mechanics is an incomplete theory. Due to this remarkable analysis by Einstein,many physicists began to explore the theory of "hidden parameters" of the linearquantum mechanics.

The "queries" to the linear quantum mechanics by Einstein and others had in-deed created quite a stir. Bohr had to respond in his own capacity to these queries.In 1935, Bohr published a short essay in Physical Review in which he argued thatif a system consists of two local particles 1 and 2, then this system should be de-scribed by a wave function ip(l, 2). In such a case, the local particles 1 and 2 are nolonger mutually independent entities. Even though they are spatially separated atthe instant the system is probed, they cannot be considered as independent entities.Thus, there is no basis for statements such as measurement of subsystem 1 couldnot influence subsystem 2 within the framework of the linear quantum mechanics,and the idea of Einstein et al. cannot be accepted. Essentially, Bohr was not re-ally against the "paradox" proposed by Einstein and others, but only confirmedthat linear quantum mechanics might not satisfy the principle of localization. Bohrfurther commented that in the final decisive steps of measurement in Einstein'sideal experiment, even though there was no mechanical interference to the systembeing probed, influence on experimental conditions did exist. Thus, Einstein's argu-ments could not verify their conclusion that the description of quantum mechanicsis incomplete.

Many scientists who followed closely the thought of localization and incomplete-ness of the linear quantum mechanics by Einstein and others believed that therecould exist a hidden variables theory behind linear quantum mechanics which mightbe able to interpret the probability behavior of microscopic particle. The conceptof "hidden variables" was proposed soon after linear quantum mechanics was born.However, it was disapproved by Von Neumann in 1932. For a long time since then,no one had mentioned this problem. After the second World War, Einstein repeat-edly criticized the linear quantum mechanics and suggested that any actual stateshould be completely described.

Motivated by this thought, Bohm put forward the first systematic "hidden vari-able theory" in 1952. He believed that the statistical characteristics of linear quan-tum mechanics is due to some "background" fluctuations hidden behind the quan-tum theory. If we can find the hidden function for a microscopic particle, thena deterministic description could be made for a single particle. But how can theexistence of such hidden variables be proved? Bohm proposed two experiments, tomeasure the spin correlation of a single proton and the polarization correlation inannihilating radiation of photons, respectively. It was realized later that in Bohm'stheory the single state ij> is essentially a slowly varying state which describes statesof a fluid with random fluctuations. Since the wave function itself cannot have suchrandom fluctuation, a hidden variable could not be introduced. Bohm's theorymentioned above was referred to as a random hidden-variables theory.

However, if the motion of particles can also be considered as a stable Markov


process. A steady state solution of the Schrodinger equation can then be given froma steady distribution of the Markov chain, and if the Fock-Planck equation was takenas the dynamic equation of microscopic particle, a new "hidden variables theories"of linear quantum mechanics can be set up. After Bell established his inequalityon the basis of Bohm's deterministic "localized variables theory" in 1966, variousattempts were made to experimentally verify which theory is the right theory and tosettle the dispute once and for all. As mentioned earlier, majority of the experimentssupported the linear quantum mechanics at that time, and it was clear that not allthe predictions by the linear quantum mechanics can be obtained from the localizedhidden variables theory. Thus the "hidden variable theory" was abandoned.

To summarize, the long dispute between Bohr and Einstein was focused on threeissues. (1) Einstein upheld to the belief that the microscopic world is no differentfrom the macroscopic world, particles in the microscopic world are matters andthey exist regardless of the methods of measurements, any theoretical descriptionto it should in principle be deterministic. (2) Einstein always considered that thetheory of the linear quantum mechanics was not an ultimate and complete theory.He believed that quantum mechanics is similar to classical optics. Both of them arecorrect theories based on statistical laws, i.e., when the probability \ip(r, t)\2 of aparticle at a moment t and location r is known, the average value of an observablequantity can be obtained using statistical method and then compared with exper-imental results. However, the understanding to processes involving single particlewas not satisfactory. Hence, il>{r,t) cannot give everything about a microscopicparticle system, and the statistical interpretation cannot be ultimate and complete.(3) The third issue concerns the physical interpretation of the linear quantum me-chanics. Einstein was not impressed with the attempt to completely describingsome single processes using linear quantum mechanics, which he made very clearin a speech at the fifth Selway International Meeting of physics. In an article,"Physics and Reality", published in 1936 in the Journal of the Franklin Institute,Einstein again mentioned that what the wave function %j} describes can only be amany-particle system, or an assemble in terms of statistical mechanics, and underno circumstances, the wave function can describe the state of a single particle. Ein-stein also believed that the uncertainty relation was a result of incompleteness ofthe description of a particle by ip{r,i), because a complete theory should give pre-cise values for all observable quantities. Einstein also did not accept the statisticalinterpretation, because he did not believe that an electron possess free will. Thus,Einstein's criticism against the linear quantum mechanics was not directed towardsthe mathematical formalism of the linear quantum mechanics, but to its fundamen-tal hypotheses and its physical interpretation. He considered that this is due to theincomplete understanding of the microscopic objects. Moreover, the contradictionbetween the theory of relativity and the fundamental of the linear quantum me-chanics was also a central point of dispute. Einstein made effort to unite the theoryof relativity and linear quantum mechanics, and attempted to interpret the atomic


structure using field theory. The disagreements on several fundamental issues ofthe linear quantum mechanics by Einstein and Bohr and their followers were deeprooted and worth further study. This brief review on the disputes between the twogreat physicists given above should be useful to our understanding on the natureand problems of the linear quantum mechanics. It should set the stage for theintroduction of nonlinear quantum mechanics.

1.4 Analysis on the Roots of Problems of Linear Quantum Me-chanics and Review on Recent Developments

The discussion in the previous section shows that the disputes and disagreementon several fundamental issues of the linear quantum mechanics are deep rooted.Almost all prominent physicists were involved to a certain degree in this disputewhich lasted half of a century, which is extraordinary in the history of science.What is even more surprising is that after such a long dispute, there have been noconclusions on these important issues till now. Besides what have been mentionedabove, there was another fact which also puzzled physicists. As it is know, theconcept of "orbit" has no meaning in quantum mechanics. The state of a particleis described by the wave function ip which spreads out over a large region in space.Even though this suggests that a particle does not have a precise location, in phys-ical experiments, however, particles are always captured by a detector placed at anexact position. Furthermore, it is always one whole particle, rather than a fractionof it, being detected. How can this be interpreted by the linear quantum mechan-ics? Given this situation, can we consider that the linear quantum mechanics iscomplete? Even though the linear quantum mechanics is correct, then it can onlybe considered as a set of rules describing some experimental results, rather thanan ultimate complete theory. In the meantime, the indeterministic nature of thelinear quantum mechanics seems against intuition. All these show that it is nec-essary to improve and further develop the linear quantum mechanics. Attempt tosolve these problems within the framework of the linear quantum mechanics seemimpossible. Therefore, alternatives that go beyond the linear quantum mechanicsmust be considered to further develop the quantum mechanics. To do this, onemust thoroughly understand the fundamentals and nature of the linear quantummechanics and seriously consider de Broglie's idea of a nonlinear wave theory.

Looking back to the development and applications of the linear quantum me-chanics for almost a century, we notice that the splendidness of the quantum me-chanics is the introduction of a wave function to describe the state of particles andthe expression of physical quantities by linear Hermitian operators. Such an ap-proach is drastically different from the traditional methods of classical physics andtook the development of physics to a completely new stage. This new approachhas been successfully applied to some simple atoms and molecules, such as hydro-gen atom, helium atom and hydrogen molecule, and the results obtained are in


agreement with experimental data. Correctness of this theory is thus established.However, besides being correct, a good theory should also be complete. Successfulapplications to a subset of problems does not mean perfection of the theory andapplicability to any physics system. Physical systems in the world are manifold andevery theory has its own applicable scope or domain. No theory is universal.

From the above discussion, we see that the most fundamental features of the lin-ear quantum mechanics are its linearity and the independence of the Hamiltonianof a system on its wave function. These ensure the linearity of the fundamentaldynamic equations, i.e., the Schrodinger equation is a linear equation of the wavefunction, all operators in the linear quantum mechanics are linear Hermitian op-erators, and the solutions of the dynamic equation satisfy the linear superpositionprinciple. The linearity results in the following limitations of the linear quantummechanics.

(1) The linear quantum mechanics is a wave theory and it depicts only thewave feature, not their corpuscle feature, of microscopic particles. As a matter offact, the Schrodinger equation (1.7) is a wave equation and its solution representsa probability wave. To see this clearly, we consider the wave function tp — f •exp{-iEt/h) and substitute it into (1.7). If we let n2 = (E - U)/(E - C) = k2/k2

a,where C is a constant, and fc2, = 2m(E - C)/h2, then (1.7) becomes

This equation is nothing but that of a light wave propagating in a homogeneousmedium. Thus, the linear Schrodinger equation (1.7) is only able to describe thewave feature of the microscopic particle. In other words, when a particle moves con-tinuously in the space-time, it follows the law of linear variation and disperses overthe space-time in the form of a wave. This wave feature of a microscopic particleis mainly determined by the kinetic energy operator, T — -(/i2/2m)V2, in the dy-namic equation (1.7). The applied potential field, V(x,t) is imposed on the systemby external environment and it can only change the wave form and amplitude, butnot the nature of the wave. Such a dispersion feature of the microscopic particleensures that the microscopic particle can only appear with a definite probability ata given point in the space-time. Therefore, the momentum and coordinate of themicroscopic particle cannot be accurately measured simultaneously, which lead tothe uncertainty relation in the linear quantum mechanics. Therefore, the uncer-tainty relation occurs in linear quantum mechanics is an inevitable outcome of thelinear quantum mechanics.

(2) Due to this linearity and dispersivity, it is impossible to describe the cor-puscle feature of microscopic particles by means of this theory. In other words, thewave-corpuscle duality of microscopic particle cannot be completely described bythe dynamic equation in the linear quantum mechanics, because external appliedpotential fields cannot make a dispersive particle an undispersive, localized parti-cle, and there is no other interaction that can suppress the dispersion effect of the


kinetic energy in the equation. Thus a microscopic particle always exhibits featuresof a dispersed wave and its corpuscle property can only be described by means ofBorn's statistical interpretation of the wave function. This not only exposes theincompleteness of the hypotheses of linear quantum mechanics, but also brings outan unsolvable difficulty, namely, whether the linear quantum mechanics describesthe state of a single particle or that of an assemble of many particles.

As it is known, in linear quantum mechanics, the corpuscle behavior of a particleis often represented by a wave packet which can be a superposition of plane waves.However, the wave packet always disperses and attenuates with time during thecourse of propagation. For example, a Gaussian wave packet given by

xP(x, t = 0) = e-a*x2'2, |V|2 = e~a»x2 (1.11)

at t = 0 becomes

rl>(x,t) = -== cf>{k)e^kx-hk l/2m)dk (1.12)V2TT J-OO

_ 1 p-(s

2/2)(l/qg+ifit/m)—— . V » ~ " 5

aO\/l/ao + iht/m

Ma = , * e->?

y/1 + {alhtlmf

after propagating through a time t, where

1 f°° 1 / n2h0(Jfc) = - ± = / i>{x,t = O)eikxdk, at = — Jl + iZSZt.

V27T J-OO Ct0 V mThis indicates clearly that the wave packet is dispersed as time goes by. Theuncertainty in its position also increases with time. The corresponding uncertaintyrelation is

. . h I oAhH2

A3;Ap=2V1+-V-'where

Ap=—^-, Ax=—F=—\/l+ u „ •y/2 V2a0V m2

Hence, the wave packet cannot be used to describe the corpuscle property of amicroscopic particle. How to describe the corpuscle property of microscopic particleshas been an unsolved problem in the linear quantum mechanics. This is just anexample of intrinsic difficulties of the linear quantum mechanics.

(3) Because of the linearity, the linear quantum mechanics can only be used inthe case of linear field and medium. This means that the linear quantum mechanicsis suitable for few-body systems, such as the hydrogen atom and the helium atom,etc. For many-body systems and condensed matter, it is impossible to solve the


wave equation exactly and only approximate solutions can be obtained in the linearquantum mechanics. However, doing so loses the nonlinear effects due to intrinsicand self-interactions among the particles in these matters. Therefore, the scope ofapplication of the linear quantum mechanics is limited. Moreover, when this theoryis applied to deal with features of elementary particles in quantum field theory, thedifficulty of infinity cannot always be avoided and this shows another limitation ofthis theory. Therefore, it is necessary to develop a new quantum theory that candeal with these complex systems.

From the discussion above, we learned that linearity on which linear quantummechanics is based is the root of all the problems encountered by the linear quantummechanics. The linearity is closely related to the assumption that the Hamiltonianoperator of a system is independent of its wave function, which is true only in simpleand uniform physical systems. Thus the linearity greatly limited the applicablescope and domain of the linear quantum mechanics. It cannot be used to study theproperties of many-body, many-particle, nonlinear and complex systems in whichthere exist complicated interaction, the self-interaction, and nonlinear interactionsamong the particles and between the particles and the environment.

Since the wave feature of microscopic particle can be well described by the wavefunction, one important issue to be looked into in further development of quantummechanics is the description of corpuscle feature of microscopic particles, so thatthe new quantum theory should completely describe the wave-corpuscle duality ofmicroscopic particles. However, this is easily said than done. To this respect, it isuseful to review what has already been done by the pioneers in this field, as we canlearn from them and get some inspiration from their work.

One can learn from the history of development of the theory of superconductiv-ity. It is known that the mechanism of superconductivity based on electron-phononinteraction was proposed by Frohlich as early as in 1951. But Frohlich failed toestablish a complete theory of superconductivity because he confined his work tothe perturbation theory in the linear quantum mechanics, and superconductivity isa nonlinear phenomenon which cannot be described by the linear quantum mechan-ics. Of course, this problem was finally solved and the nonlinear BCS theory wasestablished in 1975. This again clearly demonstrated the limitation of the linearquantum mechanics. This problem will be discussed in the next chapter in moredetails.

In view of this, in order to overcome the difficulties of the linear quantum me-chanics and further develop the theory of quantum mechanics, two of the hypothesesof the linear quantum mechanics, i.e. linearity of the theory and independence ofthe Hamiltonian of a system on its wave function must be reconsidered. Furtherdevelopment must be directed toward a nonlinear quantum theory. In other words,nonlinear interaction should be included into the theory and the Hamiltonian of asystem should be related to the wave function of the system.

The first attempt of establishing a nonlinear quantum theory was made by de


Broglie, which was described in his book: "the nonlinear wave theory". Through along period of research, de Broglie concluded that the theory of wave motion cannotinterpret the relation between particle and wave because the theory was limited toa linear framework from the start. In 1926, he further emphasized that if tp(f, t)is a real field in the physical space, then the particle should always have a definitemomentum and position, de Broglie assumed that ip(f,t) describes an essentialcoupling between the particle and the field, and used this concept to explain thephenomena of interference and diffraction.

In 1927, de Broglie put forward a "dual solution theory" in a paper published inJ. de Physique, de Broglie proposed that two types of solutions are permitted in thedynamic equation in the linear quantum mechanics. One is a continuous solution,ip = Rel9, with only statistical meaning, and this is the Schrodinger wave. Thiswave can only have statistical interpretation and can be normalized. It does notrepresent any physical wave. The other type, referred as a u wave, has singularitiesand is associated with spatial localization of the particle. The corpuscle feature ofa microscopic particle is described by the u wave and the position of a particle isdetermined by a singularity of the u wave, de Broglie generalized the formula of themonochromatic plane wave and stipulated a rule of associating the particle with thepropagation of the wave. The particle would move inside its wave according to deBroglie's dual solution theory. This suggests that the motion of the particle insideits wave is influenced by a force which can be derived from a "quantum potential".This quantum potential is proportional to the square of the Planck constant andis dependent on the second derivative of the amplitude of the wave. It can alsobe given in terms of the change in the rest mass of the particle. In the case of amonochromatic plane wave, the quantum potential is zero. In 1950s, de Brogliefurther improved his "dual solution theory". He proposed that the u wave satis-fies an undetermined nonlinear equation, and this led to his own "nonlinear wavetheory". However, de Broglie did not give the exact nonlinear equation that the uwave should satisfy. This theory has serious difficulties in describing multiparticlesystems and the s state of a single-particle. The theory also lacked experimentalverification. Thus, even though it was supported by Einstein, the theory was nottaken seriously by the majority and was gradually forgotten.

Although de Broglie's nonlinear wave theory was incorrect, some of his ideas,such as the quantum potential, the u wave of nonlinear equation which is capableof describing a physical particle, provided inspiration for further development ofquantum mechanics.

As mentioned above, de Broglie stated that the quantum potential is related tothe second derivative of the amplitude of the wave function ip = Re'e. Bohm, whoproposed the theory of localized hidden variables in 1952, derived this quantumpotential. It is independent of the phase, 6, of the wave function, and is representedin the form of V = h2V2R/2mR, where R is the amplitude of the wave, m is themass of the particle, and h the Planck constant. With such a quantum potential,


Bohm believed that the motions of microscopic particles should follow the Newton'sequation, and it is because of the "instantaneous" action of this quantum potential, ameasurement process is always disturbed. The latter, however, was less convincing.

Quantum potential and nonlinear equations were again introduced in the Bohm-Bohr theory proposed in 1966. They assumed that, in the dual Hilbert space, thereexists a dual vector, \tpi) and \a), which satisfy

n k

where a is a hidden variable and satisfies the Gaussian distribution in an equilibriumstate. They introduced a nonlinear term in the Schrodinger equation, to representthe effect arising from the quantum measurement, and determined the equationcontaining the nonlinear term based on the relation among the particles, the en-vironment and the hidden variables. Attempts were made to solve the problemconcerning the influence of the measuring instruments on the properties of particlesbeing probed, de Broglie pointed out that the quantum potential can be expressedin terms of the change in the rest mass of the particle and tried to interpret Bohm'squantum potential based on the counteraction of the u wave and domain of singu-larity. Thus, the quantum potential arises from the interaction between particles.It is associated with nonlinear interaction and is able to change the properties ofparticles. These were encouraging. It seemed promising to make microscopic par-ticles measurable and deterministic by adding a quantum potential with nonlineareffect to the Schrodinger equation, and eventually to have deterministic quantumtheory. A delighted Dirac commented that the results will ultimately prove thatEinstein's deterministic or physical view is correct.

In summary, we started the chapter with a review on the hypotheses on whichthe linear quantum mechanics was built, and the successes and problems of theory.We have seen that the linear quantum mechanics is successful and correct, buton the other hand, it is incomplete. Some of its hypotheses are vague and non-intuitive. Moreover, it is a wave theory and cannot completely describe the wave-corpuscle duality of microscopic particles. Therefore, improvement and furtherdevelopment on the linear quantum mechanics are required. The dispute betweenEinstein and Bohr, the recent work done by de Broglie, Bohm, and Bohr providedpositive inspiration for further development of the quantum theory. From the abovediscussion, the direction for a complete theory seems clear: it should be a nonlineartheory. Two of its fundamental hypotheses of linear quantum mechanics, linearityand independence of the Hamiltonian of a systems on its wave function, must bereconsidered.

However, at what level of theory will the problems of the linear quantum me-chanics be solved? What would be a good physical system to start with? Whatwould be the foundation of a new theory? These and many other important ques-tions can only be answered through further research. It is clear that the new theory


should not be confined to the scope and framework of the linear quantum mechanics.The work of de Broglie and Bohm gave us some good motivations, but their ideascannot be indiscriminately borrowed. One must go beyond the framework of thelinear quantum mechanics and look into the nonlinear scheme. To establish a newand correct theory, one must start from the phenomena and experiments which thelinear quantum mechanics failed, or had difficulty, to explain, and uses completelynew concepts and new approaches to study these unique quantum systems. This isthe only way to clearly understand the problems in the linear quantum mechanics.For this purpose, we will review some macroscopic quantum effects in the followingchapter because these experiments form the foundation of the nonlinear quantummechanics.

Bibliography

Akert, A. K. (1991). Phys. Rev. Lett. 67 661.Aspect, A., Grangier, P. and Roger, G. (1981). Phys. Rev. Lett. 47 460.Bell, J. S. (1965). Physics 1 195.Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics, Cambridge Univ.

Press, Cambridge.Bell, J. S. (1996). Rev. Mod. Phys. 38 447.Bennett, C. H., Brassard, G., Crepeau, C, Jozsa, R., Peres, R. and Wootters, W. K.

(1993). Phys. Rev. Lett. 70 1895.Black, T. D., Nieto, M. M., Pilloff, H. S., Scully, M. O. and Sinclair, R. M. (1992).

Foundations of quantum mechanics, World Scientific, Singapore.Bohm, D. (1951). Quantum theory, Prentice-Hall, Englewood Cliffs, New Jersey.Bohm, D. (1952). Phys. Rev. 85 169.Bohm, D. and Bub, J. (1966). Rev. Mod. Phys. 38 453.Bohr, N. (1935). Phys. Rev. 48 696.Born, M. and Infeld, L. (1934). Proc. Roy. Soc. A144 425.Born, M., Heisenberg, W. and Jorden, P. (1926). Z. Phys. 35 146.Czachor, M. and Doebner, H. D. (2002). Phys. Lett. A 301 139.de Broglie, L. (1960). Nonlinear wave mechanics, a causal interpretation, Elsevier, Ams-

terdam.Diner, S., Fargue, D., Lochak, G. and Selleri, F. (1984). The wave-particle dualism, Riedel,

Dordrecht.Dirac, P. A. (1927). Proc. Roy. Soc. London A114 243.Dirac, P. A. (1967). The principles of quantum mechanics, Clarendon Press, Oxford.Einstein, A., Podolsky, B. and Rosen, N. (1935). Phys. Rev. 47 777.Espaguat, B. D. (1979). Sci. Am. 241 158.Fernandez, F. H. (1992). J. Phys. A 25 251.Ferrero, M. and Van der Merwe, A. (1995). Fundamental problems in quantum physics,

Kluwer, Dordrecht, 1995.Ferrero, M. and Van der Merwe, A. (1997). New developments on fundamental problems

in quantum physics, Kluwer, Dordrecht.


Feynman, R. P. (1998). The character of physical law, MIT Press, Cambridge, Mass,p. 129.

French, A. P. (1979). Einstein, A Centenary Volume, Harvard University Press, Cambridge,Mass.

Garola C. and Ross, A. (1995). The foundations of quantum mechanics-historical analysisand open questions, Kluwer, Dordrecht.

Gleason, A. M. (1957). J. Math. Mech. 6 885.Green, H. S. (1962). Nucl. Phys. 33 297.Harut, A. D. and Rusu, P. (1989). Can. J. Phys. 67 100.Heisenberg, W. (1925). Z. Phys. 33 879.Heisenberg, W. and Euler, H. (1936). Z. Phys. 98 714.Hodonov, V. V. and Mizrachi, S. S. (1993). Phys. Lett. A 181 129.Hugajski, S. (1991). Inter. J. Theor. Phys. 30 961.Iskenderov, A. D. and Yaguhov, G. Ya. (1989). Automation and remote control 50 1631.Jammer, M. (1989). The conceptual development of quantum mechanics, Tomash, Los

Angeles.Jordan, T. K. (1990). Phys. Lett. A 15 215.Kobayashi, S., Zawa, H. E., Muvayama, Y. and Nomura, S. (1990). Foundations of quan-

tum mechanics in the light of new technology, Komiyama, Tokyo.Laudau, L. (1927). Z. Phys. 45 430.Muttuck, M. D. (1981). Phys. Lett. A 81 331.Pang Xiao-feng, (1994). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing, pp. 1-34.Polchinki, J. (1991). Phys. Rev. Lett. 66 397.Popova, A. D. (1989). Inter. J. Mod. Phys. A4 3228.Popova, A. D. and Petrov, A. N. (1993). Inter. J. Mod. Phys. 8 2683 and 2709.Reinisch, G. (1994). Physica A 266 229.Roberte, C. D. (1990). Mai. Phys. Lett. A 5 91.Roth, L. R. and Inomata, A. (1986). Fundamental questions in quantum mechanics, Gor-

don and Breach, New York.

Schrodinger, E. (1928). Collected paper on wave mechanics, Blackie and Son, London.Schrodinger, E. (1935). Proc. Cambridge Phil. Soc. 31 555.Selleri, R. and Van der Merwe, A. (1990). Quantum paradoxes and physical reality, Kluwer,

Dordrecht.Slater, P. B. (1992). J. Phys. A 25 L935 and 1359.Valentini, A. (1990). Phys. Rev. A 42 639.Vitall, D., Allegrini, P. and Grigmlini, P. (1994). Chem. Phys. 180 297.Von Neumann, J. (1955). Mathematical foundation of quantum mechanics, Princeton Univ.

Press, Princeton, N.J, Chap. IV, VI.Walsworth, R. L. and Silvera, I. F. (1990). Phys. Rev. A 42 63.Winger, E. P. (1970). Am. J. Phys. 38 1005.Yourgrau, W. and Van der Merwe, A. (1979). Perspective in quantum theory, Dover, New

York.

Chapter 2

Macroscopic Quantum Effects andMotions of Quasi-Particles

In this chapter, we review some macroscopic quantum effects and discuss motionsof quasi-particles in these macroscopic quantum systems. The macroscopic quan-tum effects are different from microscopic quantum phenomena. The motions ofquasi-particles satisfy nonlinear dynamical equations and exhibit soliton features.In particular, we will review some experiments and theories, such as superconduc-tivity and superfluidity, that played important roles in the establishment of nonlin-ear quantum theory. The soliton solutions of these equations will be given basedon modern soliton and nonlinear theories. They are used to interpret macrosopicquantum effects in superconductors and superfluids.

2.1 Macroscopic Quantum Effects

Macroscopic quantum effects refer to quantum phenomena that occur on the macro-scopic scale. Such effects are obviously different from the microscopic quantum ef-fects at the microscopic scale which we are familiar with. It has been experimentallydemonstrated that such phenomena can occur in many physical systems. There-fore it is very necessary to understand these systems and the quantum phenomena.In the following, we introduce some of the systems and the related macroscopicquantum effects.

2.1.1 Macroscopic quantum effect in superconductors

Superconductivity is a phenomenon in which the resistance of a material suddenlyvanishes when its temperature is lower than a certain value, Tc, which is referred toas the critical temperature of the superconducting materials. Superconductors canbe pure elements, compounds or alloys. To date, more than 30 single elements andup to a hundred of alloys and compounds have been found to possess such charac-teristics. When T < Tc, any electric current in a superconductor will flow forever,without being damped. Such a phenomenon is referred to as perfect conductivity.Moreover, it was observed through experiments that when a material is in the super-

23


conducting state, any magnetic flux in the material would be completely repelled,resulting in zero magnetic field inside the superconducting material, and similarlymagnetic flux produced by an applied magnetic field also cannot penetrate into su-perconducting materials. Such a phenomenon is called the perfect anti-magnetismor Maissner effect. There are also other features associated with superconductivity.

How can these phenomena be explained? After more than 40 years' effort,Bardeen, Cooper and Schreiffier established the microscopic theory of superconduc-tivity, the BCS theory, in 1957. According to this theory, electrons with oppositemomenta and antiparallel spins form pairs when their attraction due to the elec-tron and phonon interaction in these materials overcomes the Coulomb repulsionbetween them. The so-called Cooper pairs are condensed to a minimum energystate, resulting in quantmn states which are highly ordered and coherent over along range, in which there is essentially no energy exchange between the electron-pairs and lattice. Thus, the electron pairs are no longer scattered by the lattice andflow freely, resulting in superconductivity.

The electron-pair in a superconductive state is somewhat similar to a diatomicmolecule, but it is not as tightly bound as a molecule. The size of an electron pair,which gives the coherent length, is approximately 10~4 cm. A simple calculationwill show that there can be up to 106 electron pairs in a sphere of 10~4 cm indiameter. There must be mutual overlap and correlation when so many electronpairs are brought together. Therefore, perturbation to any of the electron pairswould certainly affect all others. Thus, various macroscopic quantum effects can beexpected in a material with such coherent and long range ordered states. Magneticflux quantization, vortex structure in the type-II superconductors, and Josephsoneffect in superconductive junctions, are just some examples of such macroscopicquantum phenomena.

2.1.1.1 Quantization of magnetic flux

Consider a superconductive ring. Assume that a magnetic field is applied at T >Tc, then the magnetic flux lines produced by the external field pass through andpenetrate into the body of the ring. We now lower the temperature to a valuebelow Tc, and then remove the external magnetic field. The magnetic inductioninside the body of circular ring equals to zero (B = 0) because the ring is inthe superconductive state and the magnetic field produced by the superconductivecurrent cancels the magnetic field which was in the ring. However, part of themagnetic fluxes in the hole of the ring remains because the induced current in thering vanishes. This residual magnetic flux is referred to as frozen magnetic flux. Itwas observed experimentally that the frozen magnetic flux is discrete, or quantized.Using the macroscopic quantum wave function in the theory of superconductivity,it can be shown that the magnetic flux is given by $' = n<f>0 (n — 1,2,3 • • •), wherecf>o = hc/2e = 2.07 x 10~15 Wb is the flux quantum, representing the flux of onemagnetic flux line. This means that the magnetic fluxes passing through the hole

Macroscopic Quantum Effects and Motions of Quasi-Particles 25

of the ring can only be multiple of (J>Q. In other words, the magnetic field lines arediscrete. What does this imply? If the applied magnetic field is exactly rnfo, thenthe magnetic flux through the hole is n</>0 which is not difficult to understand. Whatif the applied magnetic field is (n + l/4)</>0? According to the above, the magneticflux through the hole cannot be (n + 1/4)</>O. As a matter of fact, it should be ncpo-Similarly, if the applied magnetic field is (n + 3/4) (/>0, the magnetic flux passingthrough the hole is not (n + 3/4) <p0, but rather (n 4- l)0o- The magnetic fluxespassing through the hole of the circular ring is always quantized.

An experiment conducted in 1961 surely proved so. It indicated that magneticflux does exhibit discrete or quantized characteristics on the macroscopic scale. Theabove experiment was the first demonstration of macroscopic quantum effect. Basedon quantization of magnetic flux, we can build a "quantum magnetometer" whichcan be used to measure weak magnetic field with a sensitivity of 3 x 10~7 Oersted.A slight modification of this device would allow us to measure electric current aslow as 2.5 x 10~9 A.

2.1.1.2 Structure of vortex lines in type-II superconductors

The superconductors discussed above are referred to as type-I superconductors.This type of superconductors exhibit perfect Maissner effect when the externalapplied field is higher than a critical magnetic value Hc. There exists another typeof materials such as the NbTi alloy and NbsSn compound in which the magneticfield partially penetrates inside the material when the external field H is greaterthan the lower critical magnetic field Hc\ but less than the upper critical fieldHc2. This kind of supercondutors are classified as type-II superconductors and arecharacterized by a Ginzburg-Landau parameter of greater than l/-\/2.

Studies using the Bitter method showed that penetration of the magnetic fieldresults in some small regions changing from superconductive to normal state. Thesesmall regions in normal state are of cylindrical shape and regularly arranged in thesuperconductor, as shown in Fig. 2.1. Each cylindrical region is called a vortex (ormagnetic field line). The vortex lines are similar to the vortex structure formed ina turbulent flow of fluid.

It was shown through both theoretical analysis and experimental measurementthat the magnetic flux associated with one vortex is exactly equal to one magneticflux quantum, <f>Q. When the applied field H > HCl, the magnetic field penetratesinto the superconductor in the form of vortex lines, increased one by one. For anideal type-II superconductor, stable vortices are distributed in triagonal pattern,and the superconducting current and magnetie field distributions are also shown inFig. 2.1. For other, non-ideal type-II superconductors, the triagonal distributioncan be observed in local regions, even though its overall distribution is disordered.It is evident that the vortex-line structure is quantized and this has been verifiedby many experiments. It can be considered a result of quantization of magneticflux. Furthermore, it is possible to determine the energy of each vortex line and the


Fig. 2.1 Current and magnetic field distributions in a type-II superconductor.

interaction energy between the vortex lines. Parallel magnetic field lines are foundto repel each other while anti-parallel magnetic lines attract each other.

2.1.1.3 Josephson effect

As known in the LQM, microscopic particles such as electrons have wave propertyand they can penetrate through potential barriers. For example, if two pieces ofmetals are separated by an insulator of width of tens or hundreds of angstroms, anelectron can tunnel through the insulator and travel from one metal to the other.If a voltage is applied across the insulator, a tunnel current can be produced. Thisphenomenon is referred to as a tunnelling effect. If the two pieces of metals in theabove experiment are replaced by two superconductors, tunneling current can alsooccur when the thickness of the dielectric is reduced to about 30 A. However, thiseffect is fundamentally different from the tunnelling effect discussed above and isreferred to as the Josephson effect.

Evidently, this is due to the long-range coherent effect of the superconduc-tive electron pairs. Experimentally it was demonstrated that such an effect canbe produced in many types of junctions involving a superconductor, for exam-ple, a superconductor-metal-superconductor junction, superconductor-insulator-superconductor junction and superconductor bridge. These junctions can be con-sidered as superconductors with a weak link. On one hand, they have propertiesof bulk superconductors, for example, they are capable of carrying certain super-conducting current. On the other hand, these junctions possess unique propertieswhich a bulk superconductor does not. Some of these properties are summarized inthe following.

(1) When a direct current (dc) passing through a superconductor is smaller thana critical value Ic, the voltage across the junction does not change with current. Thecritical current Ic can range from a few tens of /xA to a few tens of mA. Figure 2.2


shows the characteristics of the dc Josephson current of a Sn-SnO-Sn junction.

Fig. 2.2 dc Josephson current of a Sn-SnO-Sn junction.

(2) If a constant voltage is applied across the junction and the current passingthrough the junction is greater than Ic, a high frequency sinusoidal superconductingcurrent occurs in the junction. The frequency is given by v = 2eV/h, in the mi-crowave and far-infrared regions (5-1000 x 109 Hz). The junction radiates coherentelectromagnetic waves with the same frequency.

This phenomenon can be explained as follows. The constant voltage appliedacross the junction produces an alternating Josephson current which in turn gener-ates electromagnetic waves of frequency v. These waves propagate along the planesof the junction. When they reach the surface of the junction (interface between thejunction and its surrounding), part of the electromagnetic wave is reflected from theinterface and the rest is radiated, resulting in radiation of coherent electromagneticwaves. The power of radiation depends on the compatibility between the junctionand its surrounding.

(3) When an external magnetic field is applied over the junction, the maximumdc current Ic is reduced due to the effect of the magnetic field. Furthermore, Ic

changes periodically as the magnetic field increases. The Ic - H curve resemblesthe distribution of light intensity in the Fraunhofer diffraction experiment, andthe latter is shown in Fig. 2.3. This phenomena is called quantum diffraction ofsuperconducting junction.

(4) When a junction is exposed to a microwave of frequency v and if the volt-age applied across the junction is varied, it was found that the dc current passingthrough the junction increases suddenly at certain discrete values of electric poten-tial. Thus, a series of steps appear on the dc I — V curve, and the voltage at agiven step is related to the frequency of the microwave radiation by nv — 2eVn/h(n — 1,2,3, • • •). More than 500 steps have been observed in experiments.

These phenomena were first derived theoretically by Josephson and each wasexperimentally verified subsequently. All these phenomena are called Josephson


Fig. 2.3 Quantum diffraction effect in superconductor junction

effects. In particular, (1) and (3) are referred to as dc Josephson effects while (2)and (4) are referred to as ac Josephson effects.

Evidently the Josephson effects are macroscopic quantum effects which can bewell explained by the macroscopic quantum wave function. If we consider a su-perconducting juntion as a weakly linked superconductor, then the wave functionsof the superconducting electron pairs in the superconductors on both sides of thejunction are correlated due to a definite difference in their phase angles. This resultsin a preferred direction for the drifting of the superconducting electron pairs, and adc Josephson current is developed in this direction. If a magnetic field is applied inthe plane of the junction, the magnetic field produces a gradient of phase differencewhich makes the maximum current oscillate along with the magnetic field and ra-diation of electromagnetic wave occurs. If a voltage is applied across the junction,the phase difference will vary with time and results in the Josephson effect. In viewof this, the change in the phase difference of the wave functions of superconductingelectrons plays an important role in the Josephson effect, which will be discussedin more details in the next section.

The discovery of the Josephson effect opened the door for a wide range of ap-plications of superconductors. Properties of superconductors have been explored toproduce superconducting quantum interferometer - magnetometer, sensitive ame-ter, voltmeter, electromagnetic wave generator, detector and frequency-mixer, andso on.

2.1.2 Macroscopic quantum effect in liquid helium

Helium is a common inert gas. It is also the most difficult gas to be liquidified.There are two isotopes of helium, 4He and 3He, with the former being the majorityin a normal helium gas. The boiling temperatures of 4He and 3He are 4.2 K and3.19 K, respectively. Its critical pressure is 1.15 atm for 4He. Because of their lightmasses, both 4He and 3He have extremely high zero-point energies, and remainin gaseous form from room temperature down to a temperature near the absolute


zero. Helium becomes solid due to cohesive force only when the interatomic distancebecomes sufficiently small under high pressure. For example, a pressure of 25 —34 atm is required in order to solidify 3He. For 4He, when it is cystallized at atemperature below 4 K, it neither absorbs nor releases heat. i.e. the entropiesof the crystalline and liquid phases are the same and only its volume is changedin the crystallization process. However, 3He absorbs heat when it is crystallizedat a temperature T < 0.319 K under pressure. In other words, the temperature of3He rises during crystallization under pressure. Such an endothermic crystallizationprocess is called the Pomeranchuk effect. This indicates that the entropy of liquid3He is smaller than that of its crystalline phase. In other words, the liquid phaserepresents a more ordered state. These peculiar characteristics are due to the uniqueinternal structures of 4He and 3He.

Both 4He and 3He can crystallize in the body-centered cubic or the hexagonalclose-stacked structures. A phase transition occurs at a pressure of 1 atm and atemperature of 2.17 K for 4He. Above this temperature, 4He is no different froma normal liquid and this liquid phase is referred to as He I. However, when thetemperature is below 2.17 K, the liquid phase, referred to as He II, is completelydifferent from He I and it becomes a superfluid. This superfluid can pass through,without experiencing any resistance, capillaries of diameters less than 10~6 cm. Thesuperfluid has a low viscosity (< 10~n P) and its velocity is independent of thepressure difference over the capillary and its length. If a test tube is inserted intoliquid He II contained in a container, the level of liquid He II inside the test tube isthe same as that in the container. If the test tube is pulled up, the He II inside thetest tube would rise along the inner wall of the tube, climb over the mouth of thetube and then flow back to the container along the outer wall of the tube, until theliquid level inside the test tube reaches the same level as that in the container. Onthe other hand, if the test tube is lifted up above the container, the liquid in testtube would drip directly into the container until the tube becomes empty. Such aproperty is called the superfluidity of 4He.

Osheroff and others discovered two phase transitions of 3He, occuring at 2.6 mKand 2 mK, respectively, when cooling a mixture of solid and liquid 3He in 1972.Further experiments showed that 3He condenses into liquid at 3.19 K and becomesa superfluid at temperatures below 3 x 10~3 K. In the absence of any external field,3He can exist in two superfluid phases, 3He A and 3He B. Under a strong magneticfield of a few thousand Gausses, there can be three superfluid phases: 3He Aj, 3HeA and 3He B. The Tc is splited into TCl and TC2, with normal 3He liquid above TCl,the 3He Ai phase between TCl and TC2, the 3He A phase between TC2 and TAB, andthe 3He B phase below TAB- 3He A is anisotropic and could be a superfluid withferromagnetic characteristics. The magnetism results from ordered arrangementof magnetic dipoles. 3He B was believed to be isotropic. However, it was knownthrough experiments that 3He B can also be anisotropic below certain temperature.Thus, 3He not only shows characteristics of superconductivity and liquid crystal, it


can also be a superfluid, like 4He. This makes 3He a very special liquid system.Experiments have shown that quantization of current circulation and vortex

structure, similar to that of magnetic flux in a superconductor, can exist in the4He II, 3He A and 3He B superfluid phases of liquid helium. In terms of the phaseof the macroscopic wave function, 6, the velocity vs of the superfluid is given byvs — hV8/M, where M is the mass of the helium atom. vs satisfies the followingquantization condition:

jvs-dr = n-^, (n = 0 , l ,2 l - - - ) .

This suggests that the circulation of the velocity of the superfluid is quantizedwith a quantum of h/M. In other words, as long as the superfluid is rotating, anew whirl in the superfluid is developed whenever the circulation of the current isincreased by h/M, i.e., the circulation of the whirl (or energy of the vortex lines)is quantized. Experiment was done in 1963 to measure the energy of the vortexlines. The results obtained were consistent with the theoretical prediction. Thequantization of circulation was thus proved.

If the superfluid helium flows, without rotation, through a tube with a varyingdiameter, then V x vs = 0, and it can be shown, based on the above quantizationcondition, that the pressure is the same everywhere inside the tube, even thoughthe fluid flows faster at a point where the diameter is smaller and slower where thediameter of the tube is larger. This is completely different from a normal fluid, butit has been proved by experiment.

The macroscopic quantum effect of 4He II was observed experimentally by Fir-bake and Maston in the U.S.A. once again. When the superfluid 4He II was setinto rotational motion in a cup, a whirl would be formed when the temperature ofthe liquid is reduced to below the critical temperature. In this case, an effectiveviscosity develops between the fluid and the cup which is very similar to normalfluid in a cup being stired. The surface of the superfluid becomes inclined at acertain angle and the cross-section of the liquid surface is in a shape of a parabola,due to the combined effects of gravitation and centrifugal forces. Fluid away fromthe center has a tendency to converge towards its center which is balanced by thecentrifugal force and a dynamic equilibrium is reached. The angular momentmn ofsuch a whirl is very small and consists of only a small number of discrete quantumpackets. The angular momentum of the quantum packets can be obtained by thequantum theory. In other words, the whirl can exist only in discrete form over acertain range in certain materials such as superfluid helium. Firbake and Mastonmanaged to obtain sufficiently large angular momentum in their experiment andwere able to observe the whirl's surface shape using visible light. They used a thinlayer of rotating superfluid helium in their experiment. While the rotating super-fluid was illuminated from both top and bottom by laser beams of wavelength of6328 A from a He-Ne laser, the whirl formed were observed. Alternate bright and


dark interference fringes were formed when the reflected beams were focused on anobserving screen. Analysis of the interference pattern showed that the surface wasindeed inclined at an angle. Based on this, the shape of the surface can be correctlyconstructed. The observed interference pattern was found in excellent agreementwith those predicted by the theory. This experiment further confirmed the existenceof quantized vortex rings in the superfluid 4He.

How to theoretically explain the superfluidity and the macroscopic quantumeffect of 4He is still a subject of current research. In the 1940s, Bogoliubov calculatedthe critical temperature of Bose-Einstein condensation in 4He based on an idealBoson gas model. The value he obtained, 3.3 K was quite close to the experimentalvalue of 2.17 K. At a temperature below Tc, some 4He atoms condense to the statewith zero momentum. At absolute zero, all the 4He atoms condense to such a state.According to the relation A = h/p, the wave length of each 4He atom would beinfinite in this case and an ordered state over the entire space can be formed whichleads to the macroscopic quantum effects. Pang believed that this phenomenoncan be attributed to Bose-Einstein condensation of the 4He atoms. When thetemperature of 4He is below Tc, the symmetry of the system breaks down due tononlinear interaction in the system. The 4He atoms spontaneously condense whichresults in a highly ordered and long-range coherent state.

3 He is different from 4He. 3He atoms are fermions and obey the Fermi statis-tical law, rather than Bose statistical law. The mechanism of condensation andsuperfluidity of 3He is similar to that of superconductivity. At a temperature belowTc, two 3He atoms form a loosely bound atomic-pair due to nonlinear interactionwithin the system. The two atoms with parallel spins revolve around each otherand form a pair with a total angular momentum of J = 1. This gives rise to ahighly ordered and long-range coherent state and condensation of 3He atomic pairs.The macroscopic quantum effect is thus observed. The mechanisms of superfluidityand vortex structure in 4He and 3He have been extensively studied recently and thereaders are referred to Barenghi et al. (2001) for a review of recent work.

2.1.3 Other macroscopic quantum effects

Macroscopic quantum effects were also observed in other materials. A few of whichare relevant to the topic of this book are briefly introduced in the following.

2.1.3.1 Quantum Hall effect

When a longitudinal electric field and a transverse magnetic field are applied toa metal, electric charges accumulate on surfaces parallel to the plane denned bythe external fields, producing an electric field and electric current in the directionperpendicular to the external fields. This phenomenon is referred to as the normalHall effect which everyone is familiar with. The Hall effect to be discussed here hasspecial characteristics and is referred to as a quantum Hall effect. At an extremely


low temperature and in the presence of a strong magnetic field, the measured Hallelectric potential and the Hall resistance show a series of steps for certain materialsin the quantum regime. It appears that they can only have values which are integermultiples of a basic unit, i.e. they are quantized. Obviously, this is a macroscopicquantum effect. The concept of quantum Hall effect was first proposed by TsuneyuAnda of Tokyo University and was experimentally verified first by Klitzing andcoworkers.

The approach by Klitzing et al. was based on the fact that the degenerate elec-tron gas in the inversion layer of a metal-oxide-semiconductor field-effect transistoris fully quantized when the transistor is operated at the helium temperature and inthe presence of a strong magnetic field (~ 15 T). The electric field applied perpen-dicular to the oxide-semiconductor interface (gate field) produces a potential welland electrons are confined within the potential and their motion in the directionperpendicular to the interface (z-direction) is limited. If a magnetic field is appliedin the ^-direction and an electric field is applied in a direction (x) perpendicular tothe magnetic field, then the electron will depart from the x direction and drift inthe ^-direction due to the Lorenz force, resulting in the normal Hall effect. How-ever, when the magnetic field is increased to above 150 KG, the degenerate electronground state splits into Landau levels. The energy of an electron occupying the nthLandau energy level is (n + l/2)/hujc, where uc is the angular frequency which isdirectly proportional to the magnetic field Hz. These Landau energy levels can beviewed as semiclassical electron orbits of approximately 70A in radius. When the"gate voltage" is modulated so that the Fermi energy of this system lies betweentwo Landau energy levels, all the Landau energy levels below the Fermi level areoccupied while all energy levels above the Fermi level are vacant. However, dueto the combined effect of the electric field Ex and magnetic field Hz, the electronsoccupying the Landau energy levels move in the ^-direction, but do not producecurrent in the direction of the electric field.

When a current flows through the sample in a direction perpendicular to thedirection of the electric field, the diagonal terms of the Hall conductivity axx =cryy = 0 and the off-diagonal terms axy is given by Ne/Hz, where N is the densityof the 2D electron gas. If n Landau energy levels are filled up, then N = nHz/h, andthe Hall resistance Rfj — h/ne2 which is meterial independent. This suggests thatthe Hall resistance is quantized and its basic unit is h/e2. Klitzing et al. conductedtheir experiment using a sample of 400 /j,m x 50 ^m in size and a distance of 130fim between the potential probes. The measured Hall resistance corresponding tothe nth plateau was exactly equal to h/ne2.

More recently, same phenomena was also observed in GaAs/AlxGai_xAs het-erojunction by scientists in Bell Laboratary under a slightly low magnetic field andtemperature. Abnormal Hall effect with n being a fractional number were also ob-served subsequently. These phenomena allow us to experimentally determine thevalue of e2/h and the fine structural constant a — e2/h(îoc/2) (no is the magnetic


permeability in vacuum) with high accuracy.Laughlin was the first to try to deduce the result of the quantum Hall effect from

the gauge invariance principles. He believes that this effect depends on the gaugeinvariance of electromagnetic interaction and the existence of a "drifting" energygap. Based on his theory, Laughlin successfully derived Rjj = h/ne2.

2.1.3.2 Spin polarized atomic hydrogen system

Hydrogen atoms are Bosons and obey the Bose-Einstein statistics. Because of itsextremely light mass, hydrgon atoms have extremely high zero-point vibrationalenergy. Thus, under ambient pressure, hydrgens remain in gaseous phase withweak interatomic interaction until the temperature approaches to 0 K. Under highpressure, its state undergoes a change when its density reaches an extremely highvalue. Above this critical density, excess atoms are transfered to a state which cor-responds to an energy minimum, and condensation occurs. In other words, whenthe temperature is below the critical value Tc and the de Broglie wave length ofatomic hydrogen is comparable to the interatomic spacing, a considerable amountof atomic hydrogens in the system occupy the same quantum states through theattraction and coherence among them. Bose-Einstein condensation of these atomsthus occurs which is acompanied by sudden changes in specific heat and susceptilil-ity, etc. Theoretical calculations show that such a condensation occurs when thedensity of atomic hydrogen p > 1016 cm"3. The critical temperature Tc increaseswith increase of p. At p = 1017 cm"3, Tc = 8 mK; and at p = 1019 cm"3, Tc = 100mK.

It was also shown theoretically that the Bose-Einstein condensation can onlyoccur when the atomic hydrogens are in a triplet states with parallel down spins,which is a stable state with an extremely low energy. Therefore, in order for thecondensation to occur, a magnetic field gradient must be applied to the system toselect the polarized hydrogen atoms with parallel spins and to hinder the probabilityof recombination of atomic hydrogens into hydrogen molecules. A magnetic fieldhigher than 9 — 11 T is normally required. At present, atomic hydrogens cannot becompressed to the density of 1019 cm"3. The methods for avoiding recombinationof atomic hydrogens are thus limited. It has not been possible to directly observethe condensation of spin polarized hydrogen atoms and the associated macroscopicquantmn effects.

2.1.3.3 Bose-Einstein condensation of excitons

It is well known that an electron and a hole can form a bound state, or an exciton,due to the Coulomb interaction between them in many materials such as silicon,germanium, cadmium sulphide, arsenical bromine, silicon carbide, etc.. An excitonhas its own mass, energy and momentum. It not only can rotate around its owncenter of mass but also move freely in the crystal lattice. At high exciton density,


it is possible for two excitons with opposite spins to form an exciton molecule withthe characteristics of a Boson, due to attractive force betweeen them.

When the temperature is below a certain critical value, a considerable amountof exciton molecules condense to the ground state with zero momentum, resultingin the Bose-Einstein condensation, and the superfluidity as in the liquid helium.Because the ratio of the average mass of an exciton or an exciton molecule to themass of an electron is 2 - 3 orders of magnitude smaller than that in 4He, thequantum effects in an exciton system is more obvious than in the liquid helium.

When the same quantum state is occupied by many excition molecules, relativelystronger lines can be observed in the emission spectrum. In 1974, scientists fromJapan, Switzerland and other countries detected a very strong line in the low energyregion of free exciton recombination radiation of a high purity AgBr sample. Alsoobserved was a stair pattern similar to that observed in the above superconductivediffraction experiment. Very sharp emission lines were later observed at Eex ~1.8195 eV using a narrow ranged excitation power (50 — 100 x 106 W/cm2) in thesame material. This is also a macroscopic quantum effect. Hang et al. predicatedthat Bose condensation in exciton systems can result in energy superfluid, similarto the above superfluid liquid helium and superconductors.

The Bose-Einstein condensation phenomenon was a subject of extensive studies,both experimentally and theoretically, in the last few decades. Numerious resultshave been published in Physical Review Letters, Physical Review, Physics Letters,etc.. For a review, the readers are referred to Barenghi et al. (2001).

Macroscopic quantum effects are also expected to occur in ferromagnets, whitedwarf and neutron stars. Recently, V. R. Khalilov investigated theoretically macro-scopic quantum effects in a degenerate strongly magnetized neutron star. However,these have not been vertified experimentally.

2.2 Analysis on the Nature of Macroscopic Quantum Effect

From the above discussion, we know that the macroscopic quantum effect is a quan-tum effect that occurs at the macroscopic scale. Such quantum effects have beenobserved for physical quantities such as resistance, magnetic flux, vortex line andvoltage, etc. The macroscopic quantum effect is obviously different from the micro-scopic quantum effect in which physical quantities depicting microscopic particles,such as energy, momentum and angular momentum, are quantized. Thus it is rea-sonable to believe that the fundamental nature and the rules governing these effectsare different. We know that the microscopic quantum effect is described by theLQM theory. But what are the mechanisms for the macroscopic quantum effects?How can these effects be properly described? These questions apparently need tobe addressed.

We all know that materials are composed of a great number of microscopicparticles such as atoms, electrons, nuclei and so on, which exhibit quantum features.


We can then infer or assume that the macroscopic quantum effect results from thecollective motion and excitation of these particles under certain conditions, such asextremely low temperature, high pressure or high density. Under such conditions, ahuge number of microscopic particles pair with each other and condense, resultingin a highly ordered and long-range coherent low-energy state. In such a highlyordered state, the collective motion of a large number of particles is the same as themotion of a single particle, and since the latter is quantized, the collective motionof the many-particle system gives rise to the macroscopic quantum effect. Thus,the condensation of the particles and the coherent state play an essential role in themacroscopic quantum effect.

What is condensation? At the macroscopic scale, the process of a gas transform-ing into a liquid, such as that of changing vapor into water, is condensation. This,however, represents a change in the state of molecular positions, and is referred toas a condensation of positions. The phase transition from a gaseous state to a liquidstate is a first order transition in which the volume of the system changes and latentheat is produced, but thermodynamic quantities of the systems are continuous andhave no singularities. The word condensation in the context of macroscopic quan-tum effect has its special meaning. The condensation being discussed here is similarto the phase transition from a gas to a liquid, in the sense that the pressure dependsonly on temperature, but not on volume during the process. Therefore, it is funda-mentally different from the first-order phase transition such as that from vapor towater. It is not the condensation of particles into a high density material in normalspace. On the contrary, it is the condensation of particles to a single energy stateor to a low energy state with a constant or zero momentum. It is thus also calleda condensation of momentum. This differs from a first-order phase transition andtheoretically it should be classified as a third order phase transition, even though itis really a second order phase transition, because it is related to the discontinuity ofthe third derivative of a thermodynamnic function. Discontinuities can be clearlyobserved in measured specific heat, magnetic susceptibility of certain systems whenthe condensation occurs. The phenomenon results from a spontaneous breaking ofsymmetries of the system due to nonlinear interaction within the system under somespecial conditions, such as extremely low temperature and high pressure. Differentsystems have different critical temperatures for the condensation. For example, thecondensation temperature of a superconductor is its critical temperature Tc, andfrom previous discussions, the condensation temperatures of superfluids 4He and3He are 2.17 K and 10~3 K respectively under the ambient pressure (1 atm.).

From the above discussions on properties of superconductors, superfluids 4Heand 3He, spin polarized hydrogen atoms and excitons, we know that even thoughthe elementary particles involved can be either Bosons or Fermions, those beingactually condensed, are either Bosons or quasi-Bosons, since Fermions are boundinto pairs. For this reason, the condensation is referred to as Bose-Einstein con-densation, since Bosons obey the Bose-Einstein statistics. Properties of Bosons are


different from those of Fermions, they do not follow the Pauli exclusion principle,and there is no limit to the number of particles occupying the same energy levels.At finite temperatures, Bosons can be distributed among many energy states andeach state can be occupied by one or more particles, and some states may not beoccupied at all. Due to the statistical attraction between Bosons in the phase space(consisting of generalized coordinate and momentum), groups of Bosons tend tooccupy one quantum state under certain conditions. Then when the temperature ofthe system becomes below a critical value, the majority or all Bosons condense tothe same energy level (e.g. the ground state), resulting in Bose condensation and aseries of interesting macroscopic quantum effects. Different macroscopic quantumphenomena are observed bacause of differences in the fundamental properties of theconstituting particles and their interactions in different systems.

In the highly ordered state the behavior of each condensed particle is closelyrelated to the properties of the systems. In this case, the wave function 4> = fel9 or<f> = yfpe%e of the macroscopic state is also the wave function of an individual con-densed particle. The macroscopic wave function is also called the order parameterof the condensed state. It was used to describe the superconductive and superfluidstates in the study of these macroscopic quantum effects. The essential features andfundamental properties of the macroscopic quantum effect are given by the macro-scopic wavefunction </> and it can be further shown that the macroscopic quantumstates such as the superconductive and superfluid states, are coherent and Bose con-densed states formed through second-order phase transitions after the symmetry ofthe system is broken due to nonlinear interaction in the system.

In the absence of any external field, the Hamiltonian of a given physical systemcan be written as

H = JdxH = J.dx [ - | | V 0 | 2 - a\<j>\2 + A|</>|4] . (2.1)

The unit system in which m = h = c = 1 will be used for convenience. If an externalfield does exist, then the Hamiltonian given above should be replaced by

H= fdxH = fdx - | | V - ie*A(j>\2 - a\4>\2 + A|</>|4 + ^

or equivalently

H= I dxH = f dx\~\(dj - ie'AjW2 - a\4>\2 + \\<j>\4 + ^Fjt • FA (2.2)

where Fji = djAi — diAj is the covariant field intensity, H = V x A is the magneticfield intensity, e is the charge of an electron, and e* = 2e, A is the vector potentialof the field, a and A are some interaction constants.

The Hamiltonians given in (2.1) and (2.2) have been extensively used by manyscientists in their studies of superconductivity, for example, Jacobs and Rebbi,


de Gennes, Saint-James, Laplae et al, Kivshar and Bullough, Nemikovskii et al,Roberts et al, Berloff et al, Huepe et al., Sonin and Svistelnov, to name only afew. The readers are referred to Barenghi et al. (2001) for further details of theirwork. The Hamiltonians given in (2.1) and (2.2) resemble that in the Davydov'ssuperconductivity theory and can be derived from the free energy expression of asuperconductive system given by Landau et al. As a matter of fact, the Lagrangianof a superconducting system can be obtained from the well known Ginzberg-Landau(Ginzburg-Landau) equation using the Lagrangian method. The Hamiltonian ofthe sytem can then be derived from the Lagrangian. The results of course are thesame as (2.1) and (2.2). Therefore, the Hamiltonians given in (2.1) and (2.2) areexpected to describe the macroscopic quantum states including superconductingand superfluid states. These problems are treated in more details in the following.

Obviously, Hamiltonians (2.1) and (2.2) possess the [/(I) symmetry. That is, Hremains unchanged under the following transformation

tf>j(x,t) -> ^j{x,t) = e-ieQt<f>j(x,t) (2.3)

where Qj is the charge of the particle. In the case of one dimension, each term inthe Hamitonian (2.1) or (2.1) contains the product of the <pj(x,t)s,

Mx,t) • ••<f>n(x,t) -»• ei«'+«»+-+O»)fl01(a;,t). ..tf>n(x,t).

Since charge is invariant under the transformation and neutrality is required for 7i.There must be (Qi + Q2 + \- Qn) — 0 in such a case. Furthermore, since 6 isindependent of x, it is necessary that V(j)j —> e~l6(^'V(f>j. Thus, each term in theHamiltonian is invarient under the transformation of (2.3), or they possess the E/(l)symmetry.

If we rewrite (2.1) as the following

n = -l-{V4>f + Ue«{<j}), ?7effW = - ^ 2 + A^4, (2.4)

we can see that the effective petential energy has two sets of extrema, <j>o = 0 and4>o = ±\/a/2A, and the minimum is located at

to = ±y^=<0M0>, (2.5)

rather than at (j>0 = 0. This means that the energy at <f>0 = ±\Ja/2\ is lower thanthat at 0o = 0 . Therefore, <j>o — 0 corresponds to the normal ground state while<f>o = ±\fa/2\ is the ground state of the macroscopic quantum systems (MQS). Inthis case, the macroscopic quantum state is the stable state of the system. Therefore,the Hamiltonian of a normal state differs from that of the superconducting state.Meanwhile, the two ground states satisfy (0|^|0) ^ — (0| >|0) under the transforma-tion (f> -» —(f>. That is, they no longer have the U(l) symmetry. In other words, thesymmetry of the ground states has been destroyed. The reason for this is obviously


due to the nonlinear term \\<fr\4 in the Hamiltonian of the system. Therefore, thisphenomenon is referred to as a spontaneous breakdown of symmetry. Accordingto Landau's theory of phase transition, the system undergoes a second-order phasetransition in such a case, and the normal ground state <f>0 = 0 is changed to thesuperconducting ground state <f>0 = ±\Ja/2\. We will give a proof in the following.

In order to make the expectation value in a new ground state zero, we make thefollowing transformation,

4>' = <S> + t o (2.6)

so that

(0\4>'\0) = 0. (2.7)

After this transformation, the Hamiltonian of the system becomes

n(4> + 4>o) = - | | V 0 | 2 + (6A0g - a)<j>2 + 4\4>O4>3 (2.8)

+ {4\<t>l - 2a<j>o)<f> + \<f>4 - a(j>l + \<j>%.

Substituting (2.5) into (2.8), we have

( f o | 4 A $ - 2 a | 0 o ) = O .

Consider now the expectation value of the variation STi/Sfi in the ground state, i.e.

then from (2.1), we get

/ o ^ o \ = (0 | -V 2 0 + 2acj> - 4A031 0> = 0. (2.9)

After the transformation (2.6), this becomes

V20o + (4A0g - 2a)0o + 12A</.0(0|</>2|0) + 4A(0|</>3|0)-

2(a-6A$)(O|0|O)=O (2.10)

where the terms <O|0310> and (O|0|O) are both zero, but the fluctuation 12A0O(O| 2|O)of the ground state is not zero. For a homogeneous system at T = 0 K, the term(0|<^2|0) is very small and can be neglected. Then (2.10) can be written as

-V2&> - to W § - 2a) = 0. (2.11)

Two sets of solutions, <po = 0 and <j)o = ±\Ja/2\ can then be obtained from theabove equation.


If the displacement is very small, i.e. 4>0 -> <j>o + S(j>o = <[>'o, then the equationsatisfied by the fluctuation S<j)0 relative to the normal ground state <f>0 = 0 is

V2<J0o - 2a6<j>0 - 0. (2.12)

Its solution attenuates exponentially, indicating that the ground state <fro = 0 isunstable.

On the other hand, the equation satisfied by the fluctuation 6<p0 relative to theground state (fi0 = ±\/a/2X is

V2<50O + 2a6<pQ = 0.

Its solution 6<f>o is an oscillatory function, and thus, the MQS ground states (f>0 =±i/a/2A are stable. Further calculation shows that the energy of the MQS groundstate is lower than that of the normal state by e0 = —a2/4A < 0. Therefore, theground state of the normal phase and that of the condensed phase are separated byan energy gap of a2/(4A), and at T = 0 K, all particles condense to the ground stateof the condensed phase, rather than filling the ground state of the normal phase.Based on this energy gap, we can conclude that the specific heat of the systemshas exponential dependence on the temperature, and the critical temperature isgiven by Tc = 1.14u;pexp[— l/(3A/a)iV(0)]. This is a feature of the second-orderphase transition. The results are in agreement with that of the BCS theory ofsuperconductivity. Therefore, the transition from the state </>o = 0 to the state4>o = ±^/a/2X and the corresponding condensation of particles are second-orderphase transition. It is obviously the results of spontaneous breakdown of symmetrydue to nonlinear interaction.

In the presence of an electromagnetic field with a vector potential A, the Hamil-tonian of the systems is given by (2.2). It still possesses the U(l) symmetry. Thatis, under the following gauge transformation

f 4>{x) —> <t>'{x) = e-ie^<j>(x)

| A(x) —> A'(x) = A(x) - \Vj9(x), or SAj(x) = \dj6{x) ^2'13^

% is invariant, since FjiF^1 /4 is invariant under this transformation. In other words,the Hamiltonian (2.2) has the U(l) invariance.

Because of the existence of the nonlinear terms in (2.2), a spontaneous break-down of symmetry can be expected. Now consider the following transformation,

<j)(x) = -^[0i(z)+i0a(aO] —> A=[M*) + <h + i<h(*)]- (2-14)


Since (0|</>j|0) = 0 under this transformation, equation (2.2) becomes

n = \idiAj - djAtf - \{V<t>2)2 - ^(V<M2 + ^ [ ( 0 i + 0o)2 + <t>l]A2

-e*<f>0AiV(j>2 + e*{4>2^(t>i ~ <piV(t>2)Ai

- | ( - 1 2 A $ + 2a)J>2 - \{l2\cj>l + 2a)cf>2 + 4Aî(<£i + <t>l)

+4A(0? + <fof _ fo(4A$ + 2a)0i - a<& + \<f>*. (2.15)

It may be seen that the classical interaction energy of </>0 is still given by

Uea{<fo) = -a<ft + \<fi, (2.16)

in agreement with that given in (2.4). Therefore, using the same argument, wecan conclude that spontaneous symmetry breakdown and the second-order phasetransition occur in the system. The system is changed from the ground state of thenormal phase, (f>o — 0, to the ground state <j>o = ±^/a/2A of the condensed phase.The above results can also be used to explain the Maissner effect and to determineits critical temperature. Thus, we can conclude that regardless the existence of anyexternal field, macroscopic quantum states, such as the superconducting state ofa superconductor, are formed through a second-order phase transition following aspontaneous symmetry breakdown due to nonlinear interaction in the systems.

We can prove that this is a condensed state using either the second quantizationtheory or the solid state quantum field theory. As discussed above, when dT-L/d(f) =0, we have

V24> - 2a<f> + 4\\(j)\2(j) = 0. (2.17)

This is the , and it is a time-independent nonlinear Schrodinger equation. Expand-ing cf> in terms of the creation and annihilation operators, 6+ and bp, we have

<t> = -jf E y b ; (bpe~iPX + b+»eiP'x) (2-18)

where V is the volume of the system. After a spontaneous breakdown of symmetry,(j)0, the ground state of <fi, is no longer zero, but (j>0 = ±yJa/2\. The operation ofthe annihilation operator on |<fo) no longer gives zero, i.e.

bp\cj>Q) ± 0 . (2.19)

A new field </>' can then be denned according to the transformation (2.6), where4>o is a scalar field and satisfies (2.11) in such a case. Evidently, (f>0 can also beexpanded into

<£o = — L E - l F ^ - i p x + tf e^*). (2-20)


The transformation between the fields <j>' and <fi is obviously a unitary transforma-tion, that is,

0 ' = U(f>U-1 = e-s<t>es = 0 + 0o (2.21)

where

S = i f dx' [<t>{x\ t)fo(x\ t) - <t>Q(x', t)<f>(x',t)]. (2.22)

<j) and <j>' satisfy the following commutation relation

[<f>'(x,t),4>(x',t)]=i6(x-x'). (2.23)

Prom (2.7) we have <O|0'|O) = <f>'0 = 0. The ground state \<j>'0) of the field 4>' thussatisfies

&p|0o) = 0. (2.24)

From (2.21), we can obtain the following relation between the annihilation operatorav of the new field cf>' and the annihilation operator bp of the <j> field,

ap = e~sbpes = bp + (,p (2.25)

where

Cp = (2^72 / ^ | = faOMK"-* + 5(x,t)e-^-] . (2.26)

Therefore, the new ground state \4>'Q) and the old ground state |0o) are relatedthrough \(f>'0) = es\(j>o). Thus we have

a,K> = (&p+ & M ) =&!#,>• (2-27)

According to the definition of coherent state, equation (2.27) shows that the newground state |0Q) is a coherent state. Because such a coherent state is formedafter the spontaneous breakdown of symmetry of the systems, it is referred to as aspontaneous coherent state. But when 4>0 = 0, the new ground state is the same asthe old state which is not a coherent state.

The same conclusion can be directly derived from the BCS theory. In the BCStheory, the wave function of the ground state of a superconductor is written as

$ o = n (u*+u*afc «i*:) ^ o = n ( U f c + v ^ t - k ) <t>ok k

where b^_k = aâ^k. This equation shows that the superconducting ground stateis a coherent state. Hence, we can conclude that the spontaneous coherent state insuperconductors is formed after the spontaneous breakdown of symmetry.

(2.28)


Recently, by reconstructing a quasiparticle-operator-free new formulation of theBogoliubov-Valatin transformation parameter dependence, Lin et al. demonstratedthat the BCS state is not only a coherent state of single-Cooper-pairs, but also thesqueezed state of double-Cooper-pairs, and reconfirmed the coherent feature of BCSsuperconductive states.

In the following, we employ the method used by Bogoliubov in the study ofsuperfluid liquid helium 4He to prove that the above state is indeed a Bose condensedstate. To do that, we rewrite (2.18) in the following form

^ ) = ^ E 9 p e ~ i p X ' 1r = -7=(bp + 1>±p)- (2-29)VV p V / £ P

Since the field <f> describes a Boson, such as a Cooper electron-pair in a supercon-ductor, Bose condensation can occurr in the system. Following traditional methodin quantum field theory, we consider the following transformation

bp = ,/No8(p) + 7 p , b+= i/No6(p) + 0P, (2.30)

where No is the number of Bosons in the system and

JO, i f p ^ O

Substituting (2.29) and (2.30) into (2.1), we can get

_ 4AiV02 2Noa ^ 4A No

4V e0 +êoePV

+ (^-^)v/]vo(7o+7o++A>+/?o+)

+ E ( ^ - £») W"-*+-w-..) <2-31>

+jy £ TP W^p+w-i.+ttit,+7,7-,+iitfp+vi%)

Because the condensed density NQ/V must be finite, the higher order terms—2

O(^/NQ~/V) and O(N0/V ) may be neglected. Next we perform the following canon-ical transformation

7P = u;Cp + vpctp, (2.32)

PP = updp + Vpdtp,


where vp and up are real and satisfy

Introducing another transformation

CP = y | K ^ ~ - vvl-v + uPP+ - vpP-p), (2.33)

vt = ~fi(up7p ~ VP1-P ~ UP@P + VPP-P)>

the following relations can be obtained

[Q, n] = gPCP + MPC+p, [rjp, U] = g'pvP + M'pvtp, (2.34)

where

(gp = Gp{ul + vl)+Fp2upvp> Mp = Fp(u2p +v2

p) + Gp2upvp

\g'p = Gp{ul + vl)+F;2upvp, Mp = F'p (u2p + v2

p) + G'p2upvp l ^

while

(GP = ep-^+6ep, Fp = - ~ + 6ep,

\ff-e aP+2? F>-° 2£> ( 2 - 3 6 )

where

,, = AiVp<p eospV-

We now discuss two cases.(A) Let Mp = 0, then it can be seen from (2.34) that 77+ is the creation operator

of elementary excitation and its energy is given by

g'p = yje% + 4e,4 - 2a. (2.37)

Using this, we can obtain the following from (2.35) and (2.36)

<"'H(1 +t) ™d « H ( - 1 + f ) - (2-38)From (2.34), we know that £+ is not a creation operator of the elementary excitation.Thus, another transformation must be made

BP = XPCP + »PCP+, |XPI2 - IMPI2 = I- (2-39)

We can then prove that

[BP,H] = EPBP, (2.40)


where

Now, inserting (2.32), (2.39), (2.40) and Mp = 0 into (2.31), and after some reor-ganizations, we have

n = U + E0 + 2 [EP(B+BP + JB+pB_p) + gp(r,+r,p + r,+pr,_p)] (2.41)p>0

where

Eo = - 2 ] [ » P | 2 = -J2(9'p-EP). (2.42)p>0 p>0

Both U and £0 are now independent of the creation and annihilation operatorsof the Bosons. U + Eo gives the energy of the ground state. iV0 can be determinedfrom the condition

8(U + EQ) _SN0 ~ U

which gives

f - 3 - l J. (,43,This is the condensed density of the ground state <£0- Prom (2.36), (2.37) and (2.40),we can get

g'p = ^Jel-a, Ep = y ^ - a. (2.44)

These correspond to the energy spectra of 77+ and B+, respectively, and they aresimilar to the energy spectra of the Cooper pair and phonon in the BCS theory.Substituting (2.44) into (2.38), we have

(B) In the case of Mp — 0, a similar approach can be used to get the energyspectrum corresponding to £/ as

EP = yje\ + a

while that corresponding to A+ = XpVp + PPV-P is

9P = \l£l + a>

(2.45)


where

oil 2e% + a \ 1 / 2el + a \

From quantum statistical physics we know that the occupation number of the levelwith the energy ep for a system in thermal equilibrium at temperature T (^ 0) isgiven by

N' = {b>»} = eÎKBT _ l (2-47)

where (• • •) denotes Gibbs average, defined as

SP[e-«/**T---]

Here SP denotes the trace in Gibbs statistics. At low temperature, or T -> 0 K,the majority of the Bosons or Cooper pairs in a superconductor condense to theground state with p = 0. Therefore,

(b+b0) « No,

where iVo is the total number of Bosons or Cooper pairs in the system and iV0 > 1,i.e.

{b+b) = 1 « (b+b0).

As can be seen from (2.29) and (2.30), the number of particles is extremely largewhen they lie in the condensed state, that is

^0 = <?Wo = -fLs (bo + b+). (2.48)V2e0V

Because 7o|<£o) = 0 and A>|</>o) = 0, 60 and 6j can be taken to be S/NQ. The averagevalue of 4>*(j> in the ground state then becomes

<<»o| </#o> = fa»o = r-^= • 4iV0 = ^ . (2.49)

Substituting (2.43) into (2.49), we can get

or

which is the ground state of the condensed phase or the superconducting phasethat we have known. Thus, the density of states, NQ/V, of the condensed phase or

(2.46)


the superconducting phase formed after the Bose condensation coincides with theaverage value of the Boson's (or Copper pair's) field in the ground state. We canthen conclude that the macroscopic quantum state or the superconducting groundstate formed after the spontaneous symmetry breakdown is indeed a Bose-Einsteincondensed state.

In the last few decades, Bose-Einstein condensation was observed in a seriesof remarkable experiments for weakly interacting atomic gases, such as vapors ofrabidium, sodium lithium, or hydrogen. Its formation and properties have beenextensively studied. These studies show that the Bose-Einstein condensation is anonlinear phenomenon, analogous to nonlinear optics, the state is coherent, and canbe described by the following nonlinear Schrodinger equation or the Gross-Pitaerskiiequation

where t' = t/h, x' = xy/2m/h.This equation was used to discuss realization of Bose-Einstein condensation in

the d+l dimensions (d = 1,2,3) by Bullough et al. The corresponding Hamillonianof a condensate system was given by Elyutin et al. as follows

n = ^ + v(x'M2-lx\<t>\i (2.5i)

where the nonlinearity parameter A is defined as A — -2Naai/al, with N being thenumber of particles trapped in the condensed state, a the ground state scatteringlength, ao and o; the transverse (y,z) and the longitudinal (a;) condensate sizes(without self-interaction), respectively. (Integrations over y and z have been carriedout in obtaining the above equation). A is positive for condensation with self-attraction (negative scattering length). The coherent regime was observed in Bose-Einstein condensation in lithium. The specific form of the trapping potential V(x')trapping potential depends on the details of the experimental setup. Work on Bose-Einstein condensation based on the above model Hamiltonian were carried out andare reported by Barenghi et al. (2001).

It is not surprising to see that (2.50) is exactly the same as (2.17), and (2.51) isthe same as (2.1). This further confirms the correctness of the above theory for Bose-Einstein condensation. As a matter of fact, immediately after the first experimentalobservation of this condensation phenomenon it was realized that coherent dynamicsof the condensated macroscopic wave function can lead to the formation of nonlinearsolitary waves. For example, self-localized bright, dark and vortex solitons, formedby increased (bright) or decreased (dark or vortex) probability density, respectively,were experimentally observed, particularly for the vortex soliton which has the sameform as the vortex lines found in type II-superconductors and superfluids. Theseexperimental results were in concordance with the results of the above theory. In the

(2.50)


following sections, we will study the soliton motions of quasiparticles in macroscopicquantum systems. We will see that the dynamic equations in MQS have such solitonsolutions.

From the above discussion we clearly understand the nature and characteristicsof an MQS such as a superconducting system. It would be interesting if a comparisonis made between the macroscopic quantum effect and the microscopic quantum effectwhich is well known. Here we give a summary of the main differences between them.

(1) Concerning the origins of these quantum effects, the microscopic quantumeffect is produced when microscopic particles which have the particle-wave dualityare confined in a finite space, while the macroscopic quantum effect is due to the col-lective motion of the microscopic particles in systems with nonlinear interaction. Itoccurs through second-order phase transition following the spontaneous breakdownof symmetry of the systems.

(2) From the point of view of their characteristics, the microscopic quantumeffect is characterized by quantization of physical quantities such as energy, momen-tum, angular momentum, etc. of the microscopic particles. On the other hand, themacroscopic quantum effect is represented by discontinuities in macroscopic quan-tities such as resistance, magnetic flux, vortex lines, voltage, etc.. The macroscopicquantum effects can be directly observed in experiments, while the microscopicquantum effects can only be inferred from other effects related to them.

(3) The macroscopic quantum state is a condensed and coherent state, but themicroscopic quantum effect typically occurs in bound states. Certain quantizationcondition is satisfied for a microscopic quantum effect and the particles involved canbe either Bosons or Fermions. But so far only Bosons or combination of Fermionsare found in macroscopic quantum effects.

(4) The microscopic quantum effect is a linear effect and motions of particlesare described by linear differential equations, such as the Schrodinger equation, theDirac equation, and the Klein-Gordon equations. On the other hand, the macro-scopic quantum effect is a nonlinear effect and the motions of the particles are de-scribed by nonlinear partial differential equations such as the nonlinear Schrodingerequation (2.17).

Thus, the fundamental nature and characteristics of the macroscopic effect aredifferent from those of the microscopic quantum effects. For these reasons, a differ-ent approach must be used to describe the macroscopic quatum effect. Also basedon these discussions, it is clear that a nonlinear quantum theory would be necessaryto describe these macroscopic quantum effects.

2.3 Motion of Superconducting Electrons

In this section, we discuss the motions of quasiparticles, such as superconductiveelectrons or Cooper pairs in superconductors and superfluid helium, in the MQS.The properties and motion of quasiparticles are not only important for understand-


ing the relevant quantum effects, but also provides the basis for establishing thenew nonlinear quantum mechanical theory.

Prom earlier discussions, we know that the superconductive state is a highlyordered coherent state or a Bose-Einstein condensed state formed after a sponta-neous breakdown of the symmetry of the system due to the nonlinear interactionin the system. A macroscopic quantum wave-function, (j), can be used to describethe superconducting electrons (Cooper pairs)

<P(r,t)^f(f,t)ct>oeie^, (2.52)

where

According to the Ginzberg-Landau theory of superconductivity, the free energydensity function of a superconducting system is given by

A = /n-^|V<A|2-a|<A|2 + AH4 (2.53)

in the absence of any external field, where /„ is the free energy of the normal state,a and A are interaction constants among the particles. On the other hand, if thesystem is subjected to an electromagnetic field specified by a vector potential A,the free energy of the system is

'• = >» - £ | ( v - 'i1) +\* ~ ^ i 2 + A ^ 4 + h*2' <2-54)where e* = 2e, H — V x A. The free energy of the system is

F. = Jf.<Px.

In terms of the convariant field, Fjt = djAi - diAi, (j, I = 1,2,3), the term F2/8TT

can be written as FjiFjl/4. Then from 5FS = 0, we get

-^-V2^-a<|!> +2A<£3 = 0 (2.55)

and

fi2 / ?>* -A 2

— V - ^~A <j>-a(f> + 2A</>3 = 0 (2.56)2m \ en J

in the absence and presence of an external field, respectively, and

Equations (2.55) - (2.57) are the well-known Ginzburg-Landau equations in a steadystate. Here (2.55) is the Ginzburg-Landau equation in the absence of external fields.

(2.57)


It is the same as (2.15) which was obtained from (2.1). Equation (2.56) can alsobe obtained from (2.2). Therefore, equations (2.1) - (2.2) are the Hamiltonianscorresponding to the free energy (2.53) - (2.54).

Prom (2.53) - (2.57), we see clearly that superconductors are nonlinear systems.The Ginzburg-Landau equations are the fundamental equations of the supercon-ductors and describe the motion of the superconductive electrons. However, theequations contain two unknown functions 0 and A which make them extremelydifficult to be solved. We first consider the case of no external field.

2.3.1 Motion of electrons in the absence of external fields

We consider only an one-dimensional pure superconductor. Let

* = f^ 2 ^ = ^ ' x' = wr (2-58)

where C(^) is the coherent length of the superconductor which depends on temper-ature. For a uniform pure superconductor,

C(T) = 0 . 9 4 0 , ^ — ^ ,

where Tc is the critical temperature and Co is the coherent length at T = 0. Then,equation (2.55) can be written as

- 0 + / - / 3 = O (2.59)

and its boundary conditions are

f(x' = 0) = 1, and f(x' -> ±oo) = 0. (2.60)

After integration, we get the following solutions:

/ = ± v 5 s e d i [ ^ i ) ] ' (2-61), , fa , \ x - x 0 ] . f a , V2ma

* = ±\Jjsech[-«TT\=±Vxsechl-ir-{x-Xo) •This is the well-known wave packet-type soliton solution. It represents a bright

soliton compatible to that in the Bose-Einstein condensate found by Perez-Garcia,et al.. If the signs of a and A in (2.55) are reversed, we then get a kink solitonsolution under the boundary conditions of f(x' = 0) = 0 and f(x' -> ±oo) = ±1,

0 = ±\[ixta-nh [ v ^ ( x ~Xo)] • (2-62)


The energy of the soliton (2.61) is given by

*„,. r w, _ r f * (f)2 - a^ - vl,«, - * £ . (2 3)We have assumed that the lattice constant TQ = 1. The above soliton energy canbe compared with the ground state energy of the superconducting state, -EgrOund =-c*2/4A. The difference is

Eso\ - Aground = ^77- ( V® + ~-J== ) > 0. (2.64)4A \ 3v2m/

This indicates clearly that the soliton is not at the ground state, but at an excitedstate of the system. Therefore, the soliton is a quasiparticle.

Prom the above discussion, we can see that in the absence of external fields, thesuperconductive electrons move in the form of solitons in a uniform system. Thesesolitons are formed by nonlinear interaction among the supercondivite electronswhich suppresses the dispersive behavior of the electrons. A soliton can carry acertain amount of energy while moving in superconductors. It can be demonstratedthat these soliton states are very stable.

2.3.2 Motion of electrons in the presence of an electromagneticfield

We now consider the motion of superconductive electrons in the presence of anelectromagnetic field. Assuming the London gauge V • A — 0, substitution of (2.52)into (2.56) and (2.57) yields

j=elA (hV6 _ £!/) f (2.65)

and

V 2 / - ( W - £ ^ ) 2 / - f?(<*-2A^/2) / = 0 (2.66)

For bulk superconductors, J is a constant (permanent current) for a certain valueof A, and it thus can be taken as a parameter. Letting

2 m2J2 2ma 1= ft2(e*)2^' ^ " " C 2 " '

equations (2.65) and (2.66) can be written as

(hVe-e-A)=-^-, (2.67)V c / e*W2


V2f=-bf3+bf+jW. (2.68)

Equation (2.68) evidently has nonlinear characteristics. When J = 0, equation(2.68) is reduced to

V2/ = -bf + bf,

which is the same as (2.55).In one-dimensional case, equation (2.68) can be written as

g = -|^(/), (2.69)

where

is the effective potential which is schematically shown in Fig. 2.4. Comparing this

Fig. 2.4 The effective potential energy defined in Eq. (2.69).

case with that in the absence of external fields, we found that the equations have thesame form and the electromagnetic field A changes only the effective potential of thesuperconductive electron. When A = 0, the effective potential well is characterizedby double wells. In the presence of an electromagnetic field, there are still twominima in the effective potential, corresponding to the two ground states of thesuperconductor. This shows that the spontaneous breakdown of symmetry stilloccurs in the superconductor, and the superconductive electrons move also in theform of solitons.

To obtain the soliton solution, we integrate (2.69) which gives

* = / ' , df (2-70)4 y/2[E - C/eff(/)]


where E is a constant of integration which is equivalent to energy, the lower limitof the integral, / i , is determined by the value of / at x = 0, i.e.

E = C/eff(/o) = t W / l ) .

Introducing the following dimensionless quantities

,2 „ b -r 4J2m\/» = «, E=-e, 2d=j^^,

equation (2.70) can be written as the following upon performing the transformationu —> —u,

-V2bx= [ . dU (2.71)./ui Vu3 - 2u2 - 3eu - 2d2

It can be seen from Fig. 2.4 that the denominater of the integrand in (2.71) ap-poaches to zero linearly when u — ui = ff, but appoaches to zero quadraticallywhen u = u2 = /o • Therefore, it can be rewritten as

\]u2 - 2u2 - leu - 2d2 = y/(u - Ul)(u - u0)2. (2.72)

Thus, we have

2u0 + ui = 2, ul + 2uoui = -2e, and uml = 2d2. (2.73)

Substituting (2.72a) into (2.71), upon completing the integral in (2.71), we can get

u(x) = f2(x) =u0- gsech2 f J-gb x j = ux + fftanh2 f J-gb x j (2.74)

where g = UQ — u\ and satisfies

(2 + <7)2(l-fl) = 27J2. (2.75)

It can be seen in (2.74) that for a large piece of sample, u\ is very small and maybe neglected, the solution u is very close to UQ. We then get from (2.74)

f{x) = fot^h(sj^gbx\. (2.76)

Substituting the above into (2.67), the electromagnetic field A in the superconduc-tors can be obtained

Jmc 1 fie Jmc 2 ( / l , | ^TIQA = -¥M-p-^d = ¥Mf!coth Iv256x) ~it™


For a large piece of superconductor, the phase change is very small. Using H =V x A, the magnetic field can be determined and is given by

s"v * x=£MI H {& *)+coth (VP *)] • pj7)

Equations (2.76) and (2.77) are analytical solutions of the Ginzburg-Landau equa-tions (2.57) and (2.58) in the one-dimensional case. The solutions are shown inFig. 2.5. Equations (2.76), or (2.74) shows that the superconductive electron in thepresence of an electromagnetic field is still a soliton. However, its amplitude, phaseand shape are changed compared with those in a uniform superconductor and inthe absence of external fields, (2.61). The soliton here is obviously influenced bythe electromagnetic field, as reflected by the change in the solitary wave form.

Fig. 2.5 The functions f(x) and H defined in (2.76) and (2.77), respectively.

This is why a permanent superconducting current can be established by themotion of electrons along certain direction in the a superconductor, because solitonshave the ability to maintain their shape and velocity while in motion.

It is also clear from Fig. 2.5 that H(x) is larger where <j>(x) is small, and viceversa. When x -> 0, H(x) reaches a maximum while <f> approaches to zero. On theother hand, when x —> oo, <j> becomes very large while H(x) approaches to zero.These are exactly the well-known behaviors of vortex lines - magnetic flux lines intype-II superconductors.

Recently, vertex solitons in the Bose-Einstein condensates were observed byCaradoc-Daries et al., Matthews et al. and Madison et ai. Tonomure observedexpermentally magnetic vorties in superconductors. These vortex lines in the type-II-superconductors are quantized. The macroscopic quantum effects are well de-scribed by the nonlinear theory discussed above, demonstrating the correctness ofthe theory.

We now proceed to determine the energy of the soliton given by (2.76) and


(2.68). From earlier discussion, the energy of the soliton is given by

which depends on the interaction between superconductive electrons and externalelectromagnetic field.

Motion of superconductive electrons in the three-dimensional case can be studiedsimilarly.

From the above discussion, we understand that for bulk superconductor, thesuperconductive electrons behave as solitons, regardless of the presence of externalfields. Therefore, the superconductive electrons are a special type of solitons. Theymove in the form of solitary wave in the superconducting state. In the presence ofexternal electromagnetic fields, a permanent superconductive current is established,vortex lines or magnetic flux lines also occur in type-II superconductors.

2.4 Analysis of Macroscopic Quantum Effects in InhomogeneousSuperconductive Systems

In this section, we discuss macroscopic quantum effects in the superconductor-normal conductor junction and superconductive junctions, based on the nonlineartheory, the corresponding Hamiltonian, and the Ginzburg-Landau dynamic equa-tions.

2.4.1 Proximity effect

Proximity effect refers to the phenomenon that a normal conductor (N) in contactwith a superconductor (S) can be superconductive. This is obviously the result oflong-range coherent property of superconductive electrons. It can be regarded as thepenetration of electron pairs from the superconductor into the normal conductor,or a result of diffraction and transmission of superconductive electron wave.

In this phenomenon, superconductive electrons exist in the normal conductor,but their amplitudes are much small compared to that in the superconductive region,and the nonlinear term A|</>|2</> in the Ginzburg-Landau equations, (2.56)-(2.57), canthus be neglected. Because of these, the Ginzburg-Landau equations in the normaland superconductive regions have different forms. On the S side of the S-N junction,the Ginzburg-Landau equation is

— (v-l-Â)<t>-a<l> +2X^=0, (2.79)2m \ ch J

(2.78)


while that on the N side of the junction is

£(V-^)*-a* = °- ^The expression for J remains the same on both sides.

J=^(4>*y<f>-^4>*)-—\<t>\2^. (2.81)Zmi me

In the S region, we have obtained the solutions of (2.79) in the previous section,and it is given by (2.74) or (2.76) and (2.77). In the N region, we assume thatthe solution is of the form of (2.52). Using the same analysis, equations (2.80) and(2.81) can be written as

J=^° fw0--/)/2> (2-82)m \ c J

and

V2/-67-yJ = 0, (2.83)

where

, _ 2ma' 1 2 _ m2./2

Here J is still treated as a parameter. If J = 0 (or B = 0), equation (2.83) has aplane wave solution. However, B ^ 0 in general. In this case, the following solutioncan be obtained by integrating (2.83)

u = )-\J{e')2 -Ad2sm(2y/b'x) + 1 (2.84)

where

f2_u E>-b'> 2d2 - 4 J ^ m A

f -u, E - -e, 2d - ( c . ) 2 a 2 .

The constant of integration has been set to zero. The solution is then given by

Graph of <f> vs. x in both the S and the N regions, as shown in Fig. 2.6, coincidewith that obtained by Blackbunu. The solution given in (2.85) is the analyticalsolution in this case. On the other hand, Blackbunu's result was obtained by ex-pressing the solution in terms of elliptic integrals and then integrating numerically.From this, we see that the proximity effect is caused by diffraction or transmissionof the superconductive electrons.

(2.85)


Fig. 2.6 Proximity effect in S-N junctions.

2.4.2 Josephson current in S-I-S and S-N-S junctions

A superconductor - normal conductor - superconductor junction (S-N-S) or a super-conductor - insulator - superconductor junction (S-I-S) consists of a normal conduc-tor or an insulator sandwhiched between two superconductors and is schematicallyshown in Fig. 2.7(a). The thickness of the normal conductor or the insulator layeris assumed to be L and we choose the x coordinate such that the the normal con-ductor or the insulator layer is located at -L/2 < x < L/2. The S-I-S junctionswas studied by Jacobson et al. based on the Schrodinger equation. We will treatthis problem using a different approach here.

Fig. 2.7 Supereonductive Junctions for S-N-S and S-I-S.

The electrons in the superconducting regions {\x\ > L/2) are described by theGinzburg-Landau equation, (2.79). Its solution was given earlier in (2.76). Aftereliminating ixi from (2.73), we have

J=±e*auO]/^(l-uo).

Setting dJ/duo = 0, we get the maximum current

_ e*a I a

This is the critical current of a perfect superconductor, corresponding to the three-fold degenerate solution of (2.72a), i.e., u\ = un-

From (2.82), we have:

y mJc hc^.n


Using the London guage, V • A = 0, we can get

dx2 ~ e*<fohdx \f2) '

Integrating the above equation twice, we get

" = ?&/(£-£)* (2"86)

where f2 = u, and /£, = u0. We have used the following de Gennes boundaryconditions in obtaining Eq. (2.86).

^ =0, ^ =0, flM-xxOôo. (2.87)d x |s|-»oo d x |*|-+oo

If we substitute (2.71) - (2.74) into (2.86), the phase shift of the wave function froman arbitrary point x to infinity can be obtained directly from the above integral.

AOdx -> oo) = - t a n " 1 . —— h t an" 1 , —^—. (2.88)y UQ — u\ y u — u\

For the S-N-S or S-I-S junction, the superconducting regions are located at \x\ > L/2and the phase shift in the S region is thus

A0S = 2A0L (^ -+ oo j Pa 2 tan"1 J Ul , (2.89)V2 / y us - m

where the factor of 2 is due to the contributions from superconductors at both sidesof the N or I layer. Therefore,

^^ - ^£tm=^ ctg(A's/2)- (2-90)According to the results in (2.80) - (2.85), the solution in the I or N region of

the S-N-S or S-I-S junction may be expressed as

u = glcos2{Vb'x) + ti (2.91)

where

^ = tti-ui, h'=u[, u[u'0 = (g' + h')h' = - ^ ± . (2.92)

Substituting (2.91) into (2.87), the phase shift over the I or N region of the junctioncan be obtained and is given by

A6W = - 2 tan x - 7 - 1 — tan ^ — - + (2.93)I J V 8mA \ 2 J \ 2e*h'u0


where mJL/2e*h'uo is an additional term introduced to satisfy the boundary con-ditions (2.87), and may be neglected in the case being studied. Then

/8Â j tan(A0;y/2)V o? 2e* t an (^L /2 ) - ( 2 ' 9 4 )

Near the critical temperature (T <TC), the current passing through a weakly linkedsuperconductive junction is very small (j <C 1), we then have

u[ = ^ ^ = 2 A 2 , and g' = 1. (2.95)

Since rjf2 and df2/dx are continuous at the boundary x = L/2, we have

VsUs\x=L/2 = VNUN\x=L/2, - J - x=L/2 = ~J— • (2-96)a a ; "Z x=Z,/2

These give

2v/67Asm(2A0Ar) = £i[l - cos(2A0s)] sin(V/67L), (2.97)

cos(v/67L) sin(2A<95) = esm{2kON) + sin(2A^s + A0N), (2.98)

where ei = T)N/T]S. From (2.97) and (2.98), we can get

sin(A0s + A6N) = 2y/2mXJVb'sm(Vb'L).e*a

Thus

j = j m a x sin(A0s + &6N) = j m a x sin(A0), (2.99)

where

jmax = -J^L=.—*—-, A0 = A6S + A6N. (2.100)2y2mAo' sin(v6 i )

Equation (2.99) is the well-known Josephson current. From Section 2.1 we knowthat the Josephson effect is a macroscopic quantum effect. We have seen now thatthis effect can be explained based on the nonlinear theory.

From (2.100), we can see that the Josephson critical current is inversely pro-portational to sin(v^'i) which means that the current increases suddenly when-ever \ftiL approaches to TITT, suggesting some resonant phenomena in the sys-tem. This has not been observed before. Moreover, Jmax is proportional toe*as/2\/2mA6' = (e*h/Am\^X)(as/aN) which is related to (T - Tc)

2. Finally, itis worthwhile to mention that no explicit assumption was made in the above onwhether the junction is a potential well (a < 0) or a potential barrier (a > 0). Theresults are thus valid and the Josephson effect (2.99) occurs for both potential welland potential barrier.


2.4.3 Josephson effect in SNIS junction

A superconductor-normal conductor-insulator-superconductor junction (SNIS) isshown schematically in Fig. 2.7(b). It can be regarded as a multilayer junctionconsists of the S-N-S and S-I-S junctions. If appropriate thicknesses for the N andI layers are used (approximately 20 A- 30 A), the Josephson effect similar to thatdiscussed above can occur in the SNIS junction.

Since the derivations are similar to that in the previous sections, we will skipmuch of the details and give the results in the following. The Josephson current inthe SNIS junction is given by

j = Jraaxsin(A0) (2.101)

where

A0 = A0,i + A9N + A0j + A0s2)

and

j 1 f £l s i n h ( ^ L ) 1 xmaX V&N 1 2[cosh(v

/^L) - cos(2A0Ar)] J1

y/\l + cos(2A0Ar)][l + cos(2A0/)] - A/[1 - COS(2A0JV)][1 - cos(2A07)] ~

1 f glv/l-cos2(2A6'^) sinh(y/b^L) \

y/b^ \ [cosh(v/6Wi) - cos(2A0w)]2 - 1 + cos2(2A0N) J X

1

V[l - cos(2A6'Jv)][l - cos(2A0/)] + y/[l - cos(2A6>jv)][l + cos(2Afl/)]"

The phase shifts A0si, A9N, Adi and A#s2 satisfy the following relations

4^/bÂsin(2A6N) = £ l 1 - cos(2A6>sl)sinh (y/b^LJ j ,

4:^1 sin(2A6>/) =Ei 1 - cos(2A0s2) sin ( ^VjL J1 ,

cosh ( Jt^Lj sin(2A0si) = sin(2A0jv) + sin(2A0sl + 2A0N),

cos (\JVIL\ sin(2A6>s2) = sin(2A0/) -I- sin(2A0s2 + 2A6i).

It can be shown that the temperature dependence of Jmax is Jmax oc {Tc — To)2,which is quite similar to the results obtained by Blackburm et al. for the SNIS


junction and those by Romagnan et al. using the Pb-PbO-Sn-Pb junction. Here,we obtained the same results using a complete different approach.

2.5 Josephson Effect and Transmission of Vortex Lines Along theSuperconductive Junctions

We have learned that in a homogeneous bulk superconductor, the phase 6(f, t) ofthe electron wave function cf> = f(r, £)el9(r>t) is constant, independent of positionand time. However, in an inhomogeneous superconductor such as a superconductivejunction discussed above, 6 becomes dependent of f and t. In the previous section,we discussed the Josephson effects in the S-N-S or S-I-S, and SNIS junctions startingfrom the Hamiltonican and the Ginzburg-Landau equations satisfied by <f>(r,t), andshowed that the Josephson current, whether dc or ac, is a function of the phasechange, ip = A6 = #i — #2- The dependence of the Josephson current on tp is clearlyseen in (2.99) or (2.101). This clearly indicates that the Josephson current is causedby the phase change of the superconductive electrons. Josephson himself derivedthe equations satisfied by the phase difference <p, known as the Josephson relations,through his studies on both the dc and ac Josephson effects.

The Josephson relations for the Josephson effects in superconductor junctionscan be summarized as the following,

(dip

Js = Jm sin <p, h— = 2e V,n»p 2ed'- X 2ed>~ (2-102)

where d' is the thickness of the junction. Because the voltage V and magnetic

field H are not determined, equation (2.102) is not a set of complete equations.

Generally, these equations are solved simultaneously with the Maxwell equation

V x H = (4TT/C) J. Assuming that the magnetic field is applied in the xy plane, i.e.,

H — (Hx,Hy,0), the above Maxwell equation becomes

^Hy(x,y,t) - ^Hx(x,y,t) = -J(x,y,t). (2.103)

In this case, the total current in the junction is given by

J = Js(x,y,t) + Jn(x,y,t) + Ji(x,y,t) + JQ.

In the above equation, Jn is the normal current density in the junction {Jn =V/R(V) if the resistance in the junction is R(V) and a voltage V is applied attwo ends of the junction), •/<* is called a displacement current and it is given byJd = CdV(t)/dt, where C is the capacity of the junction, and Jo is a constantcurrent density. Solving the equations in (2.102) and (2.103) simultaneously, we


can get

where

/ c2 1 / c2h 4JO7re*u° ~ \ A run To = ~^p;, Aj = y , IQ = —r-=—.

V 47rCd' i?(7 V 4irde* he2

Equation (2.104) is the equation satisfied by the phase difference. It is a Sine-Gordon equation (SGE) with a dissipative term. FYom (2.102), we see that thephase difference <p depends on the external magnetic field H, thus the magneticflux in the junction

$'= f Hds= IA-d= — <fipdl

can be specified in terms of <p. Equation (2.104) represents transmission of su-perconductive vortex lines. It is a nonlinear equation. Therefore, the Josephsoneffect and the related transmission of the vortex line, or magnetic flux, along thejunctions are nonlinear problems. The Sine-Gordon equation given above has beenextensively studied by many scientists including Kivshar and Malomed. We willsolve it here using different approaches.

Assuming that the resistance R in the junction is very high, so that Jn -t 0, orequivivalently 70 —» 0, setting also la = 0, equation (2.104) reduces to

^'kw^k™*- (2105)

Define

v x T Vot

A = —, 1 = — .Aj Xj

Then in one-dimension, the above equation becomes&<p d2<p .

which is the ID Sine-Gordon equation. If we further assume that

 <5 = ± 1 -

Choosing vl = 1, we have

{«'> _ ^ _ =

A kink soliton solution can be obtained as follows

±7*'= In [tan ( | )] ,

^ ')=4tan-1(e±^')>

or

<p(X',T) = 4 tan"1 p7(x'-xi-,T')j (2 106)

Prom the Josephson relations, the electric potential difference across the junctioncan be written as

where tp0 = nhc/e = 2x 10~7 Gauss/cm"2 is a quantum fluxon, c is the speed oflight. A similar expression can be derived for the magnetic field

We can then determine the magnetic flux through a junction with a length of L anda cross section of 1 cm2. The result is

4?' = / Hx{x,t)dx = B'O Hx(X',T')dX' = 6<p0.J—oo J—oo

Therefore, the kink (S = +1) carries a single quantum of magnetic flux in theextended Josephson junction. Such an excitation is often called a fluxon, and theSine-Gordon equation or (2.104) is often referred to as transmission equation ofquantum flux or fluxon. The excitation corresponding to S = —1 is called anantifluxon. Fluxon is an extremely stable formation. However, it can be easilycontrolled with the help of external effects. It may be used as a basic unit ofinformation.

This result shows clearly that magnetic flux in superconductors is quantized andthis is a macroscopic quantum effect as mentioned in Section 2.1. The transmission


of the quantum magnetic flux through the superconductive junctions is describedby the above nonlinear dynamic equation (2.104) or (2.105).

The energy of the soliton can be determined and it is given by

p _ 8 m 2

E-~F'where

The boundary conditions must be considered for real superconductors. Variousboundary conditions have been considered and studied. For example, we can assumethe following boundary conditions for a ID superconductor

<Px(0,t)=<px(L,t) = 0.

Lamb obtained the following soliton solution for the SG equation (2.105)

(f(x, t) = 4 tan"1 \h{x)g(x)} (2.107)

where h and g are the general Jacobian elliptical functions and satisfy the followingequations

[hx(x)]2=a'h4 + (l + b")h2-c',

[gx(x)}2 = c'g4+b"g2-a',

with a', 6", and c' being arbitrary constants. Coustabile et al. also gave the plasmaoscillation, breathing oscillation and vortex line oscillation solutions for the SGeqution under certain boundary conditions. All of these can be regarded as thesoliton solution under the given conditions.

Solutions of (2.105) in two and three-dimensional cases can also be found. Inthe two-dimensional case, the solution is given by

where

/ = 1 + a(l , 2)eyi+y2 + a(2,3)ey2+y3 + a(3, l)eV3+yi,

g = eyi + eV2 + a( l , 2)a(2,3)a(3, l)e»1+>«+««,

and

yi=PiX + qiY-niT-yf, p? + g?-n? = l, (i = 1,2,3),

(2.108)


fl(ffl.fa-ft)a + ( f t - t t ) a - ( n « - n j ) a (1<i<i<3)

In addition, pi, qi and fli satisfy

Pi <?i ^ idet P2 92 ^2 = 0.

P3 93 ^ 3

In the three-dimensional case, the solution is given by

where X, Y, and T are similarly defined as in the 2D case given above, and Z =z/Xj. The functions / and g are defined as

g = eyi + eV2 + eV3 + dX2dY3dZ3eVi+y2+y3

f = dX2eyi+y2 + dY3e

y2+y3 + dZ3eyi+V3 + 1

with

yt = anX + ai2Y + ai3Z + b{T + d, (i = X, Y, Z),

4 + «?2 + <4 - &i = 1,

d E U ( « * - « * ) ' - f t - » , ) » , ( 1 < j < 3 ) .

Here j/3 is a linear combination of y\ and ?/2, »-e.

2/3 ="2/i +/#Z/2-

We now discuss the SG equation with a dissipative term jod^p/dt. First wemake the following substitutions to simplify the equation

i vot _ t -YQ\2J , _ 2A = —, T — — = —, a= , B — ioAj.Xj Xj uj v0

In terms of these new parameters, the ID SG equation (2.104) can be rewritten as

The analytical solution of (2.110) is not easily found. Now let

a =~TY-, r\=-= —, q' = R , , tp = TT + y»'. (2.111)

(2.110)

(2.109)


Equation (2.110) then becomes

This equation is the same as that of a pendulum being driven by a constant externalmoment and a frictional force which is proportional to the angular displacement.The solution of the latter is well known, generally there exists an stable solitonsolution. Let Y = dtp' /dr), equation (2.112) can be written as

^ r + q'Y + sin </>' - B' = 0. (2.113)dr)

For 0 < B' < 1, we can let B' = siny>0> (0 < ifo < TT/2) and <p' = —ir — fo + <Pi,then, equation (2.113) becomes

Y-— = -q'Y + sin (po + sinfip! - <p0). (2.114)d<pi

Expand Y as a power series of <p\,

n

Inserting the above into (2.114), and comparing coefficients of terms of the samepower of ifi on both sides, we get

q' fq~^ci = - - ± y — +cosv30,

C2 = 7 T 3 ^ - 2 - ' (2'115)

1 / „ o COS<A)\

1 / siniôACA = -5c2c3 —— ,

q' + 5ci V 24 /and so on. Substituting these cn's into Y = dip'/dr) — YLn

cn(Pi> t n e solution of(px may be found by integrating rj = J dtpi/ ^ c nV"- In general, this equation hassoliton solution or elliptical wave solution. For example, when

difl o q-T- = Cxipi + C2<Pi + C3ipl

dT)

it can be found that2 / IA-B . _! lA-<pi\

(2.112)


where F{k, ip{) is the first Legendre elliptical integral, and A, B and C are constants.The inverse function tpi of F(k, cpi) is the Jacobian amptitude tpi = &mF. Thus,

. _, IA — u>i I A — CSm V A^B = a mVT^B*'

or

lA-tpx IA-C

V^B=Sn]lA-^BT1

where snF is the Jacobian sine function. Introducing the symbol cscF = 1/snF,the solution can be written as

/ /TIc\2

tpi=A-(A-B)\c8cyJ-^—^rlj . (2.116)

This is a elliptic function. It can be shown that the corresponding solution at|TJ| -> oo is a solitary wave.

It can be seen from the above discussion that the quantum magnetic flux lines(vortex lines) move along a superconductive junction in the form of solitons. Thetransmission velocity v0 can be obtained from h = ai>o/\/l - v% and cn in (2.115)and it is given by

1Vo~ Vi + [a/M^F

That is, the transmission velocity of the vortex lines depdends on the current /0

injected and the characteristic decaying constant a of the Josephson junction. Whena is finite, the greater the injection current 70 is, the faster the transmission velocitywill be; and when IQ is finite, the greater the a is, the smaller the VQ will be, whichare realistic.

2.6 Motion of Electrons in Non-Equilibrium Superconductive Sys-tems

The motions of superconductive electrons in equilibrium states, such as those dis-cussed in Sections 2.3 - 2.5, are described by the time-independent Ginzburg-Landau equation. In such a case, the superconductive electrons move as solitons.However, what are the motions of superconductive electrons in non-equilibriumstates? Naturally, the superconductive electrons in non-equilibrium states shouldbe described by the time-dependent Ginzburg-Landau equation. Unfortunately,there are many different forms of the time-dependent Ginzburg-Landau equation,under different conditions. The one given in the following is commonly used when


an electromagnetic field A is involved,

i \hjt - 2ie/i(r)] <(> = ~ (hS7 - ^ / ) <f> + a<f>- \\cf>\2<t>, (2.117)

J=^|-^-VM(r) +^(0*V0-^V0*)-^/H2, (2.118)

where

cot \ c at I c

and a is the conductivity in the normal state, 7' is an arbitrary constant, and [/, isthe chemical potential.

In certain situations, the following forms of the time-dependent Ginzburg-Landau equation are also used

ih%—lk(J'-jl*)'*+a*-w*. (»•»»>or

'(s-T)*-f(-^*)*+?(v-^)'* <2-12°>Equations (2.117)-(2.120) are nonlinear partial differential equations having soli-

ton solutions. However, these solutions are very difficult to find, and no analyticsolutions have been obtained before. An approximate solution was obtained byKusayanage et al. by neglecting the <f>3 term in (2.117) and (2.118), in the case

of A = {0,Hx,0), fJ, = -KEx, H = (0,0,H) and E = (£,0,0), where H isthe magnetic field, while E is the electric field. We will solve the time-dependentGinzburg-Landau equation in the case of weak fields in the following.

The time-dependent Ginzburg-Landau equations can be written in the followingform when A is very small,

i f i f + ^rv2* + >2*=(F-2e")*- <2'1 2 I>where a and F are material dependent parameters, A is the nonlinear coefficient,m is the mass of the superconductive electron. Equation (2.121) is a nonlinearSchrodinger equation in a potential field a/Y — 2e/x. It was used by Cai, Bhat-tacharjee et al., and Davydov in their studies of superconductivity. However, thisequation is also difficult to solve. Here we only present its solution in the one-dimensional case.


For convenience, let

n a

Then (2.121) becomes

^ + 0 + r^=[?-2e^'^- <2-1 2 2)If we let a/T — 2e/x = 0, then (2.122) is the usual nonlinear Schrodinger equationwhose solution is of the form

02 = /o(s: ' , tV'o (*'1 0, (2.123)* , / /\ /r(«? - 2vcve) , [ /A(i;2 - 2ucue) , , ,sl/o(x', 0 = y 2A sech ^ 4 r (^ - "e* ) .

where

eo(x',t') = ±ve(x'-vctl).

In the case of a/T — 2e/x ^ 0, we let /i = -KEx', where AT is a constant, andassume that the solution is of the form

^ = / ' (* ' . tV** ' " 0 - (2-124)

Substituting (2.124) into (2.122), we get

Now let

f'{x',t') = f(O, Z = x'-u(t'), u(t') = -2EKe(t')2+vt' + d (2.127)

where u(t') describes the accelerated motion of f'(x', t'). The boundary conditionat £ -» oo requires /(£) to approach zero rapidly. Equation (2.126) can be writtenas

where

duU=dl>-

(2.125)

(2.126)

(2.128)


If 288/d^ - i i ^ O , equation (2.128) may be written as

,2 . 9(t') o r de__g(n uf -(dd/dO-u/2 ° r dx>~ P + 2 " ( 2-1 2 9 )

Integration of (2.129) yields

e(x',t') = 9{?) fi Y + \x'+ h{t>) (2'130)

where h(t') is an undetermined constant of integration. Prom (2.130) we can get

Substituting (2.130) and (2.131) into (2.125), we have

IH(2 K & I '+?K»'U ( f ' ) +7 +

JO J J a;'=oJ i /Since d2f/dx'2 = cPf/dt;2, it is a function of £ only. In order for the right-handside of (2.132) to be also a functin of £ only, it is necessary that

g(t') =90 = const.

(2KEex' + ^) + Ha/ + h(t') + ^ + ^\x.=0 = V(0- (2-133)

Next, we assume that Vo(£) = V(£) — 0 where j3 is real and arbitrary. Then

2KEex> + ^ = V0(0 - \x< + [/? - ^ | - h(t') - £ ] . (2.134)

Clearly in the case being discussed, VQ(£) — 0, and the function in the brackets in(2.134) is a function oft'. Substituting (2.133) and (2.134) into (2.132), we can get

This shows that / = /(£) is the solution of (2.135) when 0 and g are constants.For large |£|, we may assume that | / | < /3'/l£|1+Ai where A is a small constant.To ensure that / and d?f/d£2 approach zero when |£| -> 00, only the solutioncorresponding to g0 = 0 in (2.135) can be kept, and it can be shown that thissoliton solution is stable in such a case. Therefore, we choose g0 = 0 and obtain thefollowing from (2.129)

£4

(2.132)

(2.135)

(2.136)


Thus, we obtain from (2.134)

2KEex' + £ = -If i i ' + 0 - &(*') - j«2, (2.137)

MO = (p - £ - i ^ ) t, _ 4 (£ê )2 ( O3 + evKE{t'f. (2.138)

Substituting (2.138) into (2.130) and (2.131), we obtain

6 = (-2eKE? + | v ) z'+f/3 - 2 - ^ 2 ) t'~(EKe)2(t')3+evKE(t')2. (2.139)

Finally, substituting the above into (2.135), we can get

^ - / ? / + f / 3 = 0 . (2.140)

When P > 0, the solution of (2.140) is of the form

f = JWsech(^/0O- (2-141)V A

Thus

/2/JT , I" /^ / /2mr 2eKEt2-wi-dV< = y—sech >//3ly-^-a;+ ^ 1

{.\(-2eKEt v\ /2mTxav\'[{—^+2J\lirx

This is also a soliton solution, but its shape, amplitude, and velocity have changedrelatively to those of (2.123). It can be shown that (2.142) indeed satisfies (2.122).Thus, (2.122) has a soliton solution. It can also be shown that this solition solutionis stable.

For the solution (2.142), we may define a generalized time-dependent wave num-ber,

k = 1& = I ~ 2eK^'' (2'143)and a frequency

u = ~ = 2eKEx' -(fi-%- \v2) + A{EKef{t')2 - 2eKEvt'at' \ F 4 /

= 2eKEx' - P - ^ + k2. (2.144)

(2.142)


The usual Hamilton equations for the supercondactive electron (soliton) in themacroscopic quantum systems (MQS) is still valid here and can be written as

I-£|.=-***• <2-"5»

This means that the frequency u still has the meaning of the Hamiltonian in thecase of nonlinear waves. Therefore,

dw _ du dk duj dx' _dP= ~dk x, dP+ M k~dF=° ( 7)

which is the same as that in the usual stationary linear medium.These relations show that the superconductive electrons move as if they were

classical particles moving with a constant acceleration in the invariant electric fieldand the acceleration is given by —4eEK. If v > 0 the soliton initially travels towardthe overdensed region, it then suffers a deceleration and its velocity changes sign.The soliton is then reflected and accelerated toward the underdensed region. Thepenetration distance into the overdense region depends on the initial velocity v.

From the above discussion, we see that the superconductive electrons behavelike soli tons in non-equilibrium state of superconductor. Therefore, we can concludethat the superconductive electron is enssentially a soliton in both equibilibrium andnonequilibrium systems.

Before ending this section, it should be pointed out that there are many formsof time-dependent Ginzburg-Landau equations which have been used and can befound in literatures. For example,

*& + 2a<^« + M2^ + 7MV = *<ty + *e|0|20 + iPtxx + *Htf|40,

^ = [i-(i + tf)M>]* + d + ^ ) 0 ,

to name only a few. These equations are essentially generalization of (2.121) inthe case of dissipation. Various studies showed that these equations have solitonsolutions. Thus superconductive electron also behaves like a soliton in media withfriction which damps the soliton motion. Even though the form, amplitude, and

(2.146)


velocity of the soliton are altered, superconductive electrons being solitons remainsa fact.

2.7 Motion of Helium Atoms in Quantum Superfluid

As mentioned in Section 2.1, liquid helium exhibits macroscopic quantum effectat temperature below 2.17 K. The helium atoms form a quantum liquid withoutviscosity, or superfluid. Studies show that this is a result of Bose-Einstein con-densation of He atoms below the critical temperature. It is very similar to thesuperconducting state, and thus also described by a macroscopic wave function <fisimilar to (2.52). Here (j> is also called an order parameter of the superfluid liquidhelium or an effective wave function of helium atoms.

It is known that the effective wave function of helium atoms, cj>, satisfies theGross-Pitaerskii (GP) equation which was derived by Gross and Pitaerskii in 1950,

ttf£ = - ^ V 2 0 + AMV-A*'0 (2.148)

where A and /x' are constants. This equation is similar to the Ginzburg-Landauequation (2.122) for superconducting electrons. This is understandable because thesuperfluid, similar to a superconducting system, is also a nonlinear system. Equa-tion (2.148) was extensively used by Ventura et al. in their studies of superfluidity.A similar equation was derived by Dewitt in 1966. According to Dewitt, A in (2.148)should be a negative value.

The corresponding Lagrangian function density of the system can be obtainedand is given by

C = \ {*!* - * * * ) " 'V*'2 " + m' (2'149)In one dimension, if we let t' — t/h and x' = y/(2m/h2) x, then (2.148) is similarto (2.122). Thus its solution can be obtained following the same procedure as thatused in Section 2.6 and the result is

4> = Jj£ sech [yfi[x' - ve(t' - t'o)}} exp {i [vex' - (/? + v2e - //) t']} (2.150)

where /? is an arbitary constant.Therefore, when the atomic helium system undergoes a second-order phase tran-

sition and changes from the normal He-I state to the superfluid He-II state at 2.17K, the helium atoms behave as a soliton due to the spontaneous Bose condensationas a result of nonlinear interaction in the system. The nonlinear interaction sup-presses the dispersion effect of the helium atoms. Because solitons can preserve theirenergy, momentum, wave form and other properties of quasi-particles throughout


their motion, the superfluidity occurs naturally when the liquid helium moves assoli tons.

As a matter of fact, we can prove that the superfluid helium atom (soliton) moveswith an uniform speed and can determine its speed from the solution (2.150) andthe Hamilton equations (2.143) - (2.147). First, from (2.150) we can find that thewave number of the soliton is k = d9/dx = ve and its frequency is u = —dd/dt =/? -f v2 — // = /3 - p! + k2. Then the acceleration of the helium-soli ton is

dk dudt dx k

That is, the speed of the helium-soliton is a constant, ve, and the helium atomsmove in the form of soliton with constant speed in the superfluid state. This is abasic property of superfluidity and the discussion above gives the phenomenon aclear physical interpretation.

The mass of the soliton can be determined from (2.150),

r°° 4M — \ \(j)\2dx ——-WmP = const.

J-oo *h

The energy of the soliton is

*-£L[|£F+>1-'H<b

= ( ^ - " ' ) M + f w ? ( 4 + " ) (2-15I)Here the first term is the kinetic energy of the soliton, the second and the thirdterms are binding energies and the fourth term is the energy of interaction.

We now discuss properties of circulation (vortex line) produced by the superfluidhelium atoms. The circulation is defined by the velocity of superfluid helium atomvs,

Q = j> vsdr.Jr

In terms of the phase, 9(x, i), of the macroscopic wave function, the velocity of thesuperfluid can be written as

Earlier we concluded that the velocity of the superfluid is equal to the group velocityof the soliton i.e., vs — ve. This indicates that the motion of soliton is the motionof superfluid, and the vortex lines in superfluid is a result of soliton motion in theliquid helium atoms. The phase difference along a closed path is given by the line


integral corresponding to the circulation of the velocity v3

A6= Iv6dr=Îvsdr.

Thus the circulation is related to the phase difference A0 of the superfluid heliumatoms. If the path of integration lies in a multiply connected domain or it enclosesa vortex line, then A0(r) ^ 0. Furthermore, if 4>(r) ^ 0 and it is single valued, wehave A0 = 27m, and

f hd> vsdr = 71—== (2.152)Jr Vim K

where n is an integer.Equation (2.152) implies that whenever the velocity of the rotatng superfluid

helium exceeds a critical velocity, vortex is produced in the liquid. The circulation(the vortex) is quantized and is given by an integer multiple of h/y/2m. Therefore,the nonlinear Gross-Pitaevskii equation indeed gives an adequate description ofsuperfluidity in helium II.

Superfluid can be viewed as a Bose condensate with local interactions (Pitaevskii1961, Gross 1963). The concept of "quantum vortex" was proposed by Ginzburg andPitaevskii (1960). Quantization of vortices in superfluids was suggested by Onsager(1949) on the basis of classical vortex flow and turbulence. The superfluidity of 4Heand its superfluid vortex lines and loops, weak turbulence and dissipative vortexdynamics in superfluids were studied by Brachet, Barenghi et al., Roberts et al.,Pismen and Rica, and many others, using the nonlinear Schrodinger equation orthe Gross-Pitaerskii equation given above (see Barenghi et al. 2001, Donnely 1991,Avenel et al. 1994, Lindensmith et al. 1996). Experimental observation of thevortices were reported by Yamchuk, Gordon, and Packard in 1979 and Zieve etal. Numerical simulations were presented by Frish, Pomeau and Rica (1992) andSchwarz. The energy of vortex lines was also measured.

If the superfluid liquid does not rotate, then V x vs = (h/m)V x V0 = 0.The superfluid velocity field is a conservative field. This suggests that when thesuperfluid liquid helium flows through a tube with a gradual decreasing diameter,the pressure inside the tube is the same everywhere and it does not depends on thediameter of the tube. This is completely different from that of a normal fluid, but ithas demonstrated experimentally. Moreover, the macroscopic wave function <f>(x, t)given in (2.150) approaches 0 when x approches oo. That is, <f>(x, t) vanishes at theboundary. This implies that the superfluid density ps(oc \(j>\2) should also approach0 at the boundary. The value of ps was measured in 1970 and it was found thatits value dropped from the value in the bulk to zero over a few atomic layers. Thisgave a direct verification of the theoretical results.

The Gross-Pitaerskii equation (2.148) is neither relativistic nor taking the grav-itational field into consideration. Anandan and others extended the theory to in-clude the relativistic effect. The generalized relativistic equation of motion for the


quantum superfluid liquid helium is given by

^ _ V V + aV = -A'H2^, (2.153)

where

, m2c2 w 2mA

Equation (2.153) is called the Gross-Pitaevskii-Anandan (GPA) equation. It is atype of the </>4-equation. Anandan did not find its solutions. Instead, he gavean order of magnitude estimate for a </> = Ae%B type of solution using the Einstein-Planck Law. As will be shown below, an exact solution of (2.153) is actually possible.

Let's assume the following trial solution,

<j>(x,y,z,t) = f(Z)ei0 (2.154)

where

Z = p- f — fit, 6 = k • f— uit = k\x + k,2y + k$z — cut.

Substituting (2.154) into (2.153), the latter can be written as

(Q2 ~ v2)% + (a2 + e~ u2)f + X>f = ° (2-155)in terms of k = (ki,k2,k3) and p = (pi,P2,P3), and UJQ = k-p. Integrating (2.155),we obtain the solution

/ 111

f(Z) = , / - sech(^Z) ,

where

_ u;2 - a2 - fc2 A'w~ n2 - p2 ' 2(ft2 - p2)'

and

<f>(x,y,z,t) = f(Z)eie = < / ^ sechUMp- ?- ilt)]^^-^. (2.156)V -ft

This is a soliton solution of the wave packet type, and its group velocity is v.In obtaining the above solution, we have set the constant of integral C" to 0. If

C # 0 but is real, let D' = 2C"/(ft2 - 01) > 0, we then have:

/ = /?sc [sjR{o? - /?2)Z] ,

and

(2.157)


where

<*2/?2 = f , a 2 - / ? 2 ^ -In (2.157), sc^ = l/cni/> and cm/' is Jacobian elliptic cosine function. Equation(2.157) is another solution of (2.153), and it is an elliptic wave. Thus, the rela-tivistic superfluid liquid helium can move as solitons or as elliptic wave. It is worthto point out that both + / and —/ are solutions of (2.153). Thus, the solutionsobtained above, (2.156) and (2.157), are both double solutions. They are actuallythe solutions of GPA equation in a flat space.

Generally, in a four-dimensional space, equation (2.153) may be written as

7 V v / x V ^ + a2(/>=-AW4>, (^,1/= 1,2,3,4). (2.158)

In this case, corresponding to (2.156) and (2.157), we have

<f>(X) = ^ 1 sech y^P»Xvrlllv\ e^-*"*" , (2.159)

and

<f>(X) = Asc [y/R(a2 + p)P»Xv'qilv~\ e^"*"*" (2.160)

where

X = (x, y, z, ict), K = (K, iu/c), P = (P, iv/c).

When gravitational force is considered, there is a transition from a flat space to acurved space which corresponds to a transformation from an orthogonal coordinate(Vnv) system to an obique coordinate (gû) system. This is because the space iscurved in the presence of the gravitational field. Here g^,, is the gauge tensor andVp is a covariant derivative. Equation (2.153) can be written as

S ^ V ^ V ^ + a2(j> = -\'\<f>\24>. (2.161)

This is the GPA equation in the space of gravitational force, g^ is the contravariantcomponent of g^v. Notice that the scalar product gÂ^B" is invariant under sucha transformation. Thus, there exists a relation r)liVKttZl/ = g^K^X" betweenthe •qllvK

liZv of the flat space and the g^K^X" of the curved space. Hence, thesolutions (2.159) and (2.160) in curved space become

0PO = M sech [y/^g^P^X"] e*-*1'*', (2.162)V •»*

4>(X) = /3 sc \y/R{o? + ^g^P^X"] jo*"*"*''. (2.163)

Equations (2.162) and (2.163) describe the relativistic motion of the superfluidliquid helium. We have shown that superfluid liquid helium can move in the form


of either a soliton or an elliptic wave, regardless the space is flat or curved. Whenthere is no gravitational force, there could exist dual solitary wave and dual ellipticwave in the surperfluid. In a gravitational space, although the form of solutionremains the same, the envelope ^w/Rsech[y/w(p • f— vt)], and the carrier wave,exp[i(k • r — wt)}, are all influenced by the gravitational field g^. If we choose9IJ,I/ 7 0 (when fi ^ v), the effect of the gravitational field g^v is mainly to changethe four-dimensional wave vectors of the enevelope wave and the carrier, i.e.,

K» -> g^K" = (KG)*, P* -> ff^J*1 = (PG)A..

Then, the solutions of solitary wave and elliptic wave found in the curved spacebecome the same as those in the flat space.

Therefore, there always exist solutions of dual solitary wave and dual ellipticwave in a superfluid regardless of the existence of gravitational force. They corre-spond to the He-I and He-II phases of the superfluid. Moreover, when the solitaryand elliptic waves in the superfluid are subject to gravitational force, their four-dimensional wave vectors are changed, K —> KG and P —> PQ. This property canbe used to detect the existence of the gravitational field and gravitational wave.Superfluid gravitational antenna have been built in many research laboratories andan important use of it is to detect the gravitational wave by making use of thequantum interference effect of superfluid liquid helium.

Bibliography

Abdullaev, F. and Kraenkel, R. A. (2000). Phys. Rev. A 62 023613.Abrikosov, A. A. (1957). Zh. Eksp. Theor. Fiz. 32 1442.Abrikosov, A. A. and Gorkov, L. P. (1960). Zh. Eksp. Theor. Phys. 39 781.Afanasjev, V. V., Akhmediev, N. and Soto-Crespo, J. M. (1996). Phys. Rev. E 53 1931

and 1190.Anandan, J. (1981). Phys. Rev. Lett. 47 463.Andenson, M. H.,et al. (1995). Science 269 198.Avenel, O.,et al. (1994). Proc. 20th Inter. Conf. on Low Temp. Phys., Physica B 194-196

491.Balckburn, J. A.,et al. (1975). Phys. Rev. B 11 1053.Bardeen, L. N., Cooper, L. N. and Schrieffer, J. R. (1957). Phys. Rev. 108 1175.Barenghi, C. F., Donnerlly, R. J. and Vinen, W. F. (2001). Quantized vortex dynamics

and superfluid turbulence, Springer, Berlin.Bogoliubov, N. N. (1949). Quantum statistics, Nauka, Moscow.Bogoliubov, N. N. (1958). J. Exp. Theor. Phys. USSR 34 58.Bogoliubov, N. N., Toimachev, V. V. and Shirkov, D. V. (1958). A new method in the

theory of superconductivity, AN SSSR, Moscow.Bradley, C. C, Sackett, C. A. and Hulet, R. G. (1997). Phys. Rev. Lett. 78 985.Bradley, C. C, Sackett, C. A., Tollett, J. J. and Hulet, R. G. (1995). Phys. Rev. Lett. 75

1687.


Bullough, R. K., Bogolyubov, N. M., Kapitonov, V. S., Malyshev, C, Timonen, J., Rybin,A. V., Vazugin, G. G. and Lindberg, M. (2003). Theor. Math. Phys. 134 47.

Bullough, R. K. and Caudrey, P. J. (1980). Solitons, Springer-Verlag, Berlin.Burger, S.,e< al. (1999). Phys. Rev. Lett. 83 5198.Burstein, E. and Lulquist, S. (1969). Tunneling phenomena in solids, Plenum Press, New

York.Cai, S. Y. and Bhattacharjee, A. (1991). Phys. Rev. A 43 6934.Caradoc-Davies, B. M., Ballagh, R. J. and Burnett, K. (1999). Phys. Rev. Lett. 83 895.Cashar, A.,et al. (1969). J. Math. Phys. 9 1312.Cooper, L. N. (1956). Phys. Rev. 104 11-89.Cornish, S. h.,et al. (2000). Phys. Rev. Lett. 85 1795.Crasovan, L. C, Malomed, B. A. and Mihalache, D. (2000). Phys. Rev. E 63 016605.Creswick, R. J. and Morriso, H. L. (1980). Phys. Lett. A 76 267.Dalfovo, F., Giorgini, S., Pitaevskii, L. P. and Stringali, S. (1999). Rev. Mod. Phys. 71

463.Davis, K. B.,et al. (1995). Phys. Rev. Lett. 75 3969.Davydov, A. S. (1985). Solitons in molecular systems, D. Reidel Publishing, Dordrecht.de Gennes, P. G. (1966). Superconductivity of metals adn alloys, W. A. Benjamin, New

York.Dewitt, B. S. (1966). Phys. Rev. Lett. 16 1092.Dirac, P. M. (1958). General theory of relativity 2, Oxford, London.Donnely, R. J. (1991). Quantum vortices in heliem II, Cambridge Univ. Press, Cambridge.Elyutin, P. V. and Rogoverko, A. N. (2001). Phys. Rev. E 63 026610.Elyutin, P. V.,e< al. (2001). Phys. Rev. E 64 016607.Fried, D. G.,et al. (1999). Phys. Rev. Lett. 81 3811.Frisch, T., Pomeau, Y. and Rica, S. (1992). Phys. Rev. Lett. 69 1644.Ginzberg, V. I. and Landau, L. D. (1956). Zh. Eksp. Theor. Fiz. 20 1064.Ginzberg, V. L. and Kirahnits, D. A. (1977). Problems in high-temperature superconduc-

tivity, Nauka, Moscow.Ginzburg, V. L. (1952). Usp Fiz. Nauk. 48 25.Ginzburg, V. L. and Pitayevsky, L. P. (1968). Sov. Phys. JETP 7 858.Ginzburg, V. L. and Sobianin, A. A. (1976). Usp Theor. Fiz. Nauk. 120 153.Gorkov, L. P. (1958). Sov. Phys. JETP 9 189.Gorkov, L. P. (1959). Sov. Phys. JETP 9 1364.Gross, E. P. (1961). Nuovo Cimento 20 454.Guo, Bai-lin and Pang Xiao-feng, (1987). Solitons, Chin. Science Press, Beijing.Hirota, R. (1972). J. Phys. Soc. Japan 33 551; Phys. Rev. Lett. 27 1192.Hoch, M. J. R. and Lemmer, R. H. (1991). Low temperature physics, Springer, Berlin.Holm, J.ef al. (1993). Physica 68 35.Huebenor, R. P. (1974). Phys. Rep. 24 127.Huepe, C. and Brachet, M. E. (2000). Physica D 140 126.Jacobs, L. and Rebbi, L. (1979). Phys. Rev. B 19 1486.Jacobson, D. A. (1965). Phys. Rev. B 8 1066; Adv. Phys. 14 419.Josephson, B. (1962). Phys. Lett. 1 251.Josephson, B. (1964). Thesis, unpublised, Cambridge University.Josephson, B. (1965). Adv. Phys. 14 419.Kaper, H. G. and Takac, P. (1998). Nonlinearity 11 291.Kayfield, G. W. and Reif, F. (1963). Phys. Rev. Lett. 11 305.Khalilov, V. R. (2002). Theor. Math. Phys. 133 1406.Kictchenside, P. W.,ei al. (1981). Soliton and condensed matter physics, eds. A. R.


Bishop, et al, Plenum, New York, p. 297.Kivshar, Yu. S., Alexander, T. J. and Turitsy, S. K. (2001). Phys. Lett. A 278 225.Kivshar, Yu. S. and Malomed, B. A. (1989). Rev. Mod. Phys. 61 763.Klitzing, K. V.,et al., (1980). Phys. Rev. Lett. 45 494.Kusayanage,et al. (1972). J. Phys. Soc. Japan 33 551.Lacaze, R., Lallemand, P., Pomeau, Y. and Rica, S. (2001). Physica D 152-153 779.Lamb, G. L. (1971). Rev. Mod. Phys. 43 99.Laplae, L.,et al. (1974). Phys. Rev. C 10 151.Leggett, A. J. (1980). Prog. Theor. Phys. (Supp.) 69 80.Leggett, A. J. (1984). in Percolation, localization and superconductivity, eds. by A. M.

Goldlinan, S. A. Bvilf, Plenum Press, New York, pp. 1-41.Leggett, A. J. (1991). in Low temperature physics, Springer, Berlin, pp. 1-93.Lindensmith, C. A.,et al., (1996). Proc. 21st Inter. Conf. on Low Temp. Phys., Czech J.

Phys. 46 131.Liu, W. S. and Li, X. P. (1998). European Phys. J. D2 1.London, F. (1950). Superfluids, Vol. 1, Weley, New York.London, F. and London, H. (1935). Proc. Roy. Soc. (London) A 149 71; Physica 2 341.Louesey, S. W. (1980). Phys. Lett. A 78 429.Madison, K. W.,et al. (2000). Phys. Rev. Lett. 84 806.Maki, K. and Lin-Liu, Y. R. (1978). Phys. Rev. B 17 3535.Malomed, B. A. (1989). Phys. Rev. B 39 8018.Marcus, P. M. (1964). Rev. Mod. Phys. 36 294.Matthews, M. R.,et al. (1999). Phys. Rev. Lett. 83 2498.Niek, I. R. and Fraser, J. C. (1970). J. Low Temp. Phys. 3 225.Ohberg P. and Santos, L. (2001). Phys. Rev. Lett. 86 2918.Onsager, L. (1949). Nuovo Cimento Suppl. 6 249.Ostrovskaya, E. A.,et al. (2000). Phys. Rev. A 61 R3160.Packard, R. E., (1998). Rev. Mod. Phys. 70 641.

Pang, Xiao-feng (1982). Chin. J. Low Temp. Supercond. 4 33; ibid 3 62.Pang, Xiao-feng (1982). Chin. J. Nature 5 254.Pang, Xiao-feng (1983). Phys. Bulletin Sin. 3 17.Pang, Xiao-feng (1985). Chin. J. Potential Science 5 16.Pang, Xiao-feng (1985). J. Low Temp. Phys. 58 334.Pang, Xiao-feng (1985). Problems of nonlinear quantum theory, Sichuan Normal Univ.

Press, Chengdu.Pang, Xiao-feng (1986). Chin. Acta Biochem. BioPhys. 18 1.Pang, Xiao-feng (1986). J. Res. Met. Mat. Sin. 12 31.Pang, Xiao-feng (1986). J. Science Exploration Sin. 4 70.Pang, Xiao-feng (1986). Phys. Bulletin Sin. 1 2.Pang, Xiao-feng (1987). The properties of soliton motion in superfluid He4, Proc. ICNP,

Shanghai, p. 181.Pang, Xiao-feng (1988). J. Xinjian Univ. Sin. 3 33.Pang, Xiao-feng (1989). J. Chinghai Normal Univ. Sin. 1 37.Pang, Xiao-feng (1989). J. Kunming Tech. Sci. Univ. 2 118; ibid 6 54.Pang, Xiao-feng (1989). J. Southwest Inst. for Nationalities Sin. 15 97.Pang, Xiao-feng (1989). J. Xinjian Univ. Sin. 3 72.Pang, Xiao-feng (1990). J. Xuntan Univ. Sin. 12 43 and 65.Pang, Xiao-feng (1991). J. Kunming Tech. Sci. Univ. 3 57.Pang, Xiao-feng (1991). J. Kunming Tech. Sci. Univ. 4 13.Pang, Xiao-feng (1991). J. Southwest Inst. for Nationalities Sin. 17 1 and 18.


Pang, Xiao-feng (1991). J. Xuntan Univ. Sin. 13 63.Pang, Xiao-feng (1994). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing, 1994.Pang, Xiao-feng,et al. (1989). The properties of soliton motion for superconductivity elec-

trons. Proc. ICNP, Shanghai, p. 139.Papaconstantopoulos, D. A. and Klein, B. M. (1975). Phys. Rev. Lett. 35 110.Pardy, M. (1989). Phys. Lett. A 140 51.Parks, R. D. (1969). Superconductivity, Marcel. Dekker.Pederser, N. F. (1993). Physica D 68 27.Perez-Garcia, V. M., Michinel, M. and Herrero, H. (1998). Phys. Rev. A 57 3837.Perring, J. K.,et al. (1963). Nucl. Phys. 61 1443.Pitaerskii, L. P. (1961). Sov. Phys. JETP 13 451.Pitaevskii, L. P. (1960). Sov. Phys. JETP 10 553.Pomeau, Y. (1992). Nonlinearity 5 707.Pomeau, Y. (1996). Phys. Scr. 67 141.Pomeau, Y. and Rica, S. (1993). Comptes Rendus Acad. Sci. (Paris) 316 (Serie II) 1523.Putterman, P. (1973). Low Temp. Phys. LT13 39.Putterman, P. (1972). Low Temp. Phys. LTB1 39.Rieger, T. J.,et al. (1976) Phys. Rev. Lett. 27 1787.Rogovin, D. (1974). Ann. Phys. 90 18.Rogovin, D. (1975). Phys. Rev. B 11 1906.Rogovin, D. (1975). Phys. Rev. B 12 130.Rogovin, D. (1976). Phys. Rep. 251 1786.Rogovin, D. and Scully, M. (1969). Ann. Phys. 88 371.Rogovin, D. and Scully, M. (1976). Phys. Rep. 25 176.Ronognan, J. P. (1974). Solid State Commun. 14 83.Schrieffer, J. R. (1964). Theory of superconductivity, Benjamin, New York.Schrieffer, J. R. (1969). Superconductivity, Benjamin, New York.Schwarz, K. W. (1990). Phys. Rev. Lett. 64 1130.Schwarz, K. W. (1993). ibid 71 259.Schwarz, K. W. (1993). Phys. Rev. B 47 12030.Scott, A. C.,et al. (1973). Proc. IEEE 61 1443.Sonin, E. B. (1994). J. Low Temp. Phys. 97 145.Sorensen, M. P., Malomed, B. A., Vstinov, A. V. and Pedersen, N. F. (1993). Physica 68

38.Suint-James, D.,e£ al. (1966). Type-II superconductivity, Pergamon, Oxford.Svozil, K. (1990). Phys. Rev. Lett. 65 3341.Tonomara, A. (1998). Foundation Phys. 28 59.Valatin, D. G. (1959). Nuovo Cimento 7 843.Ventura, I. (1990). Solitons in liquid 4He, in Solitons and applications, eds. Makhankov,

V. G., Fedyanin, V. K. and Pashaev, O. K, World Scientific, SingaporeWheatley, J. C. (1975). Rev. Mod. Phys. 47 415.Wiegel, F. W. (1977). Phys. Rev. B 16 57.Yarmchuk, E. J., Gordon, M. J. V. and Packard, R. E. (1979). Phys. Rev. Lett. 43 214.Zieve, R. J., Close, J. D., Davis, J. C. and Packard, R. E. (1993). J. Low Temp. Phys. 90

243.Zieve, R. J., Mukhjarsky, Yu., Close, J. D., Davis, J. C. and Packard, R. E. (1992). Phys.

Rev. Lett. 68 1327.

Chapter 3

The Fundamental Principles and Theoriesof Nonlinear Quantum Mechanics

3.1 Lessons Learnt from the Macroscopic Quantum Effects

In the previous chapter, we discussed some macroscopic quantum effects observedin experiments involving superconductors, superfluid and so on, and analyzed theseeffects using quantum solid theory and modern nonlinear theory. Macroscopic quan-tum effects are results of motions of quasipartides. Thus, we discussed the dynamicsof such quasiparticles as superconducting electrons and superfluid helium atoms onthe basis of soliton theory. The observed macroscopic quantum effects in certainsystems can then be easily understood based on these theories. This demonstratedthat the nonlinear theory is the proper theory for the macroscopic quantum effectsand the relevant physical quantities such as the Hamiltonian, Lagrangian as wellas the dynamic equations are in the correct forms. In this chapter, we will furtherexplore the physical insights of these concepts and give an in-depth treatment ofthe theory.

But first, let's summarize what we have learnt from the discussion on the macro-scopic quantum effects and the theoretical concepts used to understand these effects.

(1) It has become clear that the macroscopic quantum effects are fundamentallydifferent from the microscopic quantum effects. The latter can be described by thelinear quantum theory which, however, failed to describe the macroscopic quantumsystems. It is thus very necessary to establish a quantum theory that describesnonlinear systems.

(2) However, why can't the present linear quantum theory describe the macro-scopic quantum effects? On what foundation a new theory should be based? An-swers to such questions and the key for solving the existing problems of the linearquantum mechanics can only come from a clear understanding of the intrinsic prob-lems of the linear quantum mechanics and the fundamental aspects of the macro-scopic quantum effects. The behaviors of quasiparticles in such systems must playan essential role.

It is well know that the macroscopic quantum effects are nonlinear phenomena.The BCS theory of superconductivity and the modern theory of superfluidity, bothare nonlinear theories, have been well established. A basic feature of the nonlinear

81


theories is that the Hamiltonian, free energy or Lagrangian functions of the sys-tems are nonlinear functions of the wave function of the microscopic particles, andthe dynamic equation becomes a nonlinear equation, due to the nonlinearity. Thequasiparticles behave differently in such systems which results in ordered coherentstates, such as the Bose-Einstein condensed state. These states occur following asecond-order phase transition and spontaneous breaking of symmetry of the sys-tems under the nonlinear interactions. Therefore, the nonlinear interactions play avery important role in the behaviors of the microscopic particles. It also suggeststhat one should pay particular attention to the nonlinear interactions in order toestablish a correct new quantum theory, and the right direction for solving problemsencountered by the linear quantum mechanics is to establish a nonlinear quantumtheory.

(3) How can a nonlinear quantum theory be established? Again we take a lookat what we learned from the macroscopic quantum effects, and try to understandhow the superconductive and superfluid theories differ from the linear quantummechanics? Detailed examination reveals the following.

(a) The Hamiltonian, or Lagrangian function and free energy of these systems,given in (2.1) and (2.2), or (2.53) and (2.54), or (2.149) and (2.151), are all depen-dent on, and are nonlinear functions of, the wave function <f>(x, t) of the microscopicparticles, i.e. the superconductive electron or superfluid helium, respectively. Thesego directly against the fundamental hypothesis of the linear quantum mechanicsthat the Hamiltonian of the system is independent of the wave functions of themicroscopic particle.

It was exactly because of this nonlinear feature in the theories of superconduc-tivity and superfluidity that they were able to correctly describe the nonlinear be-haviors of superconductivity and superfluidity and successfully explain these macro-scopic quantum effects. On the contrary, lacking of such a nonlinearity was also thereason that other theories failed. For example, although Prohlish's superconductingtheory gave the correct superconducting mechanism, i.e., electron-phonon coupling,in 1951, he failed to establish a complete superconducting theory because his the-ory was based on the linear perturbation theory of the linear quantum mechanics.Therefore, the new nonlinear theory should abandon this hypothesis.

(b) The fundamental dynamic equations in the linear quantum mechanics is theSchrodinger or the Klein-Gordon equation which are wave equations, and are lin-ear equations of the wave function of the particles. As a result, solutions of theselinear equations cannot describe the wave-corpuscle duality of microscopic particlesas discussed in Chapter 1. On the other hand, the Ginzburg-Landau equations,(2.17), (2.55)-(2.57), (2.119), and the GP equation, (2.121), or the GPA equation,(2.148), satisfied by the quasiparticles (e.g. the superconductive electron and super-fluid helium atom), as well as the (f>4-equation (2.153) and the Sine-Gordon equation(2.104) in superconductors and superfluid are all nonlinear equations of the wavefunction of the quasiparticles. With these nonlinear equations, superconductivity

Principles and Theories of Nonlinear Quantum Mechanics 83

and superfluidity as well as other macroscopic quantum effects observed in experi-ments can be explained. This suggests that in establishing a new theory, the lineardynamic equation must be replaced with a nonlinear equation, and the superposi-tion principle of wave functions must be abandoned. Fortunately, all the nonlinearequations mentioned above are natural generalizations of the Schrodinger equationor Klein-Gordon equation which are the dynamic equations in linear quantum me-chanics. Therefore, the new nonlinear quantum theory should be developed on thebasis of the linear quantum mechanics, rather than anything else.

Certainly, it is still necessary to further examine whether these dynamic equa-tions or Hamiltonians given in Chapter 2 have the correct space-time symmetriesand what physical invariance they possess. Only nonlinear dynamic equations orHamiltonians which satisfy the required symmetries and invariances can be adoptedin the new theory.

It is known that the nonlinear dynamic equations describing superconductivityand superfluidity states admit stable soliton solutions. This shows that the normalmicroscopic particles evolve into solitons in nonlinear systems due to nonlinear in-teractions. It is therefore natural to use the concept of soliton in the descriptionof microscopic particles in nonlinear systems. A soliton is a new form of physicalentity which cannot be described by a linear theory. According to modern soli-ton theory, a soliton, which differs completely from a normal microscopic particle,possesses the wave-particle duality. Its wave property appears in the form of a trav-eling solitary wave which has all the essential features of wave motion, includingfrequency, period, amplitude, group and phase velocities, diffraction, transmission,and reflection. Its corpuscle feature is reflected by a stable shape analogous to aclassical particle, even after going through a collision with another particle, a def-inite energy, momentum and mass, and its uniform motion in free space and itsmotion with a constant acceleration in the presence of a constant external field,and so on. This suggests that modern soliton theory should be an integral part ofany new nonlinear quantum-mechanical theory.

To summarize, we see clearly that the direction for developing a new quantumtheory is nonlinear quantum mechanics. As a matter of fact, the concept of nonlinearquantum mechanics was proposed by many scientists such as Mielnik, Jordan, Gisin,Weinberg, Doebner and so on. However, the work presented in this book representsa complete different approach. Both this approach and its mathematical treatmentsare different from those in other publications in many respects. The theories andprinciples presented here are firmly based on physical foundations Its mathemati-cal basis is the nonlinear partial differential equations and the soliton theory. Itsphysical basis is the macroscopic quantum effects, and the well established moderntheories of superconductivity and superfluidity. With the macroscopic quantumeffects as the foundation, and by incorporating nonlinear interactions and solitonmotion into a generalized theoretical framework, fundamental principles and the-ories of nonlinear quantum mechanics can be established, without the hypotheses


of the linearity of linear quantum mechanical theory and the independence of theHamiltonian operator on the wave function of the particles.

3.2 Fundamental Principles of Nonlinear Quantum Mechanics

Based on the earlier discussion, the fundamental principles of nonlinear quantummechanics (NLQM) may be summarized as follows.

(1) Microscopic particles in a nonlinear quantum system are described by thefollowing wave function,

<f,(f,t)=cp(r,t)eie^ (3.1)

where both the amplitude ip(f, t) and phase 6(r, t) of the wave function are functionsof space and time.

(2) In the nonrelativistic case, the wave function (f>(r, t) satisfies the generalizednonlinear Schrodinger equation (NLSE), i.e.

ihl£ = - T - V ^ ± Wft + v<f, *)0 + M4>), (3.2)ot 1m

or

^Tt = ~ £ v ^ ± m2<t> + v{f't)<t> + A{<j>)' (3'3)

where \i is a complex number, V is an external potential field, A is a function of <f>,and 6 is a coefficient indicating the strength of the nonlinear interaction.

In the relativistic case, the wave function (p(f,t) satisfies the nonlinear Klein-Gordon equation (NLKGE), including the generalized Sine-Gordon equation (SGE)and the </>4-field equation, i.e.

*+-*!+=f,sin+ + ^+A(*), 0' = 1,2,3) (3.4)

^-^â4>±m24> = A{cj>), 0' = 1,2,3) (3.5)

where 7 represents dissipative or frictional effects, /? is a coefficient indicating thestrength of nonlinear interaction and A is a function of <j>.

These are the only two fundamental hypotheses of the nonlinear quantum me-chanics. This is quite different from the linear quantum mechanics which are basedon several hypotheses, as discussed in Chapter 1. However, the dynamic equationsare generalizations of the linear Schrodinger and linear Klein-Gordon equations ofthe linear quantum mechanics to nonlinear quantum systems. These equationswere used to study the motion of superconducting electrons and helium atoms inthe superfluid state. It has been shown that (3.2) - (3.5) indeed describe the lawof motion and properties of microscopic particles in nonlinear quantum systems.


This is the basis for having the two hypotheses as the principles of the nonlinearquantum mechanics. Obviously, the nonlinear quantum mechanics is an integrationof superconductivity, superfluidity and modern soliton theories, and its experimen-tal foundation is the macroscopic quantum effects. Prom the two hypotheses, thefollowing can be deduced.

(1) The absolute square of the wave function <j>(r,t) given in (3.1), \4>{r,t)\2 =\ip(r,t)\2 = p(f,t), is no longer the probability of finding the microscopic particle ata given point in the space-time, but gives the mass density of the microscopic parti-cles at that point. Thus, the concept of probability or the statistical interpretationof wave function is no longer relevant in nonlinear quantum mechanics. The wavefunction (3.1) has the similar form as that in the linear quantum mechanics, buttheir meaning are completely different. Here tp(r, t) is the envelope of the micro-scopic particle, and it represents the particle feature, or more precisely, the solitonfeature of the microscopic particle. Different from that in the linear quantum me-chanics, <p{r,t) has its physical meaning and satisfies certain nonlinear equation.

e%e(r,t) j s a c a r r i e r w a v e of (f>(r,t). The interpretation of (3.1) will be discussed inmore details in the following Chapters.

(2) The wave function <j>(f,t) represents a soliton or a solitary wave. It is nolonger a linear or dispersive wave. Equations (3.2) - (3.5) are nonlinear dynamicequations and have soliton solutions. Therefore, a microscopic particle is a soliton,or is described by a soliton, in nonlinear quantum mechanics. Thus the fundamentalnature of microscopic particles in the nonlinear quantum mechanics is different fromthat in the linear quantum mechanics.

(3) The concept of operator in the linear quantum mechanics is still used in thenonlinear quantum mechanics. However, they are all no longer linear operators,and thus certain properties of linear operators, such as conjugate Hermitian of themomentum and coordinate operators, are no longer required. Instead, nonlinear op-erators are constructed and used in the nonlinear quantum mechanics. For example,Equation (3.2) may be written as

ih^- = H(4>)<fi. (3.6)at

The Hamiltonian operator H{<j>) has a nonlinear dependence on (j> and is given by

HW = -^2-b\4>\2 + V(r,t) (3.7)

for A = 0.In general, equations (3.2) - (3.5) can be expressed as

<t> = (?,t), <frt = ^. = K(<t>) (3.8)at

according to Lax, where K(<f>) is a nonlinear operator or hereditary operator. Inone dimensional case, a new operator Q{<j>) which is obtained from the generator of


the translation group can be used to produce the vectorial field K{<j>), i.e.

K{<l>) = Q(4>)<l>z, (3.9)

and the operator Q{4>) is called a nonlinear recursion operator. The nonlinearSchrodinger equation (3.2) can be generalized to

Q(4>) = -iD + Ai<j>D-l^{$) (3.10)

where the operator D denotes the derivative with respect to x, and

D-lf(x) = [X f(y)dy.

From the hereditary property of K(tj>), we can obtain the following vector field

Kn{(j>) = Q{<j>)n<j>x, n = 0 , l . (3.11)

The equation of motion of the eigenvalue A' of the recursion operator Q(4>) maybe expressed as

K = = K'{<t>)[\'\ (3-12)

where K'(<j>) is the variational derivative with respect to cf> and is given by

dK'fr + e\')K (<£)lA J = w; • (3.13)

The equation describing the time variation of the recursion operator can be ob-tained,

-Q(<P) = K'(4>)Q(<t>) - Q{)K'{4>) = [K'(4>),QW}. (3.14)

This equation is very similar to the Heisenberg matrix equation in the linear quan-tum mechanics. However, both K'(<f>) and Q{(j>) here are nonlinear operators.

(4) Because the operators in nonlinear quantum mechanics are nonlinear, itis no longer necessary for their eigenvectors or states cf>(r, t) to satisfy the linearsuperposition principle, i.e. <j> = Y^n^'n't'n- This implies that superposition ofany two states is not necessarily a state of this system as shown in (1.4). As amatter of fact, superposition of two solitary waves may result in two, three or anynumber of solitary waves. Therefore, the superposition principle of waves in thelinear quantum mechanics must be modified for the nonlinear quantum mechanics.This will be discussed in the next section.

(5) There are also time-independent states and eigenvalue problems in nonlinearquantum mechanics. How the eigenvalue of the nonlinear Schrodinger equation isdefined and determined is interesting. Since nonlinear quantum mechanics differsfrom linear quantum mechanics, these concepts and the method to determine the


eigenvalues are different. The time-independent solution of (3.2) is assumed to havethe following form

<t>(f,t)=<p{r)eiEtlh. (3.15)

Substituting (3.15) into (3.2) and choose A((j>) = 0, we can get

E<p(7) = - ^ V V W + V{f)<p(r) - %(f ) |M0, (3-16)

That is,

H(vMr) = E<p (3.17)

where

H(*) = - ^ V 2 + V(r) - % | 2 = - | ^ V 2 + V(r) - bp(r). (3.18)

The energy E in (3.18) is the eigenvalue of the Hamiltonian operator H(tp). Equa-tion (3.17) is very similar to the linear Schrodinger equation in the linear quantummechanics, but the Hamiltonian operator H(<p) is a nonlinear operator of wavefunction tp, i.e.,

H(v) = H0 + bp(r), (3.19)

where

This shows that if this Hamiltonian operator acts on a wave vector of a quantumstate of a microscopic particle, an eigenvalue independent of time and positioncannot always be obtained. It is thus impossible to determine the eigenvalues ofthe Hamiltonian using the traditional method. Thus the eigenvalue of the nonlinearSchrodinger equation must be redefined.

In general, the eigenequation and eigenvalues of a nonlinear system can be de-fined and determined using the approach proposed by Lax. For a general nonlinearequation given in (3.8), we know that K{<p) is a nonlinear operator. If two linearoperators L and B, which depend on <j>, satisfy the following Lax operator equation

iLv =BL-LB = [B, L], (3.20)

where t' = t/h and B is a self-adjoint operator, then the eigenvalue E and eigen-function ip of the operator L may be derived from (3.20), i.e.

Lil) = AV>; iipv = Bip. (3.21)

Thus, the eigenvector and eigenvalue of a nonlinear system are determined by theeigenvectors and eigenvalues of the above two linear operators. It can be shown


that A is an eigenvalue that does not depend on time. In fact, if we differentiate(3.21) and multiply the resulting equation by i, we can get

i (^~ + ^ ) = * (^ + %jA = iL** + [BL - LB)^

= L(iipt' - Brp) + XBrp.

From the above, we get iip(d\/dt') = 0. Thus, A is a time-independent eigenvalue,and (3.21) is the linear eigenequation corresponding to the nonlinear equation (3.8).Therefore, for any nonlinear equation, we can always find the corresponding lineareigenequation and time-independent eigenvalue. In fact, Lax, Zakharov, Shabatand Ablowitz et al. successfully reduced a nonlinear problem into a linear one andthen obtained soliton solution by the inverse scattering method.

If both V(r, t) and A(<j>) in the nonlinear Schrodinger equation (3.2) are zeros,the above operators L and B are

f-i(1 + s ° 1 d + (0<f>*\L~l{ 0 l-sjM + U 0 ) '(3.22)

S\0l)dx* + [ ~i<pX' - | ^ | 2 / ( l - s ) J '

and s2 = 1 — 2/b, x' — Xyj2m/K2. The eigenvalues of the nonlinear Schrodingerequation (soliton eigenvalues) are thus determined by

L^ = XA * = ( £ ) •

Its solution can be obtained using the inverse scattering method which will bediscussed in detail in chapter 6.

However, it should be pointed out that if (j>(r,t) or ip(r) is further quantizedby the creation and annihilation operators of the microscopic particles, then theHamiltonian operator in (3.18) would be given in terms of the creation and anni-hilation operators in the second quantization representation, and the eigenvaluesof the Hamiltonian operator can also be determined in the same representation.This method was used by Pang et al. to obtain the eigenvalues of the Hamilto-nian operator in molecular systems, which will be discussed in Chapters 6 and 9,respectively.

(6) Compared to the linear quantum theory, two major breakthroughs weremade in the nonlinear qunatum theory, the linearity of the dynamic equationand independence of the Hamiltonian operator on the wave function of the mi-croscopic particles. In the nonlinear quantum mechanics, the dynamic equationsare nonlinear in the wave function <f>, i.e., they are nonlinear partial differentialequations. The Hamiltonian operators depend on the wave function. For example,


the Hamiltonian operator (3.7) corresponding to equation (3.2) is no longer simplyH = —(7i2/2m)V2 + V(f,t), but depends on the wave function (j>. In this respect,the nonlinear quantum mechanics is truly a break-through in the development ofmodern quantum theory.

3.3 The Fundamental Theory of Nonlinear Quantum Mechanics

Similar to the linear quantum mechanics, the fundamental theory of nonlinear quan-tum mechanics consists of the following,

(1) principle of nonlinear superposition;(2) theory of nonlinear Fourier transformation;(3) method of quantization;(4) perturbation theory.

A nonlinear quantum mechanical system or problem can be studied based on thesefundamental theories. Compare to the linear quantum mechanics, these funda-mental theories of the nonlinear quantum mechanics are much complicated. Eventhough they have been studied in soliton physics, much of it requires further im-provement. In view of this, only the fundamental concepts and processes are intro-duced here and the details will be discussed in subsequent chapters.

3.3.1 Principle of nonlinear superposition and Bdcklund transfor-mation

From earlier discussion, we know that the principle of superposition for wave func-tions of microscopic particles (soliton) in the nonlinear quantum mechanics is nota linear superposition as in the linear quantum mechanics. In the linear quantumtheory, a complicated motion can always be taken as a superposition of some basicmodes. If ij)\{r,t) and ip2(r,t) are solutions of the equation of motion, (1.4), thentheir superposition ip — aip\ {f, t) + bi\>i (f, t) is also a solution. However, microscopicparticles in the nonlinear quantum mechanics do not satisfy this relation. The non-linear interaction not only complicates the process of superposition, but also givesrise to many different forms of superposition.

(A) Konopelchenko was the first to derive completely the nonlinear superpo-sition principle of integrable nonlinear equations. The following is the differentialequations studied by Konopelchenko

dxUJ-aUiJUJ + U*,i) o JUJ (3 3)

which can be written as

9 P ^ ' f ) = 2iQ(L+)AP (3.24)at


where

L+ = -\iA^ - \i {p{x), j°Jy[P{y\A*]} ,

and fl(L+) is an arbitrary mermorphic functions. In the above,

is a two component Jost function, g(x,t) and r(x, t) are wave functions related tothe states of microscopic particle with soliton feature, A' is a constant. Equation(3.24) includes the nonlinear Schrodinger equation, Sine-Gordon equation, etc.

The Backlund transformation (BT) corresponding to (3.24) from the infinite-dimensional group P to P' for the systems consists of the following

2

Y, BiiX+^H'iP' - PH<) = 0 (3.25)»=i

where

and Bi(A+) and B2(A.+) are arbitrary entire functions. The operator A+ has thefollowing property,

A+4> = ~\AlL ~ \ f_ dy\A^y)p'^ - P(y)M{y)]P\x)

+ l-P{x) [X dy{Acj>{y)Pl{y) - P(y)A<f>(y)].

We now consider an arbitrary discrete BT (3.25), i.e. let

BI(A+)=n(A+ - ^)' B 2 ( A + ) = i i ( A + - ^)>

where Aj and /ik are arbitrary constants and rii and n-i are arbitrary integers. Then,any arbitrary discrete BT may be expressed as

J f c = l 8 = 1

where B^, and B y are arbitrary elementary Backlund transformations (EBT).

EBT B{y] is the BT (3.25) at Si = A+ - A', and B2 - 1 (A' is an arbitrary

(3.26)


constant). EBT B(£} is the BT (3.25) at Bl = 1, and B2 = A+ - y! (//' is aconstant). The explicit expressions of the EBTs are

B$\P - 4 P ' ) - ' > ¥ - \i'2r + 2AV + 2q = 0, (3.27)

B » ( f -» i") : * | j - \<?T' + 2 / / , + 2, ' = 0, (3.28)

fir' 1i^L + i r ' 2 g _ 2 y _ 2 r = o.

oa; 2

The EBTs 5 ^ and B ^ commute with each other

R(1)R(2) _ R(2)R(1) „ , R(2)R(1) _ R(l)o(2) _ -i /o 0 Q \B\' Bn' ~ Bn' B\' a n d By! B\' - B\' Bn' ~l- [6.29)

Let us consider four solutions of (3.24), {qo,fo), (<Zi,ri), (92,^2)! and (93,r3), whichare related through the commutativity of B^' and BJ,2 , respectively. Using (3.27)and (3.28), the following relations can be obtained

,3 = ,0 + ^ L , r8 = ro + 2 I e ^ . (3.30r2/2 + 2/<7i qi/2 + 2/r2

Thus, if three solutions (90,^0), (Qij?"i) and (92,^2) °f (3-24) are known, the fourthsolution ((73, rz) may be found from the relation given in (3.30). Equation (3.30)can then be considered as the nonlinear superposition formula corresponding to thenonlinear equation (3.24). As a matter of fact, equation (3.30) is applicable to allintegrable nonlinear equations.

It can be shown that an infinite number of solutions of (3.24) can be obtainedalgebraically from the relation (3.30). Let us start from the trivial solution

F(°°) = V o o J 'and apply all possible discrete BTs (3.26) on -P(oo)> solutions of the entire systemmay be obtained and can be written as

P(a^) = f[B^f[B^Pm (3.31)

where (711,712) is the vertex of P(oo)- The solutions P(on) and P(no) can be found


from (3.27) and (3.28) and are given by

n9(on) = 0, r(On) = £%xp[-2ift(/!ii)* - 2i/ij(z - Soi)],

i = 1 (3.32)9(no) = ]T]exp[2ift(A()* + 2iA,(x - rcOi)], r ( n 0 ) = 0

where xOi and xOi are arbitrary constants. Equations in (3.32) are also solutions ofthe linear equation

which is the linear part of (3.24). The nonlinear superposition formula (3.30) cor-responding to (3.31) can now be written as

2(A' - ft')9(ni + l,n2+l) — <7(ni,n2) + ~ / 0 , o /„ '

r(ni,n2+l)/^ + ^/^(m+l.na) , , o4\2Qi' - X') ( 3 > 3 4 )

r(ni + l,na+l) — r(m,-n2) + /o , 9 / . ,Using (3.34), any arbitrary solution P(ni,n2)

m a y be found from (3.31). For example,if P(oo), P{io) and P(oi) are known, then P(n) can be obtained,

= 2(At - A*I)(11^ n/2)e-2jnMlt-2i/ii(x-«oi)-)-2e~2inAl t+2 iAl(x~x°1' , ,

= 2{m-\1) ( 3 3 5 )

r ( n ) ^y2)e2inXlt~2i>vl(: i :-So1) + 2e2in>llt+2ill'i^x~Xo^

Similarly, P(12) may be obtained from P(oi)> -P(n) and -P(o2) using (3.34),

2(Ai - /ia)

""'^fe,) (3-36)'"2» = r ( ° " + 9(1.,/2 + 2 / r ( M ) '

where g(n) is given in (3.35). From P(2o), P(io) and P(n) we can get P(2i)

, 2(A2-/ii)9(21) = 9(10) + Jn.oln '

2 ( , 1 n /)

2 + 2 / g ( 2 o ) ( 3 -^r ( 2 1 ) ff(20)/2 + 2/r ( 1 1 ) '

where r(nj is given in (3.35). From P(u),-P(i2) and P(2i) one can get P(22)

, 2(A2 - /n)9(22) = 9(11) + / 0 . 0 / )^ C - j C r (3-38)••(»)=••(..)+, (M) /2+2A( IJ )-

(3.33)


Other solutions, such as -P(3i), P(32)> etc., can also be obtained from (3.34). We cansee that an arbitrary solution P(nm2) is a simple function of <7(io)) 9(20) >"" • ; 9(mO)and r(io),7"(o2), • • • ,f(on2). Thus, the nonlinear superposition principle (3.34) givesa simple algebraic structure for the family of infinite number of solutions of (3.31).

Solutions P(n), P(22) and P(nn), are soliton-type solutions. They are relatedthrough the Backlund transformation (3.32). If one of them is known, another canbe obtained using (3.34). Let r — q and r = q*, then the solution P(nn) reduces tothe n soliton solutions of the Sine-Gordon equation (fi = L*~l, r = q, A' = — fi')and the nonlinear Schrodinger equation (Q — —2L*2, r = q*, y! = A'*), respectively.

(B) The general nonlinear equation for the microscopic particle depicted by<f>(x, Y, t) may be written as

Mx,Y,t) = 2po(L,t)(j)x(x,Y,t)+an(L)[an,<p(x,Y,t)]

+pn(L)Gcrn +Y(L,t)-?-<t>(x,Y,t). (3.39)

Assume that

*(x,t)= [ dY</>(x,Y,t). (3.40)Jx

Evidently, $(z,t) satisfies the following boundary conditions

$(±oo,*) = $< (±oo,i) = 0.

In this case, the BT may be written as

$'z(x,t) + *,(*,*) = ~[9'(x,t) - *(a:,t)][4P + *'(i,t) - *(x,t)] (3.41)

where k = iP is the eigenvalue of the following linear Schrodinger equation

4>xx = [<!>- k 2 t y . (3.42)The nonlinear superposition principle corresponding to (3.41) is

(Pi - Pa)[*i2 -*o] + \ {[$12 - *o], [*i - $2]} = - ( P i + Pa)[*i - $2] (3.43)

where Pi and P2 satisfy (3.41), <J?o(z, t) is the solution of (3.39), $i(ar, t) and $2(x, t)are also solutions of (3.39) corresponding to P\ and P2 respectively, and are relatedto $0 through (3.41). $12 (z; t) is another solution of (3.39) which is related tothe solution $1 of (3.41) corresponding to P = P2, or to the solution $ 2 of (3.41)corresponding to P = Pi. The readers are referred to the book by R. H. Bullughand P. J. Caudrey for more details of this superposition principle.

(C) In regard to the nonlinear superposition principle of <j>£T — sm<f> of theSine-Gordon equation, the following is sometimes used

tan (t^t) = ^ - ^ tan {^M (3.44)


where a\ and a2 satisfy the following BT

The derivation for the above will be given in Chapter 7.

3.3.2 Nonlinear Fourier transformation

The linear Fourier transformation

often used in the linear quantum mechanics no longer holds in the nonlinear quan-tum mechanics, due to nonlinear interaction. It must be replaced by a nonlinearFourier transformation which was derived by Zakharov et al. An outline of thederivation is given in the following.

According to the Zakharov-Shabat equation, which can be obtained from thenonlinear Schrodinger equation (3.2) at V(x, t) = A(4>) — 0, we have

... . V2mb , ,ipix+i?i>i = —;—W2,

Jhr (3-46)V2x -i£,V2 = —z—<H>\-n

The corresponding time-dependent forms of the above equations are

iipu = ~ ~ £ ' 2 + ( ^ r ) <?<!> î + y/2mb i?4> - -<j>x ip2,

iiht = C'2 - U i ) > ^ - ^ ^ W + TiK î-

\_m \^<*/ J L ^ JIt can be shown that the condition for (3.46) and (3.47) to be solvable is that thefollowing nonlinear Schrodinger equation must be satisfied by <f>,

iWt = - ( J ^ \ 4>xx - b(4>*<t>)d>. (3.48)

This equation may be solved by using the inverse-scattering method in which thescattering data are

St = {t'j,Pj|/=i; F(O (£' = constant)} . (3.49)

Assume that cj> satisfy the following

/•OO

/ \<f>(x, t)\dx < oo. (3.50)J-QO

(3.45)

(3.47)


Then when x ~> ±00, if

where a and b' satisfy the following equations

<*(£')<*(£') + & ' ( £ W ) = i,

a(O = K£')]*, (3-52)

and the continuous frequency spectrum of the scattering data is given by

F ( O = ^ , (3-53)

where £' is real, then from (3.46), (3.47),(3.52) and (3.53), the nonlinear Fouriertransformation for (j>{x, t) determined by (3.47) can be obtained and is given by

F{?,t) = - ^ [°° P(x,t)e-2*'*dx + O(b*). (3.54)n J-00

The lowest order in b of (3.54) gives the normal linear Fourier transformation.

3.3.3 Method of quantization

Similar to the linear quantum mechanics, wave function in the nonlinear quantummechanics can be quantized. The commonly used methods are the canonical quanti-zation method and the path integration method. However, because of the nonlineareffect, the actual procedure becomes much more complicated. The main steps forthe canonical quantization are outlined below.

Let V(r,t) = 0, A(<j>) = 0 and using the natural unit system in which h = m =c = 1, and replacing b by 2b, the nonlinear Schrodinger equation (3.2) becomes

ijt<j>{t,x) = -JLt(t,x) + 2W(t,x)4>(t,x)</>(t,x). (3.55)

The quantization is best perceived in the Fourier transform space. If one definesapproximately the Fourier transform

-L= f (f>(t,x)ei^dx = a(t,0),V , J7& (3.56)- = / a(t,P)e-^dl3 = cl>(t,x),

one obtains the equation of motion for the amplitude in Fourier transform space

i^-ta(t,p)=p2a(t,P)+2bjdp1d/32a*(t,p1)a(t,p2)a(t,/3 + /31-/32). (3.57)

(3.51)


The field envelope can be normalized so that it represents the microscopic particleat "time" t. This enables us to identify a(t, /?) with the microscopic particle anni-hilation operator at time t, a(t,fi), and a*(t,/3) with the creation operator a+(t,fi).On the right-hand side of (3.57), the first term represents the dispersion effect andthe second term represents the third-order nonlinearity. Note that the second termis in the form of a convolution. This is because for a broad-band field one has tointegrate over the Fourier-transform space. The quantization is accomplished byassignment of the commutation relations

[a(t,p'),a+(t,p)} = 6(p-p'),

[a(t,p'),a(t,P)\ = [a+(tj'),a+(t,p)] = 0 (3.58)

The quantized equation is

ig-ta(t,P) = P2a(t,/3) +2b dp1dp2a+(t,p1)a(t,p2)a(t,p +pt - ft). (3.59)

Equation (3.59) can be derived from a well-defined Hamiltonian. That is, one canwrite (3.59) as

ih~a{t,p) = [a{t,p),H] (3.60)at

with

H = h\fp2a+(t,0)a(t,0)d/3 (3.61)

+ b fa+(t,p)a+(t,p1)a(t,p2)a(t,p + p1 -ft)d,8dftdftl .

By defining new field operators as the inverse Fourier transforms of the annihi-lation and creation operators and applying the inverse Fourier transform to (3.59),one obtains the quantum nonlinear Schrodinger equation

i%-$(t,x) = --^j>(t,x) + 2b4>+(t,x)j>(t,x)4>(t,x). (3.62)at ox*

The operators <j>(t,x) and <p+(t, x) are the annihilation and creation operators of aquantum at a "point" x and "time" t.

From the definition of the Fourier transform (3.56) and the commutation rela-tions (3.57), it is easy to prove that the field operators satisfy the following com-mutation relations

[j>(t,x"),j>+(t,x)} = 8(x-x"),

[4>(t,x"),t(t,x)] = [$+(t,x"),i+(t,x)} = 0. (3.63)

With the help of (3.63), equation (3.62) can be written as

ihjtJ>(t,x)=[4>(t,x),H] (3.64)


with

6 = h\j4>+{t,x)^>x{t,x)dx + b f j>+{t,x)j>+(t,x)j>(t,x)j>{t,x)dx\ . (3.65)

The quantum nonlinear Schrodinger equation (3.62) is the operator evolutionequation of a nonlinear quantum system with the Hamiltonian (3.65). Since thequantum nonlinear Schrodinger equation can be derived from a Hamiltonian, it isa well-defined operator equation.

Equations (3.62) - (3.64) form the representation of nonlinear quantum mechan-ics in the Heisenberg picture and corresponds to the non-relativistic case. In theSchrodinger picture, the problem is stated in terms of the time evolution of a state|$) of the system

ih±\Z) = Ht\*), (3.66)

with

Hs=h\ffc(x)4>x(x)dx + b [ 4>+(x)4>+(x)j>(x)<j>(x)dx] (3.67)

where <j){x) and 4>+(x) are the field operators in the Schrodinger picture and satisfythe following commutation relations

[t(x"),j>+(x)]=6(x-x"),(3.68)

[t(x"),4>(x)\ = [4>+(x")J+(x)} = 0.In the Heisenberg picture the quantum nonlinear Schrodinger equation (3.62)

can be solved by quantum inverse-scattering method. In the Schrodinger picture,equation (3.66) could be solved from Bether's ansatz. Lai et al. expanded thequantum state in the Fock space and substituted it into (3.66). The result is awave-function equation that has many degrees of freedom (like the equations inmany-particle systems). However, for the quantum nonlinear Schrodinger equation,this wave-function equation is in a simple form and can be solved analytically.

Any quantum state of a system can be expanded in Fock space as follows

l$) = y > « I -Lfn(Xl,---,Xn,t)j>+(x1)--4+(xn)dx1---dxn\0). (3.69)

n J vn!

The state |$) is a superposition of states produced from the vacuum state by creat-ing particle at the points x\, X2, • • • ,xn with the weighting functions fn. Since theparticles are Bosons, /„ should be a symmetric function of Xj. We require on and/„ to satisfy the following normalization conditions

£M 2 = 1, (3.70)n


J\fn(xi,--- ,xn,t)\2dx1---dxn = l. (3.71)

Substituting (3.69) and (3.67) into (3.66) and using (3.68), Lai et al. obtained ane q u a t i o n for fn(xi, • • • , xn, t)

ijfn{xi,--- ,Xn,t)= ~ Z ) ^ 2 + 2b ] C S(XJ ~ Xi) fn(xU ••• ,Xn,t). (3.72)[ 3=1 •» l<»<i<n

This is the Schrodinger equation for a one dimensional Boson system with a 6-function interaction. The t-dependence in (3.72) can be factored out by assuminga solution of the form

fn(xi,--- ,xn,t) = fn(xu--- ,xn,t)e-iEnt. (3.73)

The equation for fn(xi, • • • ,xn) is

~ Yl 7T^ + 2b H *(si ~ **) fn(xu--- ,xn) = Enfn(x1,--- ,xn). (3.74)j=l °X0 \<i<j<n

This shows that the quantum nonlinear Schrodinger equation is equivalent tothe evolution equation of an one-dimensional Boson system with a <5-function in-teraction. It is surprising that the quantum nonlinear Schrodinger equation canbe solved exactly. It was first solved by Betpher's ansatz and then by quantuminverse-scattering method.

We now introduce the quantization approach used by Lee et al. Consider realquantum field <f>1 with TV-components. Its corresponding Lagrangian function den-sity can be written as

*- i i : (g ) ' ->w>) <v«where the parameter g plays a role of a coupling constant. Now assume that

# i M ) = V ( r , t , *!,•••,**) (3-76)

is a classical solution of a single microscopic particle (soliton) and the function</?*(r, t, z\, • • • ,Zk) satisfies the following equation

For the Sine-Gordon equation, (£,*) =4tan"1 [e^*-*)] .

(3.77)


The method of canonical quantization is to expand the soliton solution (3.77) interms of the classical soliton solution <p/g, i.e.,

1 °°4>i(r,t) = -tpi(r,zi,--- ,zk)+ ^ 1n(t)fi{r,zi,--- ,zk) (3.78)

^ n=k+l

where qn{t) satisfies the canonical commutative relation

[qi(x,t),qf{y,tj\ = S^x - y). (3.79)

The JV-component function tp%n satisfies the following constraint,

| ; | ^ g d r = 0. (3.80)

Under orthogonal condition, we have

N .

5 ] \ipWn,dT = bnn,. (3.81)

After some derivations, we can get

iSkPktj,T + I Pkdzk-22[Ni + 2)OJn = 2 7 r n '

or/•MT ( 1 f 1 ^ ^ 1 "1

J ^ ) + - k - 2 E w « W \dzk=2Tm, (3.82)

where N[ is an occupation number and N't = 0,1,2, • • •, wn is a single vibrationalfrequency of static classical soliton solution tp(r, z\, • • • ,zk), under the approxima-tion of small oscillation. If we further assume that \p) is the eigenstate of bothenergy and momentum, that is

P\P)=P\P), H\p) = E\p),

where E{p) = s/p2 + M2. Then the nonlinear relation of the matrix element of thecanonical commutation relation for the quantum field <f> may be written as

(p|[0(i, t), <f>(y, t)]\p') = iS(x - y)27rS(p - p'). (3.83)

The above is the collective coordinate method of canonically quantizing theclassical soliton solution proposed by Lee, et al. The key in this approach is to findthe classical soliton solution. However, this method is only suitable to the case of anonlinear scalar field <f> with its internal symmetry G being the Abel group.

Prom the above discussion we see that the even though the two quantizationmethods are somewhat different, they all quantize essentially the wave function ofthe particle or the solution of the dynamic equation.


3.3.4 Nonlinear perturbation theory

Most dynamic equations in nonlinear quantum mechanics cannot be solved analyti-cally. Therefore perturbative approach is often used to solve these equations, whencertain terms are small relatively to the nonlinear energy or the dispersive energy inthese equations. In contrast to the linear quantum mechanics, a systematic pertur-bative approach and a universal formula are impossible to obtain in the nonlinearquantum mechanics due to nonlinear interactions. As a result, different perturba-tion methods exist in the nonlinear quantum mechanics which will be discussed inChapter 7 in details. In this section, one of such methods is briefly described.

Assume that 4>o (r, t) is the soliton solution of the nonlinear dynamic equation ofa unperturbed system. The wave function of the perturbed system can be writtenas

$ = <p0 + £ ( / > ! + e 2 f a + ••• , (3.84)

where e is a small quantity. Substituting (3.84) into the original nonlinear equa-tion, equations of fa, fa, • • • corresponding to different orders of e can be obtained,respectively. Thus fa, fa, • • • can be determined by solving these equations, and 4>is obtained from (3.84).

For example, the Sine-Gordon equation:

<Axx - 4>tt + s in <j) + Y4>t = A j , (3.85)

may be solved using the perturbation method when T and Ax are small. Its solutionis given by

</>(x,t) = 4>0(x - vt) + fa{x,t) (3.86)

where <j>i(x,t) is a small quantity. Equation for 4>\ c a n D e obtained by substituting(3.86) into (3.85),

fa,tt ~ <t>i,xx - (1 - 2sech2z) </>i + "fTfat - ^fax = AX+ 2/37rsechx. (3.87)

To solve the above equation, we expand fa(x,t) using the complete set{fb(x)Jk(x)}

rOO

0 i ( M ) - <hb(t)M*) + / dk4>x{k,t)fk(x) (3.88)J — OO

where

fb(x) = -^sechx,v2

(3.89)1 eikx

fk(x) = -y=-^-(k + itajihx).


The solutions can be found and are given by

'fr(M)= ***(*>* ,U V ^ n | sinh(7TJfc/2)

(3.90)

16(t) = 2v/2^+^)(l-e-^).

Thus, the complete solution is

fa (x,t) = 1R sin[2i?(a; - x)]e-2iqx-2i<T' (3.91)

where

Rt = 2172 - y sech ( | | ) sin x,

9t = —sech(- jS i n a ; .

We have given a very brief account of the four basic components of the nonlin-ear quantum mechanics. In the following chapters, these basic theories and theirapplications will be discussed in more details. Using these principles and theo-ries, nonlinear quantum mechanical problems can be studied. But applications toreal systems require detailed information of the nonlinear systems which will bediscussed in the next section.

3.4 Properties of Nonlinear Quantum-Mechanical Systems

In the last section we described the fundamental principles of the nonlinear quantummechanics which are used to describe nonlinear quantum systems. However, whatare nonlinearly quantum-mechanical systems? What properties do these systemshave? We will look into these in this section.

(1) The nonlinear quantum mechanics describes Hamiltonian systems. That is,behaviors of the systems can be determined by a set of canonical conjugate variables.Using these variables one can determine the Poisson bracket and write the equationsin the form of Hamilton's equations. For equation (3.2) with V(f,t) = A((j)) = 0,the variables cj> and 4>* satisfy the Poisson bracket

{â)(x),rW(y)} - iSabS(x - y) (3.92)

where


The corresponding Lagrangian density C associated with (3.2) with A(<j>) = 0 canbe written in terms of </>(z, t) and its conjugate <j>* viewed as independent variables

C = j(4>*4>t - H*t) - ^ ( V 0 • V<£*) - V{x)P4> + b{<t>* • <fr)<t>. (3.93)

The action of the system can be written as

S{4>,<j>*) = I' [ Cdxdt. (3.94)Jtb JD

and its variation for infinitesimal 6<f> and 5<f>*

6S = S{cj> + 6<p,<p* +6<p*}-S {</>,<!>*} (3.95)

can be written as

6s= f f [ ^ + - ^ v ^ + i £ t 6 A d x d t + c c - (3-96)Jto JD L d "9t J

where d£/d(Vcf>) denotes the vector with components dC/d{di(j)) (i — 1,2,3). Afterintegrating by parts, we get

»-n[s-v-(&)-*(£)MK+- <M7)A necessary and sufficient condition for a function cj)(x, t) with known values

4>(x, to) and (f>(x, ti) to yield an extremum of the action 5 is that it must satisfy theEuler-Lagrange equation

Equation (3.98) gives the nonlinear Schrodinger equation (3.2) if the Lagrangian(3.93) is used. Therefore, the dynamic equation, or the nonlinear Schrodinger equa-tion, in the nonlinear quantum mechanics can be derived from the Euler-Lagrangeequation, if the Lagrangian function of the system is known. This is differentfrom the linear quantum mechanics, in which a dynamic equation, or the linearSchrodinger equation cannot be obtained from the Euler-Lagrange equation. Thisis a unique property of the nonlinear quantum mechanics.

The above derivation for the nonlinear Schrodinger equation based on the vari-ational principle is a foundation for other methods such as the "collective coordi-nates" , the "variational approach", and the "Rayleigh-Ritz optimization principle",where a solution is assumed to maintain a prescribed approximate profile (oftenbell-type). Such methods greatly simplify the problem, reducing it to a systemof ordinary differential equations for the evolution of a few characteristics of thesystems.

(3.98)


The Hamiltonian density corresponding to (3.2) or (3.93) can be written as

•H = l4(<t>*dtcf> - <f>dt<t>*) -C = | - ( V 0 • V < n + V(x)<f>*<j> - b{<j>*cj>)2. ( 3 .99 )2 2m

Introducing the canonical variables

1 , , , , . dC

(3.100)1 , , . . . dC

the Hamiltonian density takes the form

n = Y,PidtQi-£ (3-101)

i

and the corresponding variation of the Lagrangian density can be written as

6C = H ir5qi + ir£~^Vqi) + ATSTT*^*)-

(3-102)

*-? oqi o(Vqi) 0{Otqi)Prom (3.102), the definition of pi, and the Euler-Lagrange equation

% d(Vqi)

one obtains the variation of the Hamiltonian in the form

SH = ^ [(dtqidpi - dtpi6qi)dx. (3.103)

Thus the Hamilton equations can be derived

dqi_6ji dVi_jE_dt - sPi' dt - sqi

[6-im)

or in complex form

*a*=!p (3-105)

This is also interesting. It shows that dynamic equations, such as the nonlinearSchrodinger equation, can also be obtained from the Hamilton equation, if theHamiltonian of the system is known. Obviously, such method of finding dynamicequations is impossible in the linear quantum mechanics.

The Euler-Lagrange equation and the Hamilton equations are important equa-tions in classical theoretical (analytic) mechanics, they were used to describe mo-tions of classical particles. Now these equations are used to depict motions ofmicroscopic particles in the nonlinear quantum mechanics. This shows clearly theclassical nature of microscopic particles in the nonlinear quantum mechanics. From


these, we obtain new ways of finding the equation of motion of microscopic parti-cles in the nonlinear quantum mechanics. If the Hamiltonian or the Lagrangian of asystem is known in the coordinate representation, then we can obtain the equationof motion from the Euler-Lagrange or Hamilton equations.

(2) The nonlinear dynamic equations such as the nonlinear Schrodinger equationcan be written either in the Lax form, given in (3.20) and (3.21), or in the Hamiltonform of a compatibility condition for overdetermined linear spectral problem. Gen-eral nonlinear equations with soliton solutions are often expressed as (3.8), whereK(<j>) is defined as a nonlinear operator in some suitable function space. If linearoperators L and B which depend on solution <f> satisfy the Lax operator equation(3.20), where B is a self-adjoint operator, the eigenvalue A and eigenfunction ipof operator L may be derived from (3.21). Thus, eigenvectors and eigenvalues ofnonlinear dynamic equations can be determined by the eigenvectors and eigenval-ues of linear operators. Then, A is a time-independent eigenvalue. Equation (3.21)is the corresponding linear eigenequation of the nonlinear equation (3.8). For thenonlinear Schrodinger equation (3.2) with V{f) = A(<f>) = 0, the operators L andB are of the forms of (3.22).

We now assume that the eigenfunction of the operator L satisfies the followingequation

ii/>v=Bil> + f(L)il> (3.106)

where the function f(L) may be chosen according to convenience. In this case wecan write the overdetermined system of linear matrix equations as

t/y =[/(*',*' , A) V,(3.107)

rPt.=V'{x',t',\)rl>,

where U and V are 2x2 matrices, t' = t/h, x' = (y/2m/h)x. The compatibilitycondition of this system is obtained by differentiating the first equation of (3.107)with respect to t' and the second one with respect to x' and then subtracting onefrom the other

Ut.-Vj.,-[U,V'] = 0. (3.108)

We emphasize that the operators U and V depend not only on t' and x' butalso on some parameter, A, which is called a spectral parameter. Condition (3.108)


must be satisfied by A. In this case the operators U and V have the form

—[%5VV.]+U)V(1T1.

+2i\ , (l + s ) ( l - s 2 ) 1-S2 1 + s

L ( l - S ) ( l - S 2 ) J V l - S 1 - S 2 /

The presence of a continuous time-independent parameter is a reflection of thefact that the nonlinear Schrodinger equation (3.2) with V(f, t) = A((f>) — 0 describesa Hamiltonian system with a set of infinite number of conservation laws, which willbe discussed in chapters 4. In the case of a system with a finite number (N1) ofdegrees of freedom one can succeed sometimes in finding 2N' first integrals of motionbetween which the Poisson brackets are zero (they are said to be in involution).Such a system is called completely integrable. In such a case, equation (3.108)with (3.109) is referred to as the compatibility condition, and its consequence isthe nonlinear Schrodinger equation (3.2) with V(x,t) = A((p) = 0. This meansthat for each solution, <f>(x',t'), of (3.2), there is always a set of basis functionip, parameterized by A, which can be obtained through solving the set of linearequations (3.107) - (3.109). Therefore, the nonlinear quantum systems describedby the nonlinear quantum mechanics is completely integrable. This concept wasgeneralized by Zakharov and Faddeev in quantum field theory.

(3) The systems described by the above equations have infinite countable (orin the case of the internal symmetry, sets of) conservation laws (local or non-local)and integrals of motion.

We now write the wave function in terms of the phase and amplitude, cj> = \fpz%k'•This transformation is usually called Madelung's transformation. After substitu-tion in the nonlinear Schrodinger equation (3.2) with V(x,t) = 0 and A{(f>) = 0and separation of the real and imaginary parts of the equation, one obtains (afterrescaling the time by a factor of 2)

pt + V • (pV0) = 0, or pt = -V-j, J=pV0, (3.110)

and

0t + ^|W|2-^=^=Av^. (3.111)

Taking p = |^|2 = |y>|2 as a density and 6 as an hydrodynamic potential, equations(3.110) - (3.111) with b < 0 can be identified with the equations for an irrotationalbarotropic gas. Hence, (3.110) is a continuum equation that occurs in macrofluidhydrodynamics. Thus, equation (3.110) is also called the fluid-dynamical form of

(3.109)


the nonlinear Schrodinger equation. This shows the features of classical particlesof the microscopic particle described by the nonlinear Schrodinger equation in thenonlinear quantum mechanics. Prom this interpretation of p — ]<$*• we know clearlythat the meaning of |</>|2 = |< |2 is in truth far from the concept of probability inthe linear quantum mechanics. It essentially represents the mass density of themicroscopic particles in the nonlinear quantum mechanics. If (3.110) is integratedover x one finds the integral of motion N = J^ p(x, t)dx with j ->• 0 when x ->•±oo. N is then the mass of the microscopic particle, and (3.110) is nothing butthe mass conservation for the microscopic particle. This discussion enhances ourunderstanding on the meaning of wave function of the microscopic particle, 4>(f, t),in nonlinear quantum mechanics.

(4) In some cases, the inverse scattering problem for the operator L, in (3.20),can be solved, and a potential can be obtained from the scattering data. The roleof the potential is played through the function <\>. This means that for the equationsdiscussed one can solve the Cauchy problem and the behavior of the integrablesystem is strictly determinate. The localized solutions of the integrable equationswhich correspond to the discrete spectrum of the operator L are usually related tothe solitons. In the case of the simplest integrable systems the soliton dynamics istrivial.

(5) For integrable systems, an exact quantum approach can be developed thatallows one to determine the ground state and excitation spectra of the system.This was studied by Faddeev et al. and Thacker et al.. It thus connects classicaldescription and quantum objects with sufficient rigor. For integrable equations, onemay construct a perturbation theory and investigate the structural and "initial"stability problems. In the first case the equation itself is weakly perturbed and theperturbation can be non-Hamiltonian. In the second case the initial state studied (inparticular it may be a one-soliton solution) is weakly perturbed. For Hamiltoniansystems with a finite number of degrees of freedom there is a rigorous theory ofstructural stability. These problems were studied in details in the literature.

Bibliography

Ablowitz, M. J., Kaup, D. J. Newell, A. C. and Segur, H. (1973). Phys. Rev. Lett. 301462; ibid 31 125.

Ablowitz, M. J., Kaup, D. J. Newell, A. C. and Segur, H. (1974). Stud. Appl. Math. 53249.

Alfinito, E., et al. (1996). Nonlinear physics, theory and experiment, World Scientific,Singapore.

Bullough, R. K. and Caudrey, P. J. (1980). Solitons, Springer-Verlag, Berlin.Burt, P. B. (1981). Quantum mechanics and nonlinear waves, Harwood Academic Pub-

lishers, New York.Calogero, F. (1978). Nonlinear evolution equations solvable by the spectral transform,

London, Plenum Press.


Calogero, F. and Degasperis, A. (1976). Nuovo Cimento 32B 201.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1996). Chin. Phys. Lett. 13 660.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1997). Chin. J. Atom. Mol. Phys. 14 393; Chin.

J. Chem. Phys. 10 145.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1998). Acta Phys. Sin. 7 329; J. Sichuan

University (nature) Sin. 35 362.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1999). Acta Phys. Sin. 8 1313; Chin. J. Chem.

Phys. 11 240; Chin. J. Comput. Phys. 16 346; Commun. Theor. Phys. 31 169.Christ, N. H. (1976). Phys. Rep. C 23 294.Doebner, H. D., et al. (1995). Nonlinear deformed and irreversible quantum systems, World

Scientific, Singapore.Doebner, H. D., Manko, V. I. and Scherer, W. (2000). Phys. Lett. A 268 17.Faddeev, L. (1979). JINR D2-12462, Dubna.Faddeev, L. D. and Takhtajan, L. A. (1987). Hamiltonian methods in the theory of solitons,

Springer, Berlin.Friedberg, R., Lee, T. D. and Sirlin, A. (1976). Phys. Rev. D 13 2739.Gisin, N. (1990). Phys. Lett. A 143 1.Gisin, N. and Gisin, B. (1999). Phys. Lett. A 260 323.Gisin, N. and Rigo, M. (1995). J. Phys. A 8 7375.Jordan, T. F. and Sariyianni, Z. E. (1999). Phys. Lett. A 263 263.Karpman, K. I. (1979). Phys. Lett. A 71 163.Konopelchenko, B. G., (1979). Phys. Lett. A 74 189.Konopelchenko, B. G., (1981). Phys. Lett. B 100 254.Konopelchenko, B. G., (1982). Phys. Lett. A 87 445.Konopelchenko, B. G. (1987). Nonlinear integrable equations, Springer, Berlin.Konopelchenko, B. G. (1995). Solitons in multidimensions, World Scientific, Singapore.Lai, Y. and Haus, H. A. (1989). Phys. Rev. A 40 844 and 854.Lee, T. D., et al. (1975). Phys. Rev. D 12 1606.Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1.Mielnik, B. (2001). Phys. Lett. A 289 1.Miura, R. M. (1976). Backlund transformations and the inverse scattering method, solitons

and their applications, Springer, Berlin.Pang, X. F. and Chen, X. R. (2000). Chin. Phys. 9 108.Pang, X. F. and Chen, X. R. (2001). Commun. Theor. Phys. 35 323.Pang, X. F. and Chen, X. R. (2001). Inter. J. Infr. Mill. Waves 22 291.Pang, X. F. and Chen, X. R. (2001). J. Phys. Chem. Solids 62 793.Pang, X. F. and Chen, X. R. (2002). Commun. Theor. Phys. 37 715.Pang, X. F. and Chen, X. R. (2002). Inter. J. Infr. Mill. Waves 23 375.Pang, X. F. and Chen, X. R. (2002). Phys. Stat. Sol. (b) 229 1397.Pang, Xiao-feng (1983). Physica Bulletin Sin. 5 6.Pang, Xiao-feng (1984). Principle and theory of nonlinear quantum mechanics, Proc. 3rd

National Conf. on Quantum Mechanics, p. 27.Pang, Xiao-feng (1985). Chin. J. Potential Science 5 16.Pang, Xiao-feng (1985). J. Low Temp. Phys. 58 334.Pang, Xiao-feng (1985). Problems of nonlinear quantum theory, Sichuan Normal Univ.

Press, Chengdu.Pang, Xiao-feng (1990). The elementary principle and theory for nonlinear quantum me-

chanics, Proc. 4th APPC (Soul Korea) p. 213.Pang, Xiao-feng (1991). The theory of nonlinear quantum mechanics, Proc. ICIPES, Bei-

jing, p. 123.


Pang, Xiao-feng (1992). Proc. IWPM-6 (Shenyang), p. 83.Pang, Xiao-feng (1993). The theory of nonlinear quantum mechanics, in research of new

sciences, eds. Lui Hong, Chin. Science and Tech. Press, Beijing, p. 16.Pang, Xiao-feng (1994). Acta Phys. Sin. 43 1987.Pang, Xiao-feng (1994). The theory for nonlinear quantum mechanics, Chongqing Press,

Chongqing.Pang, Xiao-feng (1995). Chin. J. Phys. Chem. 12 1062.Pang, Xiao-feng (1995). J. Huanghuai Sin. 11 21.Pang, Xiao-feng (1997). J. Southwest Inst. Nationalities, Sin. 23 418.Pang, Xiao-feng (1998). J. Southwest Inst. Nationalities, Sin. 24 310.Pang, Xiao-feng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu.Sabatier, P. C. (1990). Inverse method in action, Springer, Berlin.Sulem, C and Sulem, P. L. (1999). The nonlinear Schrodinger equation: self-focusing and

wave collapse, Springer-Verlag, Berlin.Toda, M. (1989). Nonlinear waves and solitons, Kluwer Academic Publishers, Dordrecht.Weinberg, S. (1989). Ann. Phys. (N.Y.) 194 336.Weinberg, S. (1989). Phys. Rev. Lett. 62 485.Wright, E. M. (1991). Phys. Rev. A 43 3836.Zakharov, B. E. and Shabat, A. B. (1971). Zh. Eksp. Teor. Fiz. 61 118.Zakharov, B. E. and Shabat, A. B. (1972). Sov. Phys. JETP 34 62.Zakharov, V. and Fadeleev, I. (1971). Funct. Analiz 5 18.

Chapter 4

Wave-Corpuscle Duality of MicroscopicParticles in Nonlinear Quantum

Mechanics

In this chapter, we will look into physical properties of microscopic particles whichare described by the nonlinear quantum mechanics. In particular, we focus onthe wave-corpuscle duality of microscopic particles and its description in nonlinearquantum mechanics. Whether microscopic particles have wave-corpuscle dualityis a key and central issue of quantum mechanics. Unless the nonlinear quantummechanics can give a correct description for the wave-corpuscle duality of micro-scopic particles, it cannot be accepted as a proper theory and a different theoryfrom the linear quantum mechanics which only describes the wave feature of mi-croscopic particles properly. Therefore, in this chapter we present an extensive andin-depth study on this feature of microscopic particles, from their static state todynamic properties, based on the fundamental principles of the nonlinear quantummechanics. Hence, this chapter is the center of this book.

In accordance with the significance of the corpuscle feature of microscopic parti-cles, we focus our discussion on the following issues: (1) the mass, momentum andenergy of microscopic particles and the corresponding conservation laws; (2) thesize, position and velocity of microscopic particles and the laws of motion of micro-scopic particles in free space, in the presence of an external field, and in a viscousenvironment; (3) interaction and collision between microscopic particles, or betweenmicroscopic particles and other particles or objects; (4) the stability of the micro-scopic particles under external perturbations; and so on. Prom these discussions,we will see clearly that microscopic particles in the nonlinear quantum mechanicsbecome solitons, which have properties of a classical particle. It can move over amacroscopic distance, retaining its shape, energy, momentum, and remains stableafter collisions and interactions with other particles or external fields.

We will also examine the wave features of a microscopic particle, including itswavevector, velocity, amplitude, the superposition principle of waves, scattering,diffraction, reflection, transmission and tunneling phenomena. We will see that eachmicroscopic particle has a determinate wave vector, amplitude and width, and obeysthe physical laws of scattering, diffraction, reflection, transmission, tunneling, etc.Therefore, the microscopic particles has not only corpuscle but also wave features,i.e., it has evident wave-corpuscle duality in the nonlinear quantum mechanics.

109


4.1 Invariance and Conservation Laws, Mass, Momentum and En-ergy of Microscopic Particles in the Nonlinear Quantum Me-chanics

We have learned in earlier chapters some conservation laws in a system of micro-scopic particles which are described by the nonlinear Schrodinger equation in thenonlinear quantum mechanics. In practice, these conservation laws are related tothe invariance of the action with respect to groups of transformations through theNoether theorem, which was studied by Gelfand and Fomin (see Sulem and Sulem1999) and Bluman and Kumei (1989) (See Sulem and Sulem, 1999 and referencestherein). Therefore, before these conservation laws are given, we first discuss theNoether theorem for the nonlinear Schrodinger equation which was given by Sulem.

To simplify the equation, we introduce the following notations: £ = (t, x) —( l o . l i , • • • ,ia), do = dt,d= (do,du- • • ,dd) and $ = ( $ ! , $ 2 ) = (cf>, f). According

to the Lagrangian (3.93) corresponding to the nonlinear Schrodinger equation, theaction of the system

5{</>} = / " f C{<t>,V<f>,<f>t,p,V<l>\ti)dxdtJto J

now becomes

5 W = f /°°£($,a$K. (4.1)JD Jx'

Under the transformation T£ which depends on the parameter e, we have£ -+ £'(£, $, e), $ -> $(£,$, e), where | and <l are assumed to be differentiate withrespect to e. When e = 0, the transformation reduces to the identity. For infinites-imally small e, we have £ ' = £ + £ £ , $ = $ + J$. At the same time, $(f) -»• $(£')by the transformation TE, and the domain of integration D is transformed into D,

S{0}->5{£}= / / C(*,d*)d£',JD JX'

where d denotes differentiation with respect to £'. The change 5S = S{(p} — S{(j)}in the limit of e under the above transformation can be expressed as

6S = ID II [£ (^} " £^'9$)] d$ + fD /" A*,^) E ^fdtwhere we used the Jacobian expansion

d&>--->£d) =1 , y - ^ f rd(io,---,id) ôdiv'

and £(#,5$) in the second term on the right-hand side has been replaced by theleading term £($,9$) in the expansion.

Wave-Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics 111

Now define

(4.2)

BMi) - 9VHO = 0v - dv)*id)+ev[hd) - miwith

* = §*,= (^ + U = a. + KWe then have

= a*;6**+ww ~cî+dv [wm s*{ ~d" [m*T)\6**-

Equation (3.96) can now be replaced by

where we have used

Using the Euler-Lagrange equation, the first term on the right-hand side in theequation of SS vanishes. We can get the Noether theorem.

(I) if the action (4.1) is invariant under the infinitesimal transformation of the de-pendent and independent variables <f> -> <f>+6(f>, f -> £+6£ where f = (t, x\, • • • , Xd),the following conservation law holds

or

4 r I £ * + s o ^ ( * ' - ^ ' 6 ' ) 1 - 0 (4-3)in terms of 6$i defined above.


(II) If the action is invariant under the infinitesimal transformation

t-¥ i=t + St(x,t,<j>),

x -> x = x 4- Sx(x, t, cf>),

<P(x, t) -> 4>(t, x) = <f>(t, x) + 6<j)(t, x),

then

f \^(dt<j>dt + Vcf>-5x- S<j>) + ^{dt<f>*dt + V<f>* • Sx - 6<j>*) - £St] dxJ [d(t>t d(f)*t \

is a conserved quantity.For the nonlinear Schrodinger equation in (3.2) with A((j>) — 0, we have

dC i dC iWt=2^' a n d Wt^'^'

where C is given in (3.93). Several conservation laws and invariances can be obtainedfrom the Noether theorem.

(1) Invariance under time translation and energy conservation law.

The action (4.1) is invariant under the infinitesimal time translation t -> t + Stwith Sx = 6(f> = 8(f>* = 0, then equation (4.3) becomes

dt [V<A• v f - b-{<t>*<t>? + v{x,t)r<A - v • (<£tvr + ptv<p) = o.

This results in the conservation of energy

E= f \v<t> • V^* - (</>*</>)2 + V(s : , t )0v | dx = constant. (4.4)

(2) Invariance of phase shift or gauge invariance and mass conservation law.

It is very clear that the action related to the nonlinear Schrodinger equation isinvariant under the phase shift <j> = elB4>, which for infinitesimal 6 gives 6tfi = iO<j>,with 5t = Sx = 0. In this case, equation (4.3) becomes

dt\<t>\2 + V • {*(W* - <pV0)} = 0. (4.5)

This results in the conservation of mass or number of particles.

N = / \<j)\2dx = constant

and the continuum equation

dt h

Wave- Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics 113

where j is the mass current density

J=-i(0V0"-0*V0.

Equation (4.5) is the same as (3.110).

(3) Invariance of space translation and momentum conservation law.

If the action is invariant under an infinitesimal space translation x -t x + 5xwith St = 5<j> = 8cf>* = 0, then (4.3) becomes

& [ i ( W - <f>*V<j>)] + V • {2(V<£* x Vcj> + V(f> x Vcj>* + C)] = 0.

This leads to the conservation of momentum

P = i (</>V<£* - <j>*V4>)dx = constant. (4.6)

Note that the center of mass of the microscopic particles is defined by

(x) = ^jx\^dx,

we then have

A T ^ M = f xdt\<j>\2dx = - f xV • [t(0V0* - <f>*V<p)]dx

(4.7)

= / i(cj)V(t>* - <j>*V<t>)dx = P = -J= - fjdx.

This is the definition of momentum in classical mechanics. It shows clearly thatthe microscopic particles described by the nonlinear Schrodinger equation has thefeature of classical particles.

(4) Invariance under space rotation and angular momentum conservation law.

If the action (4.1) is invariant under a rotation of angle 58 around an axis / suchthat St = 8<j> = 6(f>* = 0 and Sx = 561 x x, this leads to the conservation of theangular momentum

M = i I f x (4>*V<t> - 4H4>*)dx.

Besides the above, Sulem also derived other invariance of the nonlinearSchrodinger equation from the Noether theorem.

(5) Galilean Invariance


If the action is invariant under the Galilean transformation

x -» x" = x — vt,

* - > * " = t,

<f>(x,t) -4 cf>'\x",t") = -i ^-vx" + \v-vt"]^ 4>{x,t),

then the nonlinear Schrodinger equation can also retain its invariance. For aninfinitesimal velocity v, 5x" = -vt, St = 0 and 5<j> = <f>"(x",t") — <f>(x,t) =

~(i/2)vx(f>(x,t). After integration over the space variables, equation (4.3) leadsto the conservation law (4.7) which implies that the velocity of the center of massof the microscopic particle is a constant.

(6) Scale invariance of power law nonlinearities in the nonlinear Schrodingerequation with V(x, t) = A((f>) = 0

If \(j)\2 in the nonlinear term, b\<j>\2()), in the nonlinear Schrodinger equation (3.2)is replaced by 6| >|2<T, where a is a constant, then the nonlinear Schrodinger equationis invariant under the scale transformation

<f>(x,t) -> <j>"(x",t") = X'V'Wx",*2?).

But the action scales as \l2/<'~1ld under the same transformation, where d is thesize of the system. In the critical case, a = 2d, the action is invariant under thescale transformation while the nonlinear Schrodinger equation remains invariant.a = 2d also happens to be the critical condition for a singular solution and isusually referred to as the critical dimension. In such a case, the invariance underthe scale transformation leads to an additional conservation law. For an infinitesimaltransformation, A' = 1 — e, with |e| < 1, fe = ex, St = 2et and 6(f> = —(e/a)<j>(x,t).Then (4.3) results in

f UetjC - </>* (^<t> + ex-V<j> + 2e*&) + %-<j> (^4>* + eS • Vcf>* + 2 e t # ) j dx = eCi

or

i ( x • {<j)V(t>* - 4>*V4>)dx - AEt = 2Ci

where C\ is a constant, E is the energy of the system.

(7) Pseudo-conformal invariance at critical dimension


Talanov (see Sulem and Sulem 1999) proved that when the nonlinearity is apower law b\(f>\2(f> at the critical dimension a = 2d, the nonlinear Schrodinger equa-tion has other additional symmetry. This invariance is given here.

Obviously, the nonlinear Schrodinger equation (3.2) with V(x, t) = A{<f>) = 0 atthe critical dimension is invariant by the pseudo-conformal transformation:

l(t)' Jo I2(s) t*(t*-t)'

<P(x,t) -> 0"(x",t") = Id/2<j>(x,t) • e-il^lil = Id/2(j,(x,t)eia^2^12

withdl dl _t* -t

a~ dt~~ldt'~ t20 '

where to appears as an arbitrary time unit. This shows that there exists certainsymmetry in the explicit solution of the nonlinear Schrodinger equation at thecritical dimension which is singular at some finite time t*. It is referred to aspseudo-conformal invariance of the nonlinear Schrodinger equation at the criticaldimension.

Expressing </>" in terms of <f>, the transformed Hamiltonian

H1 = J [ (VT) 2 - YÎ<A"I2 C T + 2] dx,where V" corresponds to x", becomes

H' = j Q/V* + if*|2 - ~l\<t>\2aA dx.

We can show that the energy corresponding to the Hamiltonian is conserved. Infact, choosing t* = 0, it yields the conservation of

C2= f (\x<t> + 2i*V</>|2 - ~^\<f>\2°+2) dx.

Kuznetsov and Turitsyn (1985) (see Sulem and Sulem 1999) pointed out thatthe above transformation preserves the action, and the invariance given by theabove equation is a result of Noether theorem. This can be seen by considering atransformation close to identity by choosing to = t*, and by assuming that 8A' =1/t* is infinitesimally small. In such a case Sx = xtSX', St = t26X', and 5<j> =(—dt/2 + i\x\2/A)(j>8X'. Upon evaluation of the space integration, equation (4.3)leads to the conservation of

I [t2 (-|V«/>|2 + ^ J - l ^ + 2 ) + l-t2-(W - <A'V«/,) - ^|£|2M2] dx = C2.

Kuznetsov and Turitsyn (see Sulem and Sulem 1999) further proved that thepseudo-conformal transformation also holds for the two dimensional nonlinear


Schrodinger equation (in the natural unit system)

where t' = t/h, x' = (V2m/h)x, y' = (y/2rn/ti)y.It is well known from classical physics that the invariance and conservation laws

of mass, energy and momentum and angular momentum are some fundamental anduniversal laws of matters in nature, including classical particles. In this section wedemonstrated that the microscopic particles described by the nonlinear Schrodingerequation in the nonlinear quantum mechanics also have such properties. This showsthat the microscopic particles in the nonlinear quantum mechanics also have corpus-cle feature. Therefore the proposed nonlinear quantum mechanical theory reflectsthe common rules of motions of matters in nature.

Fig. 4.1 Non-topological solitary wave of nonlinear Schrodinger equation and its frequency spec-trum

In the one-dimensional case with V(x, t) = A(<j>) = 0, Zakharov et al. found thesolution of (3.2) using the inverse scattering method. The solution is a bell-typenon-topological soliton

4>(x,t) = Aosech | ^ ^ [(x -xo)-vet}\ e«-.-.[(-..)-.*]/>» (4.8)

where

I{v2e-2vcve)m

A°-\l 26 •Therefore the microscopic particle in nonlinear quantum mechanics is a soliton.Here ip(x,t) = Ao sech.{Aoy/bm[(x - x0) - vet]/h} is the envelope of the soliton,and exp{ivem(x — x0) - vet)]/2h} is its carrier wave. The form of the soliton isshown in Fig. 4.1. The envelope ip(x,t) is a slow varying function and Ao is itsamplitude. The width of the soliton is given by W = 2Trh/(y/mbA0). Thus the sizeof the soliton, A0W = 2nh/Vmb, is a constant. It depends only on the nature ofthe microscopic particles, but not on its velocity. This shows the corpuscle featureof the microscopic particles. In (4.8), ve is the group velocity of the soliton, and


vc is the phase speed of the carrier wave. From Fig. 4.1 and (4.8), we see thatthe carrier wave carries the envelope to propagate in space-time. The envelopeis typical of a solitary wave, its changes in the space-time are slow and its formand amplitude are invariant in the transport. Thus the frequency spectrum of theenvelope has a localized structure around the carrier frequency wo, as shown inFig. 4.1(b). This figure shows the width of the frequency spectrum of the envelopeip(x,t). For a certain system, ve and vc and the size of the soliton are determinateand do not change with time. This shows that the soliton (4.8) or the microscopicparticle moves freely with a uniform velocity in space-time, and it has certain mass,momentum and energy, which are given by

p= -i ((t>*</>x- - (j>(f>*x.) dx' = 2V2Aove = ^-y=Nsve = const. • ve,J-oo 2 v 2

/

oo r hi 1

\\<l>x-\2 - l\\4\dx' = Eo + -Msoiv2

e,where Msoi = Na = 2\/2Ao is the effective mass of the microscopic particles andit is a constant. These results clearly show that the energy, mass and momentumof a microscopic particle are invariant in nonlinear quantum mechanics and theyare not dispersed in its free motion. Thus the microscopic particles can moveover a macroscopic distance with the group velocity ve and retain its energy andmomentum. It shows that the microscopic particle has corpuscle characteristics likea classical particle.

4.2 Position of Microscopic Particles and Law of Motion

Besides mass, momentum and energy of microscopic particles mentioned above,position of a microscopic particle can also be precisely determined in nonlinearquantum mechanics. We know that the solution (4.8) shown in Fig. 4.1, of thenonlinear Schrodinger equation (3.2) with V(x',t') = A((j>) = 0, has the behavior

*+- - 0x —x0

Thus we can infer that the position of the soliton is x' = x'o at t' = 0. Next, at\x'\ —> co, <p(x',t') —> 0, this means that

Q /-OO r

— / <p*(j)dx' = 0, or / p(x')dx' — constant."t J-oo J

which is the mass conservation of the microscopic particle. Therefore, (fffidx' =p(x')dx' can be regarded as the mass in the interval of x' to x' + dx. In the light


of definition in (4.7), the center of mass of the microscopic particle at x'o + vet' isgiven by

/•OO

/ <f>*x'4>dx'(x1) = x'g = ^ (4.9)

/ <j>*<j>dx'J-oo

where x' = x/y/h2/2m, t' — t/h. We can thus obtain the law of motion for thecenter of mass of the microscopic particles described by (3.2) with A((f>) = 0 and itsconjugate equation. Under such a condition, we have

d f°° f°° r°° 8— <P4>t,dxr= <t>*f^dx'+ P^tt'dx'a t J — oo J — oo J-oo "x

= i { I (4>*<Px'x'x> - (f>X'<t>x'x')dx'+ (4.10)(J — oo

/

°° f°° BV 1

with/•OO /-OO

/ {fj>*<}>x'x'x' - Î'x'^x^dx' = 4>*4>x>x'\°ôo~ I 4>*xi<j>x'x>dx'J—oo J—oo

-K'^x'\-oo + f°° Px'<Px'x>dx' = 0, (4.11)J — oo

/

OO -I pOO / QL \

[b4>*4>2^ + W ) V * ' ] dx = - ^ J W*)2 [j&) dx' = °. (4-12)where b is constant. Using (4.11) and (4.12), equation (4.10) becomes

*L* t *" f c l = ' / .*•»**'• (4-13)From (4.7) and (4.9), the acceleration of the center of mass of the microscopic

particle can be defined as

/•oo roo

J — oo J—oo

Since/•OO

/ (j>*(j>dx' = cons t ,J—OO

(4.14)


d f°° r°° f°°

— / (j)*x'<f>dx' = / (fcx'^ + px'^dx1 = -2i / ffodx'

we can obtain the following from (4.13) and (4.14)

7-OO

where V = V(x') is the external potential field experienced by the microscopicparticle.

We now expand dV/dx' in (4.15) around the center of mass x' — (x1) as

dV{x') _ dV((x')) ^ _ d2V({x'))dx> ~ d(x>) j ^ j { j)) d{x))d{x'j)

Taking the expectation values on both sides of the above equation, we can get

ldV\=dV{{x')) 1 ^ d3V((x'))

\dx'/ d(x') 2 f c ^ >-kdtydtydw

where

Ajlfc = ([(4 - (4»(^- - ( ))]> = (x'A) - (^)<4)-For the non-topological soliton solution, (4.8), the position of its center of mass

is determined as mentioned above, thus Aj^ = 0. Therefore the above equationbecomes

/dv(x')\^dv({x'))\ dx' I d(x') '

Thus the acceleration of the center of mass of the microscopic particles in nonlinearquantum mechanics, (4.15), can be expressed as

Returning to the original variables, equation (4.16) becomes

cP(x) dV((x))

This is the equation of motion of the center of mass of the microscopic particle inthe nonlinear quantum mechanics. It is analogous to the Newton's equation for aclassical object. It shows that the acceleration of the center of mass of a microscopic

(4.16)

(4.17)

(4.15)


particle depends directly on the external potential field. It also states that motionof a microscopic particle in nonlinear quantum mechanics obeys the classical laws.For a free particle, V = 0, thus dP{x)/dt2 = 0. Hence, the center of mass of themicroscopic particle moves with a uniform speed, which is consistent with results ofearlier discussion. We can find from the definition (4.9) that the speed is the groupvelocity of the microscopic particle (soliton), ve in (4.8). From (4.9) and (4.13), thevelocity of the center of mass of the microscopic particle can be expressed as

M I / PW*' I <f>*(i>dx'

We have used the following,

— / (f>*x <f)dx'dt J-oo

J — OO

/•OO

= t / { t f>V [4>x,t, + bcj>*4>2 - Vd>] - <t>x' [4?x,x, + b<f>(d>*)2 - V<t>*]} dx'

= - 2 t f <p*<f>x'dx',J-oo

lim (t>*<f>(x',t')= lim <f>x.(x',t') = lim </>>x- = lim 4>*x,x'<j) = 0,|x'|-+oo |cc'|->oo |a:'|-foo |x'|->oo

/•CO

/ cj)*(j)dx' = const.

and the boundary condition <fi(x',t') = 0 at |x'| -> oo. Substituting (4.8) into (4.18)we can obtain

vg = ^ = ve = const. (4.19)

which is the same as the results obtained from (4.16) above. It shows clearly that fora microscopic particle described by the nonlinear Schrodinger equation with V = 0,not only its velocity of the center of mass is the group velocity of the soliton, butit also moves with a uniform speed ve in the space-time.

The above equation of motion of microscopic particles can also be derived fromthe nonlinear Schrodinger equation (3.2) with A(<f>) = 0 by means of anothermethod. As it is known from Chapter 3, the energy E and momentum P of amicroscopic particle described by the nonlinear Schrodinger equation (3.2) with

(4.18)


A(<j>) = 0 takes the form of

E=L Vx><t>xi ~ l^*®2+w&w*]dx>'( 4 . 2 0 )

r°°P = - i (<t>*4>x, -<j>*x,<f>)dx'.

J — oo

For this system the energy E and quantum number Ns = J^ {^dx' are inte-gral invariant. However, the momentum P is not conserved and has the followingproperty

where the boundary condition is <p(x') -> 0 as |x'| -» oo. For slowly varyinginhomogeneities (in comparison with soliton scale), i.e., Ws 3> L, where L is theinhomogeneity scale, Ws the soliton width, expanding (4.21) into a power series inWs/L and keeping only the leading term, we can get

df ~ 2 dx'o Ns ( 4 - 2 2 j

where x'o is the position of the center of mass of the microscopic particle. Equation(4.21) or (4.22) is essentially consistent with (4.17) which is in the form of the equa-tion of motion for a classical particle. Indeed, if we write the one-soli ton solutionas

<f>(x',t') = y{x' - x'o,t')eip^'-x'^+i0. (4.23)

It is easy to verify that (4.23) is a solution of (3.2) with A(<f>) = 0. Inserting (4.23)into (4.20), we get

P=pNs. (4.24)

Let p = dx'0/dt' for the center of mass of the particle. Then we can obtain from(4.22) and (4.24) that

^ = -J ., o, J£—g-. (4.25)dt12 dx'o dt2 oxo

This is the same as (4.16) or (4.17) and it is the Newton's equation for a classicalparticle. Equations (4.22) and (4.25) indicate that the center of mass of the micro-scopic particle moves like a classical particle in a weakly inhomogeneous potentialfield V(x'Q). Let the function t',

(4.21)


where u = (N/2)2. Equation (4.25) coincides with the equation obtained from theadiabatic approximation of inverse scattering transformation (1ST) based pertur-bation theory for solitons (see Chapter 7).

We now consider a particular one-dimensional case. Let V(x') = ax' in (3.2),where a is constant, and make the following transformation

4>(x',t') = <j>'{x',i')e-ia2''t'-ia'2^3l\

x' = x'-ai'2, t'=i',

then (3.2) becomes

i # , + ^ . s , + 2 | 0 # | V = O, (4.26)

where 6 = 2. The exact and complete solution of the above equation is well known.We can thus obtain the complete solution of the nonlinear Schrodinger equationwith V(x') = ax'. Its single soliton solution is given by

<t>(x',t') = 2r) sech [2ri(x' - 4ft' + 2at'2 - x'o)] x (4.27)

exp | -i ktf - erf V + ^j~ - 4a^'2 + 4(£2 - r,2)t' + 80Y\.

This is the same as the solution (2.142) of (2.122) obtained by Pang et al.. How-ever, Chen and Liu first obtained the above soliton solution using the inverse scat-tering method. The same authors also gave the soliton solution of the nonlinearSchrodinger equation with Vo(x') = a2x'2, which is

<j> = 2// sech i2r}x' - — sin[2a(i' - t'0)]\ x (4.28)

exp fi mx' cos 2a(t' -Q-^ sin[4a(t' - to)] + V(*' ~ *o) + o] } •

Detailed derivation of these solutions will be given in Chapter 8.In each of the above two cases, with two different external potential fields, the

characteristics of motion of the microscopic particle can be determined according to(4.17) or (4.22). The acceleration of the center of mass of the microscopic particleis given by

for V(x') = ax', and

^ = - 4 a 2 4 (4.30)

for V(x') — a2x'2, respectively.These results can also be obtained using the following method. From de Broglie

relations E = hv = huj and P = hk for microscopic particles which have the

(4.29)


wave-corpuscle duality in quantum theory, the frequency ui retains its role as theHamiltonian of the system even in this complicated, nonlinear systems and

du_cko dk dui dx' __~dt' ~ ~dk x, ~dl + d~x' kl)F ~

as in the usual stationary media. From Chapter 3, we also knew that the usualHamiltonian equations for nonlinear quantum mechanical systems remain valid formicroscopic particles (solitons). At present, the Hamilton equations are

dk _ dui dx' _ du> .dt'~~dx~'k'~d7=~dkx,' ^ ' '

where A; = 80/dx' is the time-dependent wave number of the microscopic particle,u> — -dO/dt' is its frequency, 9 is the phase of the wave function of the microscopicparticle. Equations (4.31) are essentially the same as the Hamilton equation (3.104).Prom (4.27) and (4.28), we know that

6 = 2(Z- at')x' + ^ L L . _ 4a£f'2 + 4(£2 - r,2)t' + 0o,

for V(x') = ax', and

0 = 2£x' cos[2a(t' - t'o)] + (£-\ sin[4a(t' - t'o)} + Ar]2{t' - t'o) + ff0,

for V(x') = a2x'2, respectively. From (4.31) we can find that for V(x') = ax',

k = 2(£ - a t ' ) , w = 2ax' - 4(^ - at')2 + (2ry)2 = lax' - k2 + (2r))2.

Thus, the group velocity of the microscopic particle is

dx' dui ,. ,.v° = -dF=dkx,

=^-at)>

and the acceleration of its center of mass is given by

cPx' dk-r-fi- = -377 = - 2 a = constant. (4.32)

Here x' = x'o.For V(x') = a2x'2, we have

fc = 2£cos2a(i'-to),

w - âix' sin 2a(t' - t'o) - 4^2 cos 4a(t' - t'o) - 4T)2

= 2ax'(<ke - k2)1'2 - 2k2 + 4(£2 - rf).

Thus the group velocity of the microscopic particle is

Vg= ~dk , = T ^ l - f c W 1 ~ = 2ax'ct^2a^' ~ o)] - 4^cos[2a(t' - f0)],


while its acceleration is

Since

dPx' _ dfcUp ~~ dt7'

we have

f = g—«-i»[2««'-<;>i,and

x'= ^ sin[2a(t'- t'o)]. (4.33)

Finally the acceleration of the microscopic particle is

dk d2x' , , ,dt = -dt* = ~ 4 a X • ( 4 - 3 4 )

Equations (4.32) and (4.34) are exactly the same as (4.29) and (4.30). It showsthat (4.17) or (4.22) and (4.25) have same effects and function as (4.31) and (3.104)in nonlinear quantum mechanics. On the other hand, it is well known that a macro-scopic object moves with a uniform acceleration, when V(x') = ax' which corre-sponds to the motion of a charged particle in an uniform electric field, and whenV(x') = a2x'2 which is a harmonic potential, the macroscopic object performslocalized vibration with a frequency LJ = 2a and an amplitude x'o, and the corre-sponding classical vibrational equation is x' = x'Qsinujt'. The equations of motionof the macroscopic object are consistent with (4.17) and (4.29) - (4.30) or (4.32)- (4.34) for the center of mass of microscopic particles in the nonlinear quantummechanics. These correspondence between a microscopic particle and a macroscopicobject shows that microscopic particles in the nonlinear quantum mechanics haveexactly the same properties as classical particles, and their motion satisfy the classi-cal laws of motion. We have thus demonstrated clearly from the dynamic equations(nonlinear Schrodinger equation), the Hamiltonian or the Lagrangian of the sys-tems, and the solutions of equations of motion, in both uniform and inhomogeneoussystems, that microscopic particles in the nonlinear quantum mechanics really havethe corpuscle property.

We now look into the motion of microscopic particles described by the Sine-Gordon equation (3.4). In the absence of external fields and damping (i.e., j = 0,A = 0), the Sine-Gordon equation can be written as

4>tt ~ v\4>xx + WQ sin (/> = 0. (4.35)


Under certain perturbation, the state of the particle is changed and can be denoted

by

(j>(x,t) = <f>0{x,t) + <p(x,t) (4.36)

where tfi°(x,t) = Atan~1[e(-x~vt^(-1~v )] represents the state of the microscopic par-ticle in the absence of the perturbation, <p(x, t) is its change resulting from theperturbation which is assumed to be small in the case of weak perturbation. Insert-ing (4.36) into (4.35), we get

ftt - vfoxx + ul [l - 2sech2 (^j ] V = 0, (4.37)

where Z = (x — vt)/(l - v2). Now assume that ip(x, t) = f(x)e~tut, and insert itinto (4.37), we can get

-v20fxx - u* [l - 2sech2 (jY-)]f = "2f- (4-38)

Since the Lagrangian function L and the Hamiltonian function H of the systemsare invariant under translation, the frequency spectrum of a single microscopicparticle (soliton) must contain the translation mode of u = 0 (Goldstone mode),i.e., UJ = Ub,i = 0. Thus the solution of (4.38) can be represented by

, I OJO , fuoZ\/ M = v s e c H~J 'which satisfies

V Mfc dx J_0O

Thus the complete wave function of the system is

0(x,t) = 4>°(x) + efbil(x) = <fr°+ z^fj^^r

~^° x+£\[wk\ =<&x>x>>> (439)

where e is small quantity and X = —ey/A/Mk. Therefore, the microscopic particlemoves a distance X = -e^/A/Mk when it experiences a small perturbation. Thisshows that the motion of the center of mass of a microscopic particle described by theSine-Gordon equation follows the same law of motion of classical particles, similarto that of microscopic particles described by the nonlinear Schrodinger equation.


4.3 Collision between Microscopic Particles

The corpuscle feature of particles can be clearly manifested in the collision process.In the following, we consider collision between microscopic particles described bythe nonlinear Schrodinger equation (3.2) with V(x', t') = A(0) — 0, i.e.,

i<t>t' + 4>x'X' + b\cf>\2(t> = 0 (4.40)

where x' = x/^/h2 /2m, t' = t/fr. This problem has been investigated by manyscientists including Gorss, Pitavski, Tsuxuki, Zakharov et al., and Pang et al. Thediscussion below follows closely the approach of Zakharov et al.

4.3.1 Attractive interaction (b > 0)

We first consider the case of b > 0. Equation (4.40) can be solved exactly using theinverse scattering method. As discussed in Chapter 3, the nonlinear Schrodingerequation can be written in the form of Lax equation (3.20), where L and B arelinear differential operators containing the unknown function <f>(x',t') in the formof a coefficient. Their explicit forms are given in (3.22), where b = 2/(1 — s2) > 2and s2 > 0. By virtue of lim|a;/|_>oo <f>(x',t') —¥ 0, we can examine the scatteringproblem for operator L. To this end, we consider (3.21), where

- ( * ) •

and A is an eigenvalue of L, and make the following change of variables

{ (4-41)

Equation (3.21) can be rewritten in the form of the Zakharov-Shabat equation (3.46)which may be written as

{ (4.42)

1*2.' -*'C*2 = -«**!,

where

i<f> , ft ti ^s » c . •q=vr^> C = ( ^ ) ^ e = r ^ ' C = ^+M?-

For real £ = £, the solutions of (4.42) are the Jost functions * ' and ip with the


following asymptotic forms

* ' - + f J J e " * ' , when x' ->• - o o ,

ip -> ( ) eiix', when x' -> +00.

The pair of solutions \p and i/> form a complete system and therefore

¥ '(* ' . O ^ - V O c ' , £)» j = l,---,JV, (4.43)

i«2(e)i2 + |62«)i2 = i,

where a(£), 6(4) and Cj are referred to as scattering data of the nonlinear Schrodingerequation (4.40). From (4.42) and idip/dt1 = Bip in (3.21), we can find that

b(U')=b(Z,0)e-i4t2t',

o(0 = «*(-£), (4-44)

Ci = & + «'O,

Cj(t')=Ci(0)ei4«2t '.

Let us now find <f>(x',t') from these scattering data a(£), 6(^!^), (—00 < f < 00)and Cj(t'), j = 1, • • • ,N. The values of these quantities at t' = 0 are determinedfrom the initial conditions. In the inverse scattering problem, the time t' playsthe role of a parameter. Therefore, it suffices to consider the reconstruction of thecoefficient q(x') in (4.42) from a(£), &(£) and Cj. We introduce the function

f-irSV.Oe**', ImC>0*(C)=*(C,a:')=< } L*(x> C)\ ,

I W2KX ,(, ) \ Kx> I m / - < 0

and use $(£) to denote the discontinuity of this function across the real axis

¥ ( 0 = *(£ + iO)-$(£- iO) .

Assuming that the zeroes Ci> C2i • • •) CAT of a(C) are simple, we can get the formulathat reconstructs the piecewise-analytic function $(£) from the discontinuity $(£)and the residues at the poles Cj

/ l \ ^ W(x' Ci,)eiikX' 1 f+°° $(f)

^J w + fctt (c - c*Mco 2« ;_„ ^ - cd^'


or

$ = $(!) -(- $(2); (4.45)

where

*(1)(0 = (J) + E j-r^*«*',«,

•o>(0 = i /+L~ Wand

The tilde over Cfc will henceforth be omitted. From (4.43) we get

* ( O = ^ | e ^ > ( x ' , O - (4.46)

According to the inverse scattering theory, the equations for the function *!/(£),f < +oo and for the parameters it>(x',£j) (x' is fixed), (j = 1, ••• ,7V), can beobtained by putting £ = Q, (j> = 1, • • • ,N) in (4.44). For £ = £ - iO, equation(4.44) yields

Here J is the Hilbert transformation

(j*)* = - j * * .

Zakharov et ai. then obtained the following

-Vi(z',fle-*x' + 1(1 + 7)*; = ^1}*(e).

Multiplying the first of the above equations by c*(x',£) and the second by c(x'',£) =[b(Q/a(£)]e2ltx', and using again (4.46), we get

J f c = l ^ ^

(4.47)


In addition to (4.47), we obtain 2JV equations for fa (x1, Q) and $j ix\ 0)i by puttingC = C;, U = 1,---,N), in (4.45)

UAQe-^ + £ £^L<^V,C*) = r^ /+°° p@-dt, (4.48)fc=1 S j — <,fc z m J-oo Z — Sj

Equations (4.47) and (4.48) make it possible to obtain *(£) or ^(x1, £) and ip(x',Cj)from the scattering data. From (4.45) and (4.42), we have

\-fa(x',o) ~\o) atik IM*'W +

From these two equations we get

q(x') = -2i^24e-^x'r2(x', <*)-\j *2(fl#. (4-49)

[°°\q(p)\2dp= -2*5]cfcexp(iCfcil)î(a:',C*) + - / * i ( 0 « -

Equation of the Marchenko type can then be obtained from (4.47) - (4.48) by aFourier transformation with respect to £. Let

fc— 1

then

ffitx'.y) = -F*(x' +j/) + [°° K;(x',p)F*(p + y)dp, (4.51)/•OO

K!&,y) = - K1(x',p)F(p + y)dp, (4.52)A'

where the kernel K(x',y) is connected to tp(x',£) by

•4>(x',0 = ei(x' + [ K(x,p)ei(x'dp, ImC > 0.7z'

(4.50)


Then (4.49) may take the form

q(x') = -K1(x',x'),/•oo

/ \q{p)\2dp=-2K2{x',x').Jx'

We consider the inverse scattering problem in the case of &(£, t) = 0. Thesolution of the inverse problem then reduces to the solution of a finite system (4.48)of linear algebraic equations. We rewrite this system in a more symmetrical form

^ + Er§fe = 0, (4.53)

Here

1>j = ( ^ ) = y/c-jW, 0 ) , XJ = Vci**"' •

Equation (4.49) also assumes a simpler form in this case

q(x') = -2iJTK1>2k, r\q(p)\2dp=-2if2*k /"%!*. (4.54)*=1 Jx> fc=l Jx'

If N — 1 and a(£) has only one zero in the upper half-plane, then (4.53) becomes

| \ | 2

(4.55)

It can be easily verified that (4.55) describes a soliton of the following form

<j>(x', t') = 2Vy/2jb sech[2r}(x' - x'o) + ^ f ]e"4 i«a ' -^ t ' '~2 i i x ' '+ i B , (4.56)

where

In general (4.53) describes an ./V-soliton solution. This system is non-degenerateand

£ \q(p)\2dp = i ( £ Xki>ik - X ) A^J*) = £ 7 l n ( d e t II ID (4-57)


where \\A\\ is the matrix of (4.53). For (4.57), we have

Î n ( d e t«=deTW|d e tîl

where the matrix ||.Ajfe|[ differs from \\A\\ in their kth column, the fcth column of\\Ak\\ is the derivative of the corresponding column of \\A\\. For I < k < TV, thecolumn of the aforementioned matrix \\Ak\\ is

Using Cramer' rule, Zakharov et al. obtained

detail

Thus,

To prove (4.56), it remains to verify that

-f 'X>2i« = d£Pii S detPfcl1- (4-58)

To this end, Zakharov et al. rewrote (4.53) as

fc=i ^ ' ^*

The matrix of this system relative to {ip2i,--- ,ifoN,~ip*ir" >~1PIN} coincideswith m | | , from which (4.57) - (4.58) follow. For the iV-soliton solutions theyfinally obtained explicitly

\4>{x\ t')\2 = V2b-^ ln(det \\A\\) = VttJ^ ln(det \\B'B» + 1||)


where

B< = V^^ e <(0-«)* '

The time-dependence of CJ has been given in (4.45).We now study the behavior of the iV-soliton solution at large \t'\ to determine

the rules for collision of microscopic particles (solitons) described by the nonlinearSchrodinger equation (4.40). We confine ourselves to the case where all the £,- aredifferent, i.e., there are no two solitons having the same velocity. In this case theiV-soliton solution breaks up into diverging solitons as t' ->• ±oo. To verify this,Zakharov et al. arranged the £,• in decreasing order, £i > £2 > • • • > £N- From(4.44), one can get

and

\\j(x',t')\ = \\j(0)\e-r>iVi,

where yj — x' + 4^-t'.Let us consider the asymptotic form of the JV-soliton solution on the straight

line ym = const, as t' ->• oo. Then

yj ->• +00, |Aj| -> 0, for j < m;

yj -> -oo , |Aj| ->• oo, for j > m.

It follows from (4.53) that ipij,tp2j -*• 0, when j < m. In this limit, we have areduced system of equations for the 2(N — m — 1) functions of ym: i/>im, fam, *ifc =lfcAfc, # ^ = t/>2*A*M k > m, namely

2lTlm k=m+l U ~ C*(4.59)

and

(4.60)w 1 A

c^7^- = -1 - câ^'


Solving the last system with respect to *ifc and $2fc> w e obtain

^ _ 2ir)m ajW 2 * - _ , A* f* V2mAm-

Here

n &-«> „ , „p = m + l _ . TT S>m <jp

afc = —77 , am = 2it)m I I _ *

(Cfc - CP)m<p#fe

Substituting the expressions for \fu and *2fc in (4-59), Zakharov et al. obtained

| \+ |2 IX+I2

i>lm + y^-r2m = o, L^Lv-im + ^ m = (A+r, (4.61)

with

\+ _ \ TT w k.Am- *m 1 1 , _ ,„ •

p=m+l S m ^PEquation (4.61) coincides with (4.55) and describes a soliton with a displaced centerZQ"1" and phase 6+, given by

(4.62)

p=m+l V W ^P/

The case of t' -> -00 can be analyzed similarly and we can get from (4.55)

m - l f _f

A = A-=Amn^377- (4-63)p = 1 W (,p

On the straight lines y — x' + £t', where £ does not coincide with any of the£m's as t' ->• ±00, the reduced system becomes asymptotically homogeneous, andthe solution approaches to zero at an asymptotic rate, thus proving the asymptoticbreakdown of the JV-soliton solution into individual solitons.

The above results make it possible to describe the soliton scattering process. Ast' —> 00, the iV-soliton solution breaks up and the resulting solitons move in such away that the fastest soliton is in the front while the slowest at the rear. However,this is reversed as t' —> —00. As time t varies from — 00 to +00, the quantity


Xm changes by a factor of A+/A~, corresponding to a total change, &x'Qm, in thecoordinate of the soliton center, given by

A*^-^(Jjn|^|-|\»|^||), ,4.64)

and a total change in its phase

A9m = fl+ - 9~ = 2 £ arg (%^g) - 2 £ arg ( £ ^ £ ) . (4.65)k=1 \ U <»k/ k=m+1 Um (,*/

Equation (4.64) can be understood by assuming that the microscopic particles (soli-tons) collide pairwise and every microscopic particles collides with all others. Ineach paired collision, the faster microscopic particle moves forward by an amountof r)^1 In |(Cm — Cfc)/(Cm — C*)l> Cm > Ck, and the slower one shifts backwards by anamount of r)^1 In |(£m — Cfc)/(Cm - Cfc)l- The total shift of the soliton is equal to thealgebraic sum of those of the pair during the paired collisions, so that the effect ofmulti-particle collisions is insignificant. The situation is the same with the phases.This rule of collision of the microscopic particles in nonlinear quantum mechanicsis the same as that of classical particles. This demonstrates the feature of classicalmotion of the microscopic particles, or, corpuscle feature of the microscopic particlesin nonlinear quantum mechanics.

Using the above properties of two-particle collision and following the approachof Zakharov and Shabat, Desem and Chu obtained a solution for the interactingtwo-particle system from (4.50), it is given in terms of the solution of the nonlinearSchrodinger equation corresponding to two discrete eigenvalues Ci,2)

. . , tl) = |a i | coshfa + i0i)ei6* + \g2\ cosh(a2 4- zfl2)e<' «3 cosh(ai) cosh(a2) - a4[cosh(ai + a-x) — cos(A')]'

where

01,2 = 2 [ 2 « 2 - 8,2)* - *'fc,2] + (O'0)l,2,

A'^9'2-0[+(92-01),

<*i,2 = 2T/I,2(X' + 4t '6 l 2 ) + (ao)i,2,

|Qll2|e"lia = ± { fc " ( A ^ ) ] ± i ( A ^ ) } '_ 1 _ 1

Q 3 - 4 W a4-2(r?2 + AO '

Cl,2 =6,2+«?l,2, A^ = ^ 2 -6 i V = Vl+V2,

(4.66)


77 and £ are the same as those in (4.56), and represent the velocities and amplitudesof the microscopic particle (soliton), (ao)i,2 the position, and (#0)1,2 the phase.They are all determined by the initial conditions.

Of particular interest here is an initial pulse waveform,

0(0, x') = sech(a;' - x'o) + sech(z' + x'0)eie (4.67)

which represents the motion of two soliton-like microscopic particles into the system.Equation (4.67) will evolve into two solitons described by (4.66) and a much smallernon-soliton part which decays like a dispersive tail. The interaction between thetwo microscopic particles given in (4.67) can therefore be analyzed through the two-soliton function, equation (4.66). Given the initial separation x'o, phase difference6 between the two microscopic particles, the eigenvalues £1,2, ao and 6'0 can beevaluated by solving the Zakbarov and Shabat eigenvalue equation (4.42), using(4.67) as the initial condition. Substituting the eigenvalues obtained into (4.66), wethen obtain the description of the interaction between the two microscopic particles.

Fig. 4.2 Interaction with two equal amplitude microscopic particles. Initial microscopic particlesseparation = 3.5 pulse width (pw).

The two microscopic particles described by (4.67) interact through a periodicpotential in t1, through the cos A' term. The period is given by TT/(TJ| - VI)- Thepropagation of two microscopic particles with initial conditions 6 = 0, £1 = £2 =0, (6'0)i = (6'0)2 = 0, obtained by Desem and Chu is shown in Fig. 4.2. Thetwo microscopic particles, initially separated by x'Q, coalesce into one microscopicparticle at A' — n. Then they separate and revert to the initial state with separationx'o at A' = 2TT, and so on. An approximate expression for the spacial variation ofthe separation between the microscopic particles can be obtained provided the twomicroscopic particles are well resolved in such a case (Gordon 1983; Karpman andSolovev 1981). Assuming that the separation between the solitons is sufficientlylarge, one can obtain the separation Ax as

Ax' = In I"-1 cos(at') |1 , a = 2e-*o.


Thus the period of oscillation is approximately given by t' — (n/2)ex'° (Blow andDoran 1983; Gordon 1983).

4.3.2 Repulsive interaction (b < 0)

We now discuss the case of b < 0. In this case, equation (4.40) also has soliton so-lution, when limij/iôo |0(a;',£')|2 -> constant, and Vna^.^^^, = 0. The solitonsare dark (hole) solitons, in contrast to the bright soliton when b > 0. The brightsoliton was observed experimentally in focusing fibers with negative dispersion, andthe hole soliton was observed in defocusing fibers with normal dispersion effect byEmplit et al. and Krokel. In practice, it is an empty state without matter in lightor a hole in microscopic world. Therefore b > 0 corresponds to attractive interac-tion between Bose particles, and b < 0 corresponds to repulsive interaction betweenthem. Thus, reversing the sign of b not only leads to changes in the physical pictureof the phenomenon described by the nonlinear Schrodinger equation (4.40), but alsorequires considerable restructuring of the mathematical formalism for its solution.Solution of the nonlinear Schrodinger equation must be analyzed and collision rulesof the microscopic particles must be obtained separately for the case of b < 0, us-ing the inverse scattering method. Again, the approach of Zakharov and Shabat isoutlined in the following. Readers are referred to the original paper by Zakharovand Shabat published in Soviet Physics, JETP, 37 (5) (1973) 823 for details.

In the direct scattering problem of (4.40) with b < 0, equations (4.41) - (4.44)are still valid, as long as the factors (1 — s2) and (1 — s) in these equations arereplaced by (s2 — 1) and (s — 1), respectively, a and b now satisfy \a\2 — \b\2 = 1.The corresponding the scattering data are

a(\,t') = a(\,0),

&(A,i') = KA,0)e-4iA^',

Cj-(f) = c,-(0)e4W,

The above equations enable us to calculate the scattering matrix at an arbitraryinstant of time from the initial data. This yields <j>(x',t') at an arbitrary instant oftime. The integral equations (Marchenko equations) can then be reconstructed todetermine the potential q(x') = <j>(x', t')/(s2 -1) from the scattering data. Zakharovet al. obtained the following

^)=i-^,,8(x-,,-)=(î^y'. («.)where


Here y is an arbitrary argument.Of particular physical interest is the state \<j)(x',t')\2 -> const., (j>x> ->• 0 as

x' -> ±00, which corresponds to the propagation of a wave through a condensateof constant density. In this case the soliton

fb (\-iv)2 + eM"'~x'0)-2Xt'

can move with a constant velocity. Thus

W , i ' ) | 2 = 1 - . 2 . , ,"' , ..,,,• (4.70)2 cosh \v(x' -x'Q- 2Xt)]

The parameter A characterizes the amplitude and velocity of the microscopic par-ticles, and x'o is the position of its center at t' = 0, where d(lnfi)/dt' = 4\v orln/i = 2i/(a;{) + 2At')-

We can verify that the eigenvalue A corresponds to a soliton moving with velocity2A and that the collision between these microscopic particles satisfies the rule for6 > 0 .

Let us consider the interaction of two microscopic particles (solitons) with veloc-ities 2A2 and 2Ai. To this end it is necessary to obtain the corresponding two-solitonsolution. Owing to the complexity of the explicit formula for this solution, we limitour discussion to the asymptotic behavior as t' -> ±00.

Following the same procedure as before, we can show that as t' -> ±00, thetwo-soliton solution breaks up into individual solitons

<j>{x',t') -> cj)0{x' - 2\lt\\1,x'+) + <t>o{x' - 2A2i ' ,A2,4+) , t1 -» +00,

<f>{x',t') ->^0(ar'-2Ai<',Ai,a;i") + ô(a;'-2A2<',A2,a;2~), *' -J- -00.

The scattering of the solitons by each other gives rise to the following displacementsof their centers

Sx[ =x'+ -x'~,X™' _ _'+ T ' ~

To determine these displacements, we note first that the following discontinuity inthe phase of the wave function of the condensate occurs for a microscopic particle(soliton) with velocity 2Aj

a'j = arg^(-oo) — arg </>(+co) = 2 tan"1 I -p ) •

The total phase discontinuity for N eigenvalues is

N

k=l

(4.69)


which, however, does not depend on the relative positions of the solitons. In the caseof two solitons, the Jost function ViO 'i A) takes the following form as x' -¥ — oo,

^ ' ' ^ ^ ( . - A ) ^ ) ] ' (-KA<1).

Thus, the asymptotic form of ip[(x', A) as x' -> +00 is

\a\X,iv)evx'\. l J , A#A1>A2>

V4(*',A)^ " •y ' - A J , (4.71)[ 6 i , a W e — ' [ ^ - a - A l - a ] , A = A1;A2.

Zakharov et al. represented the coefficient a in the form of a' = a ' ^ , where

, . iv + A - ivifi - Ai,2a, , (A, v) = • —2-

1)2 « / + A + ii/1,2 + Ai>2

is t he component of "single-soliton" sca t ter ing mat r ix . Let Ai > A2. As t' -¥ - 0 0 ,

the microscopic part icles move a p a r t and separa ted by a large dis tance and \<j>\ -> 1.

Assume t h a t part ic le 2, with velocity 2X2, is located to t he r ight , t h e n <f> -> exp( ia2)

in t he region between t he microscopic part icles. As t' -> 00, par t ic le 1, with velocity

2Ai, is located to the right, t hen </> -+• exp(io;'1) in the region between t he microscopic

part icles . We consider in this region t he asympto t ic form of t he Jos t functions

il>'i(x',X) as t' —> —00. We can get

U(A,^)e-'[e ia, (i_A)], (AÂO,[bT(t')e-^ 1 j , (A = Ax),

here b^(t') is an unknown function. To determine &J~(£')> w e u s e a(\,—iv) =l/a*(X,iv). Through microscopic particle 2, the variation of the function ^ inaccordance with this rule, and comparison with (4.71), we finally get

b^(t') = a'*(X1,iu1)b1(t').

Using the same procedure, bît') = b2(t')/ai(X2,iu2) can be determined. b^(t')and b2(t') determine the positions of the microscopic particles as t' —> —00. Similarquantities, b*(t') and b^ (£'), can be introduced to describe the positions of thesolitons as t' —>• 00. A similar analysis as above results in the following

bt(t') = , y } y bt(t') = b2(t')a'1(X2,iu2).


Thus, the displacements of the microscopic particles (solitons) after the collisionwere found to be

x > - 1 i fbi\ _ i , ( i \ _ x_dXl ~ 2u, ln UrJ " 2 i iMA!,^)!2 j - 2^'

(4.72)

^=2^ln(|)=iln^^^|2)=-^where

L ( A i - A 2 ) 2 + ( ^ 1 - l / 2 ) 2 J -The microscopic particle which has the greater velocity acquires a positive shift, andthe other a negative shift. The microscopic particles repel each other, like classicalparticles. Prom (4.72) we get

uiSx\ + 1/26x2 = 0.

This relation was obtained by Tsuzuki directly from (4.40) for b < 0 by analyzing themotion of the center of mass of a Bose gas. It can be interpreted as the conservationof the center of mass of the microscopic particles during the collision. This showssufficiently the classical feature of microscopic particles described by (4.40).

The case of JV-solitons can be studied similarly and it can be shown that ast' -»• ±00, an arbitrary iV-soliton solution breaks up into individual solitons. Thearrangement of the solitons according to their velocity is reversed as the changesfrom —00 to +00, so that each soliton collides with each other soliton. Goingthrough the same analysis given above, we can verify that the total displacementof a soliton, regardless of the details of the collisions, is equal to the sum of thedisplacements in individual collisions

°j ~ xj xj - 2-j°li'i=\

where

Prom the above studies we see that collisions of many microscopic particlesdescribed by the nonlinear Schrodinger equation (3.2), with both b > 0 or b < 0,satisfy rules of classical physics. This shows sufficiently the corpuscle feature ofmicroscopic particles in nonlinear quantum mechanics.

4.3.3 Numerical simulation

The discussion on collision of microscopic particles presented above are based onanalytical analysis using the inverse scattering method, and as a result, only the

(4.73)


asymptotic behaviors of the particles can be obtained. The same process can bestudied by numerically solving (3.2). Numerical simulation can reveal more de-tailed features of collision between two microscopic particles in nonlinear quantummechanics. We begin by dividing (3.2) with V = constant and A(<f>) = 0 into thefollowing two-equations

• * + & * * = * " •(4.74)

d t ' 2 d x ' 2 d x ' { ] 9 1 1 '

Obviously, if we assume £ = x' — vt' in (4.74), we can get the nonlinear constantb = 1/(1 - v2) in (3.2) at lim|a.,|_>oo<£' = Wm^^^cj)^ = 0. Equation (4.74) issimilar to that in Davydov's or Pang's model for bio-energy transport in the a-helix protein and molecular crystalline-acetanilide (see Chapters 5 and 9), in whichthe soliton is formed by self-trapping of excitons interacting with lattice phonons. Inthese models, </>' represents the wave function of the exciton, u is the longitudinaldisplacement of the lattice molecule, b is the nonlinear coefficient of the systemresulting from the exciton-phonon interaction. Evidently b is related to the excitonvelocity v. The soliton solution of (4.74) can now be written as

<f>' = > / 2 ( l - « 2 ) V sechWz' -x'o- vt')} exp \^-vx' - i{~ - r,2)t' + tf] , ( 4 J 5 )

u = -27j2sech2[7j(a;' - x'Q - vt')].

The soliton in (4.75) is a self-trapped state of exciton plus deformation of thelattice in such a case. The properties of the soliton depend on three parameters:77, v and 6, 77 determines the amplitude and width of the soliton, 6 is the phase ofthe sinusoidal factor of ft at t'= 0. Tan et al. carried out numerical simulationof the collision process between solitons using the Fourier pseudo-spectral methodwith 256 basis functions for the spatial discretization, together with the fourth-orderRunge-Kutta method for time-evolution. The system given in (4.74) has two exactintegrals of motion

/•oo roo

Ns= \<t>'\2dx', and Ex = / udx',J—00 '—00

which can be used to check the accuracy of the numerical solutions.For the collision experiments, the initial state consists of two solitary waves


separated by distance x'Q,

$ = yj2(1 - v\) 771 sechîz') exp ( ^-x' + i(9i J +

\/2(l - vl) r/2 sech[tB(a/ + x'o)}exp I - y ( z 2 + x'o) + i62\,

u = ~2r1lsech2(r]1x') - 2r?|sech2[772(x' + x£)],

where the first term in each expression represents one soliton (1) while the secondterm represents the other soliton (2). It can be shown that the post-collision state ofthe solitons is strongly dependent on both the initial phases and the initial velocitiesof the solitons. Since (/)' can be multiplied by an arbitrary phase factor, exp(i#)where 0 is an arbitrary constant, and still remains a solution (with u unchanged),one of the phases is arbitrary, and only the difference of the two initial phases issignificant. Therefore, we can set 6\ = 0 for the convenience of discussion.

Fig. 4.3 Fast collisions of solitons. The initial ratio of velocities of the fast and slow solitons is1.8.

Figure 4.3 shows the fast collisions obtained by Tan et al., in which the initialratio of velocities of the fast and slow solitons is fixed to be 1.8. The absolutevalue of the <j>' are shown using contours on the left in each pair of plots, with x'being the horizontal coordinate and t' increasing upward. The right panel showsthe absolute value of u. The relative phase increases from 0 (top) to 7r/2 (middle)to 7T (bottom). All cases are identical except that 62 is increased by TT/2 in eachcase, beginning with 62 = 0 at the top. Before the collision, each of the initialsoliton contributes 0.7600 to N3 in all three cases (62 = 0, TT/2 and TT). When


the relative phase is zero (top), the solitons penetrate each other freely and thenemerge with their shapes and velocity unchanged. When 62 = TT/2 (middle graphs),</>' emerges from the collision asymetrically, and a large soliton, which contributes1.4272, moving to the left, at the same velocity as the initial speed of soliton 2.Another small pulse, contributing 0.0928, travels to the right at the speed which isthe same as the initial speed of soliton 1. The post-collision energies are the sameas those of pre-collision for </>' when 6\= 0 and 62 = ""• For all values of 62, there islittle change in the contributions of the solitons in their u-field to energy Ei, andthey are not shown here. When 02 = n, as shown in the bottom panel of Fig. 4.3,the u-components penetrate freely, but the (//-components bounce off each otherand change their directions, without interpenetration. The fourth case, 82 = 3?r/2,is not shown here because it is just the mirror image of the middle figure. That is

<t>'(x',t>,62 = ^=(-x',t',e2 = ^ ) ,

u{x\t',e2 = ^=u(-x',t',92 = l).

The same is true for intermediate and slow collisions processes. However, thereflection principle cannot be generalized to all solutions which are different in theirinitial phases by n because the cases of 62 = 0 and 92 = n are quite different in thegeneral case.

Pang et al. simulated numerically the collision behaviors of two solitons basedon the improved Davydov model with a two-quantum quasi-coherent state and anadditional interaction term in the Hamiltonian for the a-helix protein molecules,using the fourth-order Runge-Kutta method. This result is shown in Fig. 4.4 (seeChapter 9 for details).

Fig. 4.4 The collision of two microscopic particles described by nonlinear Schrodinger equationin the improved Davydov model.

From Figs. 4.2 - 4.4, we see clearly that the collision between microscopic par-ticles described by the nonlinear Schrodinger equation show features of classicalparticles. Thus we confirm again that microscopic particles in nonlinear quantum


mechanics has the corpuscle property.

4.4 Properties of Elastic Interaction between Microscopic Particles

We now consider further the properties of elastic interaction of microscopic particles(solitons) described by the nonlinear Schrodinger equation (3.2) with A((f>) = 0,following the approach of Gorshkov and Ostrovsky (1981) and Abdullaev et al.. Weassume that the interaction among the microscopic particles is weak.

For the Euler-Lagrange equation (3.101) corresponding to the nonlinearSchrodinger equation, we further assume that it has a soliton solution </>°(£ =x' - vt', v) satisfying

The interaction between the microscopic particles (solitons) in the system is weak,provided they are separated by a sufficiently large distance, and their relative ve-locity is small. The motion of each soliton can then be analysed by assuming thatit is either in the weak field or on the "tail" of other solitons, and expanding theconcerned quantities in terms of a small parameter e. Based on this approach, thesolution of (4.76) or (3.2) with A(4>) — 0 in such a case can be expressed as a series

#r',i') = ^K"-«n(O,«]+5Z*(0)[A-ui(*')'i;]+Ee^(*)K"-u«(*')'-x''r] (4-77>

where X = ex', T — et', n is the number of microscopic particles, dun/dt' = O(ev),and e « (vn — Vj)/(vn + Vj) <C 1. This perturbation method is self-consistent if weassume that the field produced by the tail of the jth microscopic particle at the siteof the nth microscopic particle is of the order e2. Inserting (4.77) into (4.76) andequating the terms of the same orders in e2, we obtain a set of linear equations for<f>^. In order for the solutions to remain finite at £ -> ±oo, it is necessary to havea suitable orthogonality condition which, to the first order in e, has the followingform

J —OO

where

R = l f [d£dfo + dj \ d 4 % + d4>xld<f>t<) ** ~ d(f>d(t>t> ^ J '

In a similar way, Abdullaev et al. obtained the equation of motion for the centerof mass coordinates of the microscopic particles by setting the determinant of the

(4.76)


secular equation up to the second order in e to zero

m ^ T = Z ) / ( U n " Uj)- (4-78)

3

These equations are similar to (4.25) and those of classical particles with aneffective mass m, and an interaction potential U(un) w f f(un)dun, given by

(4.80)where P is the momentum of the field.

The above method and results can be easily extended to the multi-dimensionalcase, where the solution depends on several variables, £i, £2, • •• ,&• For the non-linear Schrodinger equation (3.2) at A((j>) = 0, 0 is a decreasing function of £1 as£1 —> 00, and a periodic function in £2- Details of derivation of the correspondingequations for the generalized Pq and U are given by Gorshkov and Ostrovsky (1981).Corresponding to (4.78) - (4.80), we have

fcPun\dPq r n . . .l"djo~) -Q^- = 2-,ûnU(uln - Ulj,- • • ,umn-umj),^ ' 9 j=n

' - l o o ^ ^ '

/•OO

U(uin - Uij, • • • ,Umn ~ Umj) = / d£0(O)(£i - Ui, • • • , £ m - Um) XJ—00

[^ {""e* + e^J ' H^(0)(Ci_,...,u__)'For the (jfi -field equation

<Att - 0XZ - </• + =tanh[± . ' - * 1.

Next, we study the kink-kink and the kink-antikink interactions, by substitutingthe expression for C of the ^4-field equation into (4.80)

u(Un - Uj) = r d&w3(z-un)<t>Mti- Uj)J — OO

where the asymptotics < °H£ ± oo) = $± have been subtracted from the kink(antikink) fields. We thus obtain an equation of motion for the equivalent classicalparticle with an effective mass m and moving in the effective potential U

J2,. .

m l ^ L = ±8 x/2(l -t,)2e-v/2(i-«2)«n.

Here the + sign corresponds to the kink-kink or the antikink-antikink interaction,and the — sign corresponds to that of the kink-antikink pair. In other words,the kink and antikink interact through an exponential attractive potential. Con-sequently, a bound state could be formed between a kink and an antikink. Note,however, that this is valid only for particles moving with low speed (v < 0.2). It wasshown by Campbell et al. (1986), that when the particles move with high speeds,there may be a resonant interaction between the kink and antikink, as well as otherinelastic processes (Makhankov, 1979).

For the nonlinear Schrodinger equation (4.40) with b = 2, the soliton solutioncan be written as

0(°) = v/2A"sech(Aêi"^2+(',

where £ = x' - vt1, 6 = ut', (A")2 = a/2 - v2/4. Inserting the above into (4.80), wecan obtain the components for the field function and the interaction potential asthe following

Pc = 2™ A", Pe = 4TTA",

U(uX',ug) = 47rA' 3exp(-A"wI/)cos !-(«!-) - ug .

Making use of these components, the equations of motion can be written as

df> ~ Pe '

where

R = z + iO, /? = 32(A")2.


In the above,

z = -\"uxi, 6 = --ux> +ue-

Thus we also got an equation of motion for the equivalent classical particle with aneffective mass 1 and moving in the effective potential U, which is consistent withthe results of (4.16), (4.17) and (4.25). The solutions depend on four parameters,t'j, t'j, Roo, and RQ, which satisfy the following

e(R-Ro)/2 = _ c o s h [ f i o o ( i / + t,j + - ^

We now consider the interaction between microscopic particles based on theinverse scattering transformation, as it was done by Karpman and Solovev (1981);Anderson and Lisak (1986). Consider again the interaction between microscopicparticles (solitons) described by (4.40). We seek a solution with well-separatedsolitons

4>(x',t') = M^',t') + <l>2(x',t').

Then, the interaction between the microscopic particles is described by the following

emRm(.i) = i{4>m4>f + 2<j>m<t>j<t>*), (m,j = l , 2 , r o ^ j).

Treating emi?m as perturbation and applying the perturbation theory, Desem andChu obtained a set of equations for the parameters of the j- th soliton

^ f = {-iyi6ri3e-2r>dcos(2ij,d + 6),

^ f = (-iyi6T)3e-2T>d sin(2/id + 6), (4.81)

^^2fxj+4Ve-2"dsm(2fid + e),

^ f = 2(t# + n2j) + Snr)e-2i)d sm(2nd + §)+ 2^2e~^d cos(2/xd + 0)Civ

where

<* = 6 - 6 > 0 , 9 = Si-S2,

V= 2^1+^2) , fi= j , ,

and Afj, < fi, Ar) — r)2 - 771 < 771, r\d » 1, Ar\d C 1, A/x = Hi - H2.Equations (4.81) have three constants of motion:

/i = constant,

77 = constant, (4.82)Y'2 = lQv2e-2rid+i(2iJ.d+S) _ ^ 2 )


where

Y' = An + iArj.

From this, a soliton solution is readily found

y ' = -Atanh(2jjAt' - ax - ia2).

Setting A = m + ij, we obtain the following solution of (4.82)

1 [cosh(4r?m^/ - 2QI) + c o s ^ r - 2a2) ]d(t) - d(0) = - log ^ c o s h ( 2 a i ) + c o s ( 2 a 2 ) j •

0(i) - 6»(0) = -2tan-1[tanh(2j?mt' - c*i) tanh{2r]jt' - a2)] (4.83)

+2 tan"1 (tanh Q2) - 2jj,[d - d(0)].

The interaction between microscopic particles in the nearly integrable perturbednonlinear Schrodinger equation differs from that of the Sine-Gordon equation, forexample, in that the binding energy of the two-soliton state of the former is zero.The two-soliton state is unstable with respect to small perturbations. However,there might be situations when perturbation stabilizes a two-soliton state, whichenables us to consider multi-particle effects for solitons in nonlinear quantum me-chanics, such as interaction between a two-soliton state and another soliton.

One of the features of solitons is their stability against a fairly broad class ofperturbations and their elastic interaction upon colliding. The inverse scatteringmethod can be used to study the interaction of solitons in elastic collsion of micro-scopic particles (solitons). Parameters of the microscopic particles (solitons) do notchange, but simply acquire a phase shift Sm (fast soliton) or -8P (slow soliton),

Sm,P = ^ - In ^ — ^ , R{A,} < U{\p)

as mentioned in the previous section. For nearly equal velocities the solitons forma bound state.

Using results of inverse scattering method for the two-soliton solution, Gordon(1983) derived expressions for interaction between microscopic particles (solitons).Assuming large separation between solitons, he found that the interaction betweenthe particles decreases exponentially with the distance between the microscopicparticles, and depends only on the relative phase. This is consistent with resultsof the analysis based on the direct perturbation theory (Gorshkov and Ostrovsky,1981). Gordon showed that with an initial state in the form of

4>o = sech(z' - x'o) + sech(x' + x'o), (x'o > 1),

the distance d between the microscopic particles is given by

d = do + 21n|cos(2t'e-a!o)|. (4.84)


It follows that the distance d has an oscillating character and that a bound stateof the microscopic particles exists. Numerical simulation (Blow and Doran, 1983;Hermansson and Yervick, 1983) showed that a nonlinear interaction between mi-croscopic particles leads to an attraction between pulses for any value of x'o, with arelative phase of zero.

We now derive the interaction between two microscopic particles from the systemof equations for the soliton parameters (4.81). A general solution is given by (4.83).If the microscopic particles (solitons) have the same amplitude and they are initiallyat rest, this solution simplifies to

d(t') = d(0) H In - cosh(4r]mt') + cos(4r]jt') ,2r] [2 J

where

n ^ - ^ e - ^ s i n ^ , and j = ê""^0) cos ^ .

If the initial phase of the microscopic particles is zero, 0(0) = 0, then m = 0 andj = 4r)e~nd(0\ The distance between the microscopic particles is

d(t) = d(0) + - In I cos(27jjt')|.V

Thus the microscopic particles undergo oscillatory motion. This expression coin-cides with (4.84) if 2rj = 1. The separation between the particles d(t') is zero attime

(; = 257icos"1[e""Jl0)1-Since r]d(0) » 1, the period of the oscillation is

t' - w rvd(o)

c~16r?2 '

On the other hand, if the initial phase difference is 0(0) = TT, then j — 0,m = -4r]e-r'd^\ and

d{t) - d{0) = - ln[cosh(2»jmf)].

In this case, the distance between the microscopic particles increases monotonically.The separation is doubled after a time of

For r)d(0) S> 1, we have


Desem and Chu gave the changes of d(t')/d(Q) with increasing 4i]mt', as shownin Fig. 4.5, where three different situations are shown: a monotonic increasingseparation between the microscopic particles (solitons), a quasi-periodic oscillationwhich gradually reduced to a monotonic increasing function, and an oscillatorymode. Hence, binding of microscopic particles (solitons) can be suppressed byappropriate choice of initial phase difference.

Fig. 4.5 The dependence of distance d between the miroscopic particles (solitons) on the initialphase difference.

4.5 Mechanism and Rules of Collision between Microscopic Parti-cles

The fact that two microscopic particles can survive a collision completely unscatheddemonstrates clearly the corpuscle feature of the microscopic particles. This prop-erty is frequently used in investigations to separate nonlinear quantum mechanicalmicroscopic particles (solitons) from particles in the linear quantum mechanicalregime. During the collision, the microscopic particles interact and exchange posi-tions in the space-time trajectory as if they had passed through each other. Afterthe collision the two microscopic particles may appear to be instantly translated inspace and/or time but otherwise unaffected by their interaction. This translationis called a phase shift as mentioned in section 4.3. In one dimension, this processresults from two microscopic particles colliding head on from opposite directions,or in one direction between two particles with different amplitudes. This is possiblebecause the velocity of a particle depends on the amplitude.

In the following, we describe a series of laboratory and numerical experimentsdedicated to investigation on the detailed structure, mechanism and rules of colli-sion between microscopic particles described by the nonlinear Schrodinger equationin the nonlinear quantum mechanics. The properties and rules of such collision be-tween two microscopic particles have been studied by Aossey et al. Both the phaseshift of the microscopic particles after their interaction and the range of the inter-action are functions of relative amplitude of the two colliding microscopic particles.The microscopic particles preserve their shape after the collision.

For the microscopic particles described by the nonlinear Schrodinger equation


(4.40), we will limit our discussion to the hole (dark) spatial solitons with b < 0which, at present, is given by

<t>(x',t') = <j>0yJl-B2sech2{Z') e±ie«'>,

where

e(O = sin- [ . gtanh(O 1, ? = Kx - vtt>)._^/l-52sech2(£').

Here, B is a measure of the amplitude ("blackness") of the solitary wave (holeor dark soliton) and can take a value between - 1 and 1, vt is the dimensionlesstransverse velocity of the soliton center, and fi is the shape factor of the soliton.The intensity (Id) of the solitary wave (or the depth of the irradiance minimum ofthe dark soliton) is given by B2fy\. Zakharov et al. and Hasegawa et al. showedthat the shape factor /j, and the transverse velocity vt are related to the amplitudeof the soliton, which can be obtained from the nonlinear Schrodinger equation inoptical fibre (see Chapter 5 for its description).

S = no\n2\nlB*<t>l vt*±^(l~B>)^

where no and n2 are the linear and nonlinear indices of refraction of the opticalfibre material. We have assumed |n2|< >o ^ no- When two microscopic particles(solitons) collide, Aossey et al. expressed their individual phase shifts as

, , = &--L-ln [ (^p±^!±fcM| . (4.85)V|n2|0§2/iOnoB> [ ( ^ 1 - B\ + y/\ - B\)2 + (Bi - B2)

2\

The hole spatial soliton interaction can be easily investigated numerically byusing a split-step propagation algorithm which was found, by Skinner et al. andThurston et al., to closely predict experimental results. The results of a simulatedcollision between two equi-amplitude microscopic particles (solitons) are shown inFig. 4.6(a), which are similar to that of general (bright) solitons as shown in Figs. 4.3and 4.4. We note that the two solitons interpenetrate each other, retain their shape,energy and momentum, but experience a phase shift at the point of collision. Inaddition, there is also a well-defined interaction length in z along the axis of timet that depends on the relative amplitude of the two colliding microscopic particles(solitons).

This case occurs also in the collision of two KdV solitons. Cooney et al. studiedthe overtaking collision, to verify the KdV soliton nature of an observed signal inthe plasma experiment. In the following, we discuss a fairly simple model whichwas used to simulate and to interpret the experimental results on the microscopicparticles (solitons) of nonlinear Schrodinger and KdV solitons.


Fig. 4.6 Numerical simulation of an overtaking collision of equiamplitude dark solitons. (a)Sequence of the waves at equal intervals in the longitudinal position z. (b) Time-of-flight diagramsof the signals.

The model is based on the fundamental property of soliton that two solitonscan interact and collide, but survive the collision and remain unchanged. Ratherthan using the exact functional form of sech£ for the microscopic particles (solitons)of the nonlinear Schrodinger equation, the microscopic particles are represented byrectangular pulses with an amplitude Aj and a width Wj where the subscript jdenotes the j'th microscopic particles.

An evolution of the collision of two microscopic particles is shown in Fig. 4.7(a).In this case, Aossey et al. considered two microscopic particles with different am-plitudes. The details of what occurs during the collision need not concern us hereother than to note that the microscopic particle with the larger-amplitude has com-pletely passed through the one with a smaller amplitude. In regions which can beconsidered external to the collision, the microscopic particles do not overlap as thereis no longer interaction between them. The microscopic particles are separated bya distance, D = D\ + D2, after the interaction. This manifests itself in a phaseshift in the trajectories depicted in Fig. 4.7(b). This was noted in the experimentaland numerical results. The minimum distance is given by the half-widths of thetwo microscopic particles, D > Wi/2 + W2/2. Therefore,

D i > ^ and D2>^-. (4.86)

Another property of the microscopic particles (solitons) is that their amplitudeand width are related. For the microscopic particles described by the nonlinearSchrodinger equation with b < 0 in (4.40) (W « \j\x ), we have

BjWj = constant = Kx. (4.87)


Fig. 4.7 Overtaking collision of two microscopic particles, (a) Model of the interaction just priorto the collision and just after the collision. After the collision, the two microscopic particles arephase shifted, (b) Time-of-flight diagram of the signals. The phase shifts are indicated.

Using the minimum values in (4.86), we find that the ratio of the repulsive shifts forthe microscopic particles described by the nonlinear Schrodinger equation is givenby

D2 B\

Results obtained from simulation of this kind of microscopic particles (solitons) arepresented in Fig. 4.8(a). The solid line in the figure corresponds to (4.88).

In addition to predicting the phase shift that results from the collision of twomicroscopic particles, the model also allows us to estimate the size of the collisionregion or the duration of the collision. Each soliton depicted in Fig. 4.7 travelswith its own amplitude-dependent velocity Vj. For the two microscopic particles tointerchange their positions during a time AT, they must travel a distance L\ andL2, respectively, where

Li=viAT and L2 = v2AT. (4.89)

The interaction length must then satisfy the relation

L = L2 - Lx = (v2 - ui)AT >Wi+ W2. (4.90)

Equation (4.89) can be written in terms of the amplitudes of the two microscopicparticles (solitons). For the nonlinear Schrodinger solitons, by combining (4.86) and

(4.88)


Fig. 4.8 Summary of the ratio of the measured phase shifts as a function of the ratio of theampliudes. (a) Microscopic particles (solitons) described by the nonlinear Schrodinger equation.The solid line corresponds to (4.88) and the symbols are results of Skinner et al. (b) KdV solitons.The symbols represent experimental results of (1) Ikezi et al., (2) Zabusky et at., and (3) Lamb.The solid line corresponds to (4.92).

(4.90), Aossey et al. obtained

L>Kl[l- + j - ] . (4.91)

In Fig. 4.9(a), the results for the microscopic particles described by the nonlinearSchrodinger equation are presented. The dashed line corresponds to (4.91) withS2 = 1 and Kx = 6. The interaction length (solid line) is the sum of the widthsof the two microscopic particles, minus their repulsive phase shifts, and multipliedby the transverse velocity of microscopic particle 1. Since the longitudinal velocityis a constant, this scales as the interaction length. From the figure we see that thetheoretical result obtained using the simple collision model is in good agreementwith that of numerical simulation.

The discussion presented above and the corresponding formulae reveal the mech-anism and rule for collision between microscopic particles depicted by the nonlinearSchrodinger equation in the nonlinear quantum mechanics.

In order to verify the validity of this simple collision model, Aossey et al. studiedcollision of solitons using the exact form of sech2£ for the KdV equation, ut + uux +d'uxxx = 0, and the collision model shown in Fig. 4.7. For the KdV soliton theyfound that

AJ(WJ)2= constant = K2 and ^ = | ^ = j j , (4.92)L>1 VV2/A V •ril

where Aj and Wj are the amplitude and width of the jth KdV soliton, respectively.Corresponding to the above, Aossey et al. obtained

L > jfc ( * + * ) = * L ( l + . / £ ) (4.93)


Fig. 4.9 Summary of the measured interaction length as a function of the amplitudes, (a) Non-linear Schrodinger solitons, the dashed line corresponds to (4.91) with B2 — 1 and K\ = 6. (b)KdV solitons, the symbols represent (1) experimental results of Ikezi et aL, and (2) numericalresults of Zabusky and Kruskal. The dashed line corresponds to (4.93) with K2 = Ai = 1.

for the interaction length.Aossey et al. compared their results for the ratio of the phase shifts as a function

of the ratio of the amplitudes for the KdV solitons, with those obtained in theexperiments of Ikezi et a/., and those obtained from numerical work of Zabuskyand Kruskal and Lamb, as shown in Fig. 4.8(b). The solid line in Fig. 4.8(b)corresponds to (4.92). Results obtained by Aossey et al. for the interaction lengthare shown in Fig. 4.9(b) as a function of the amplitudes of the colliding KdV solitons.Numerical results (which were scaled) from Zabusky and Kruskal are also shownfor comparison. The dashed line in Fig. 4.9(b) corresponds to (4.93), with Ax = Iand K% = I.

Since the theoretical results obtained by the collision model based on macro-scopic bodies in Fig. 4.7 are consistent with experimental data for the KdV soliton,shown in Figs. 4.8(b) and 4.9(b), it is reasonable to believe the validity of themodel of collsion presented above, and results shown in Figs. 4.8(a) and 4.9(a) forthe microscopic particles described in the nonlinear Schrodinger equation whichwere obtained using the same model as that shown in Fig. 4.7. Therefore, theabove colliding mechanism for the microscopic particles shows clearly the classicalcorpuscle feature of the microscopic particles in the nonlinear quantum mechanics.

4.6 Collisions of Quantum Microscopic Particles

In previous sections we considered collision of microscopic particles described bythe classical nonlinear Schrodinger equation (3.2). However, a microscopic particleis itself quantized. Therefore, it is necessary to consider also collision betweensuch quantum microscopic particles which are described by the quantum nonlinear


Schrodinger equation (3.62).We have learned that soliton solutions and its properties for the classical non-

linear Schrodinger equation (3.2) can be obtained analytically using the inversescattering method. FVom the correspondence principle, it is natural to expect thequantum nonlinear Schrodinger equation (3.62) to serve as an elementary dynamicequation for the quantum microscopic particles in nonlinear quantum mechanics.The quantum nonlinear Schrodinger equation can be solved analytically. In statisti-cal physics, the quantum nonlinear Schrodinger equation is the equation describingthe evolution of a one-dimensional Boson system with a (5-function interaction inthe second quantized form, as mentioned in (3.72) - (3.74). It was first solved usinga method based on the Bethe's ansatz in the 1960s. Since the work of Bethe onthe isotropic Heisenberg spin chain in the 1930s, this method has been successfullyapplied to many models in statistical physics and quantum-field theory. The in-verse scattering method has been also applied to the solution of quantum nonlinearSchrodinger equation. Both methods can be used to construct the eigenstates ofthe Hamiltonian. The quantmn inverse scattering method constructs the creationoperators of these eigenstates and derives their commutation relations, while themethod based on Bethe's ansatz achieves this by solving the wave equation.

When the coefficient of the nonlinear term in the quantum nonlinear Schrodingerequation is negative, there are bound-state solutions that are the eigenstates of theHamiltonian with bound wave functions. Surprisingly, many treatments of thisproblem ended at this stage, leaving important problems unsolved such as howthese bound states are related to the soliton motions. Nohl was the first one togive an answer to this question, and later Wadati and Sakagami improved Nohl'sresults. They introduced a wave packet which is a time-dependent superpositionof the fundamental bound states and showed that the main element of the fieldoperator for this wave packet approaches the classical fundamental soliton withzero velocity when the photon number is large. They then generalized their resultsto moving solitons by a Galilean transformation. Although their results provideda good basis for quantization of nonlinear Schrodinger equation, their approachstill leaves some questions open. (1) A soliton state should be a time-independentsuperposition of the bound states so that it is a solution of the governing equation.(2) It is the expectation value of the field operator that corresponds to the classicalsoliton field, not the matrix element of the field operator. (3) The constructionshould be generalized to higher order soliton states to provide infromation aboutsoliton collisions.

Lai and Haus, and Wright constructed soliton states that meet the above crieria.This construction enables us to study the quantun effects of soliton propagationand soliton collisions. An approximate solution was obtained by these authorsusing the time-dependent Hartree approximation. In this approach, the quantumnonlinear Schrodinger equation is equivalent to the equation of evolution of a one-dimensional Boson system with a 5-function interaction. Using this approach, they


constructed approximate fundamental and higher-order soli ton states. It was foundthat a soliton experiences phase-spreading when it propagates. The soliton collisionswere also studied by these authors using the same approach.

The collision between quantum microscopic particles described by the nonlinearSchrodinger equation (3.2) is treated in the following using the approach of Lai andHaus. Obviously, in the second quantization representation, the quantum nonlinearSchrodinger equation corresponding to (4.40) can be (3.62), where <j>(x',t') and<j)+{x' ,t') are the annihilation and creation operators, respectively, at a point x' =xy/2m/h and time t' = t/h, which satisfy the commutation relations (3.63), and(3.64). Any quantum state of this system can be expanded in the Fock space by(3.69) - (3.72). Equation (3.72) can be solved approximately by the time-dependentHartree approximation. This approximation is valid when the number of particlesis large. The basis of the Hartree approximation is the assumption that everyparticle "sees" the same potential which is due to its interaction with other particles.Therefore we can use a single-particle wave function to describe a system of particles.

Lai and Haus denned explicitly a Hartree wave function based on the followingansatz

ffW(x[,...,x'n,t') = f[^n(x'j,t'). (4.94)

The function $ras are determined by minimizining the following function

i=frn{H){x'1,.--,x'n,t') i± + J2-^-2bY,wJ-*'i) *

J yOT j = 1 OXj !<;<.;<„tiH)(x'1,---,<,t')dx'i,---,dx'n (4.95)

= "/*" \jW + 7 ^ ~ (U - ^ n * - * - ] dX'-It turns out that the above function reaches its minimum value if $ n obeys theclassical nonlinear Schrodinger equation with the nonlinearity scaled by n — 1

i w = -^+2<>n-vb*»*«*»- (496)

This is one of the connections between quantum theory and classical theory. Equa-tion (4.96) has the following fundamental soliton solution according to (4.56)

*„(«',t') = 2r] e - ^ - ^ ' - ^ ' - ^ s e c h p ^ s ' - x'o + 4£t')]- (4.97)VW*-1)I

Contrary to the classical case, 7? cannot be arbitrary because </>„ has to satisfy thenormalization condition

J\$n(x',t')\2dx' = l. (4.98)


This leads to the following quantization condition

V=^\b\*^\b\. (4.99)

Inserting (4.99) into (4.97) and setting £ = -p/2, where p plays the role ofmomentum, one has

$ n p = Ijin - l)\b\ei(n-D2\l>ft'/4~ipH'+iP(x'-x'o)

xsech I i (n - l)|6|(i' -x'Q- 2pt')] . (4.100)

With (4.100), we can construct the Hartree product eigenstates according to (4.94)

\n,p,t')H = -j= ^J #np(*\*')^lV)d*'] |0>. (4.101)

A superposition of these states using the Poisson distrbution of n gives the funda-mental soliton state

i^>" = E-7V~K|2/2in>^'>H„ Vn!

= E^T e H a ° | 2 / 2 [/*nP(z',O0+W*']n|O>. (4.102)If the photon number is large, n0 = \ao\

2 » 1, the nonlinearity is not excessive,|6| <C 1, and the time of observation is limited, no,/no\b\2t < 1, then the summationin (4.102) can be approximated by an exponential term and \<ps)ff can be identifiedas a coherent state

M* « E ^ T e H a ° | 2 / 2 [J*noP(x',t')j>+(x>)dx'J\O)

= e-i«°(2/2 exp y$nop(x>,t')4>+(x')dx'j |0>. (4.103)

Here we have ignored the n dependence of $ n p by replacing the variable n by itsaverage n0. The mean field is

H^s\k^'Ms)n * ao$noP(x',t') « ^ V ^ x

eHn0-if]b\H'/4~iPH'+iP(x'-x'0)sech [1 ( n Q _ I ^ I ^ / _afQ_ 2^)1 ( 4 1 0 4 )

which is the classical solution.


If the time of observation is long enough, then the n dependence of the phasecannot be ignored. The mean field then becomes

B(+.\fa)\*.)B * Ee"ia°|2 i a°r a°fvw\ *Til Zi

n

jn»W»t>/4-ip>t+iP(x'-x'0)gedl ^ | 6 | ( x / _ ^ _ 2^ ) ] . (4.105)Equation (4.105) shows that the expectation value of the field is the average of aset of classical soli tons. This is surprising. It shows that a simple superpositionof solutions of the dynamic equation as the expectation value of the field is notanticipated in nonlinear quantum mechanics when the field propagates in a nonlinearmedium. Since in (4.105) components of different n's have different phase velocities,a microscopic particle (soliton) is expected to experience phase spreading during itsmotion.

Note that here we used a single value of the "momentum" p, rather than asuperposition. However, \n,p, t')jj is not an eigenstate of the momentum operatorp and thus a momentum distribution is associated with the state. We can also findthat a momentum distribution is necessary to construct a soliton state. Classicallythe nonlinearity and the dispersion balance exactly to form a soliton. Quantummechanically only the mean values of the two effects are in balance. There arestill higher-order phase-spreading effects and higher-order dispersion effects. Thequasi-probability-density method can be used to visualize these effects.

We now use the Hartree approximation to construct two-microscopic particle(soliton) states and study their collision. The construction is not as straightfor-ward as that of the fundamental soliton states because the two-microscopic particle(soliton) states during the collision and other times have to be treated differently.During the collision, the microscopic particles (solitons) are in the same spatial re-gion and interact strongly. All particles behave in the same way and therefore canbe represented by the same wave function. However, before and after the collision,the system consists of two independent groups of particles. The particles belong todifferent groups behave differently and therefore are represented by different wavefunctions, although particles in the same group still interact and can be representedby the same wave function. Based on this argument, Lai and Haus constructed atwo-soliton state that consists of two groups of n\ and ni particles, respectively.They assumed the following total wave function

m+n2

f$nM>---,<1+naJ)= II *mn2K-,*') (4-106)j=l

during the collision and

m ni+n

/s)«,(«i.-.<+n1.*i)=i;iI*£)(«bo).*1) n *£?(»w> (4-107)\Q\ j=l J=ni+1

Wave-Corpuscle Duality of Microscopic Particles m Nonlinear Quantum Mechanics 159

otherwise. In the latter expansion the summation is over Q, all possible per-mutations of [1,2, • • • , rii + n2]. The grouping of particles into [1,2,- • • ,n\] and[ni + 1, rii + 2, • • • , Hi + TI2] does not change the results of the summation because/ni n 2 is symmetric with respect to the x'-'s. All the wave functions $ni!n2! $nV>$ni satisfy the normalization condition (4.98). The connection between $ni,n2 and$ni > $ni can be established by noting that $n i , n 2 is the "mean" wave function of aparticle. When the two-microscopic particle (soliton) state is not in collision, sincethere are ni particles with wave function 3>nV and n2 particles with wave function

(2)$m , we can conclude that the asymptotic approximation of $m,n2 should be

We shall use (4.108) to establish the connection between the wave functionsbefore and after collision similarly to the WKB method in the linear quantummechanics. Inserting (4.106) into (4.93) and minimizing the functional, one gets

{ ^ i ^ = ' ^ ^ + 2 ( n i +n2~ l)b\*ni,n2\2$nun2. (4.109)

Substituting (4.107) into (4.95) and minimizing the functional, we have

i?W~ = J - S - + 2(ni" ^l*^!2*^- (4-m)If (4.108) is substituted into (4.109) and $ V and $£^ are separated, equations(4.110) - (4.111) can be obtained as well. This shows that (4.108) is consistentwith the criteria of the Hartree approximation. Moreover, equations (4.110) and(4.111) are the same equations as (4.96). This indicates that the non-collision two-microscopic particle (soliton) state is a product of two fundamental soliton states.The solution of (4.110) and (4.111) can been obtained as earlier and they are

$W) = J-(nj - 1)|6| ^-ifibft'/i-iplt'+ip^x'-x'^+ie, x

sech 7^l\b\(x'-x'jO-2Pjt')\, (j = l,2). (4.112)

However, the phases and mean positions before and after the collision can be dif-ferent. The difference can be determined by noting that before and after collision,

, / n i $W + . / n2 $(2) (4113)yni+n2

ni + V ni + n2 m { '

is the asymptotic approximation of the same $m,n2> *-e-i t n e asymptotic solution ofthe classical nonlinear Schrodinger equation (4.109). It was shown that the classical

(4.108)


nonlinear Schrodinger equation has two-soliton solutions. Before the collision, a two-soliton solution is similar to that of two fundamental solitons. After the collision,the solution is of the same form but their phases and positons have been changed.According to Zakharov and Shabat, the magnitudes of these shifts for one of themicroscopic particles (solitons) are given by

Mi(ni,Pi,n3>P2) = -2arg ( ^ 3 % ) (4.114)

„ _2 (tan-i [fe±i*>i _ tan-! [y*-"i>i \,I L2(p2-P i ) J L 2(pa-Pi) Jj *

^(n^.n^^llnd^H)w ^ i { l n h ~ p i ) 2 + 1 T ( n 2 - n i ) 2 ] (4-n5)

- l n ^ - p i ^ + ^ n a + m)2]},

where Ci = £i + ir/i, C2 = £2 + J/fc- Similar shifts occur for the second microscopicparticle (soliton). Therefore, the characteristics of collision between quantum mi-croscopic particles described by the quantum nonlinear Schrodinger equation are thesame as those of classical microscopic particles described by the classical nonlinearSchrodinger equation in nonlinear quantum mechanics. This result also containsthe quantum fluctuations produced in the collision. The S8iS and Sx^s (i = 1,2) arefunctions of rij (j = 1,2) and are determined probabilistically.

In the above calculation, we used the time-dependent Hartree approximation inthe constructino of approximate eigenstates of the particle-number operator, insteadof the exact eigenstates of the momentum and Hamiltomian operators. The solitonsolution thus obtained experiences phase spreading when it propagates. The Hartreeapproximation suppresses the effect of the momentum uncertainty. A distributionof momentum must be associated with a soliton with a mean position, just as theuncertainty of particle number causes a phase-spreading of its own. Lai and Hausused Bethe's ansatz to construct the exact eigenstates of the Hamiltonian, andsoliton states, by superimposing these eigenstates

l ) = E a » [9n(p)\n,p,t')dp, (4.116)n

where

a - _ô_e-l«o|2/2

^=(Ap)i /W/<e x p H i i # - M=9{p)e~inpx'°


and the an's satisfy the following

n

Hence both fundamental and higher order soliton states were constructed andtheir mean fields were also calculated to justify the construction. Classical resultscan be recovered by taking the limit of large particle number. It was found that dueto the uncertainty of momentum a soliton experiences dispersion when it propagatesand such effect is very small when the average particle number of the solitons is muchlarger than unity. Phase and position shifts due to a collision and the uncertaintyof these shifts were also obtained by these authors as follows

801 = 6{nw + l,Pio,n2Q,P2o) -^(nio,Pio,«2o,P2o),$x> ~ 1 dO(nio,pio,n2o,p2o)

1 ~ nw dpi

for the phase and position shifts, respectively, of the first microscopic particle (soli-ton) and

662 = 0(nio,PlO,™2O + l>P2o) - #(»1lO,PlO>ft20,P2o),

, 1 dO(nw, pio,n2o,P2o)OX2 ~ S

"20 OPifor the second microscopic particle (soliton), respectively. Therefore property ofquantum collision of microscopic particles in quantum nonlinear Schrodinger equa-tion is basically the same as that of the microscopic particles in the classical non-linear Schrodinger equation.

4.7 Stability of Microscopic Particles in Nonlinear Quantum Me-chanics

Stability is another important property of macroscopic particle. In this section wewill demonstrate that the microscopic particles in nonlinear quantum mechanicshas similar property as macroscopic particles.

Let us first define the stability of microscopic particles in nonlinear quantummechanics. Usually three types of stability of microscopic particles should be con-sidered, (1) with respect to a perturbation of the initial state, (2) with respect toa perturbation of the dynamic equation governing the system dynamics (structuralstability); (3) with respect to minimal energy state under an externally appliedfield.

In the first case, the problem has been investigated in detail using various tech-niques and approaches (linear and nonlinear). In the linear approximation the sta-bility problem is usually reduced to an eigenvalue problem of linearized equations.In the nonlinear case it is reduced to the study of Lyapunov inequalities.


Study of structural stability of microscopic particles can be done in the frame-work of dynamic equations under different types of perturbations. Attempts havebeen made to construct a general perturbation theory of the microscopic particlesbased on the dynamic equations by using the Green function method and spectraltransformation (i.e. a transformation from the configuration space (x,t) to thescattering data space based on the well-known two-time formalism). However, thelatter technique is suitable only for a rather restricted class of perturbation func-tionals. Although some results were obtained in this direction, the studies cannotbe considered as complete. In some cases numerical studies are very effective tools.Prom the computational point of view, both problems can be investigated in theframework of a unified approach. In the first case one studies the dynamics, de-scribed by an unperturbed dynamic equation, of a perturbed or unperturbed initialstate given in the form of a soliton solution, the stability of which is examined.In the second case the initial state evolution is governed by a perturbed equation.In both cases the solutions depend on some parameters which are slowly varyingfunctions of time.

In the first case, a solution is considered stable if initial perturbations are notmagnified as the initial state evolves with time. In accordance with this defini-tion, weakly radiating soliton-like solutions which are not destroyed under initialperturbations can be considered stable. Obviously, structure-stable solutions arethe solutions which conserve their shape for sufficiently long time. The notion of"sufficiently long time" is relative to the time scale of physical processes occurringin the systems. In the following, a few examples illustrating investigations in bothdirections will be given. The examples are selected for the purpose of illustratingthe problems. They are by no means complete. More details on these problems canbe found in the relevant publications of Makhankov et al..

4.7.1 "Initial" stability

It is known that in the case of Lagrangian relativistically invariant equations describ-ing a complex field, besides the conventional energy and momentum conservations,there is an additional "charge" conservation which is connected to the LagrangianU(l) symmetry, Q = 0 with Q = 9 / 4>*t, <j>dx'. For non-relativistic models of thenonlinear Schrodinger equation type, instead of Q, the wavefunction normalization(particle number) is conserved, N = 0 and N — J \<j)\2dx'. By the variationalprinciple Q = const, or N = const., one may prove a theorem (the Q-theorem) thatformulates sufficient conditions for the stability of complex soliton-like solution. Thestability region of these solutions is determined by the inequality d\nQ/dlnu> < 0for the nonlinear Klein-Gordon equation and dN/du> < 0 (LO < 0) for nonlinearSchrodinger equation (w < 0 is necessary for the existence of soliton-like solutions).

To obtain these formulae, we consider the soliton-like solutions of the generalizednonlinear Klein-Gordon equation, (D+l-\<f>\n)<t> = 0, where D = d^-d^-d^-d^,


and the generalized nonlinear Schrodinger equation, (idf + d2, + d2, + d2, + \(p\n)(j> =0, with a |< |"0 type nonlinearity, in the rest frame with a time dependence of4>(x',t') = <p(x')e~zut , which minimizes the Hamiltonian of these systems. Thecommon form for both equations in the 1 + 1 dimensional case is

-Vx'x' + k2<p - <pn+1 = 0 (4.117)

where k2 = 1 - LJ2 for the nonlinear Klein-Gordon equation and k2 = —w for thenonlinear Schrodinger equation. The w dependence of Q and N is easily obtainedby the scale transformation x' -t k'1^', k2lny. In terms of these variables,equation (4.117) has the following form (free of k)

-yvc + y-yn~1=^ (4-ii8)

with

N = I <p2dx' = fc(4-")/" Iy2dt;t = k^-n^nC'(n), Q = uN.

Calculating the derivatives duN and d^Q, we find that the stability regions are

(i) i > " 2 > ^ ;(2) n < 4 .

where w = -fc2 is arbitrary. That is, in both cases, stable soliton-like solutionsexist only for n < 4. This result shows that the dynamic equations (3.2) - (3.5) innonlinear quantum mechanics have stable soliton solutions.

We can also study the initial stability of microscopic particles following theapproach of Zakharov and Shabat. As a matter of fact, in the above discussion, wedid not consider the general evolution of the initial conditions, where an importantrole may be played by the "nonsoliton" part of the solution, which is connected tothe scattering quantity b(£,t') in (4.44). Here we consider only the case when thisquantity is small, i.e.,

and the coefficient a(£) has only one zero, £ = £i, in the upper half-plane. Such achoice for the initial conditions corresponds to posing the problem of the stability ofa microscopic particle (soliton) with parameters ft + irji = £i when the microscopicparticle (soliton) is perturbed by a field with a continuous spectrum.

Let us consider the systems (4.47) and (4.48) with N — 1 and express thefunctions ^>|(a;',î) and tpi(x',£i) in (4.48) in terms of

*(2)(Cl') = ^ / r f ^ (4'120)

(4.119)


After substituting it in (4.47), we obtain a system only for the quantities * iand \&2- This system contains as a coefficient the small quantity c(z',£,t') —b(£,t') exp(i£x')/a(£). Keeping terms up to quadratic in c, we have

*1=0, and ^ c ^ U ^ l + ff l^l- j—f)], (4.121)

where A = ci(0)e2lf i :c '. From (4.49) we now obtain

1 f+°°cj>{x',t') = <j>0{x',t') - - *S(sU,O, (4-122)

* J-oo

fo(a/,O = (A*)2/l + |A|4, (4.123)

where 4>o is the soliton (4.56) with £ + 277 — £1. Recognizing that

we find that the integral in (4.122) decreases as 1/Vt*, as t; ->• 00. This means thatthe soliton is stable under the perturbation by a field with a continuous spectrum.As t' -¥ 00, the solution develops asymptotically into a soliton.

A general perturbation would shift the position of the zero of a(£) on the complexplane and by the same token perturbs the parameters of the microscopic particle(soliton). As t' — 00 the solution evolves asymptotically into this perturbed micro-scopic particle (soliton).

4.7.2 Structural stability

The first studies of structural stability concerned the effect of weak dissipation inthe nonlinear Schrodinger equation. The behavior of microscopic particles (solitons)depends essentially on the form of the perturbation term. In the case of power lawdamping 7 w ekn (k is the wave number, e is a small number, n is a constant),Makhankov et al. demonstrated that only when n = 2 the soliton does not changeits shape with time (which is connected to the scaling properties of the nonlinearSchrodinger equation). If n ^ 2, evolution of the microscopic particle causes itsshape variation proportional to the coefficient of the damping term. For n = 2, 3,and 4 the inequalities e < 0.2, e < 0.03 and e < 0.01 hold, respectively. Structuralstability of microscopic particles for nonlinear Schrodinger equation were studied byYajima et al.. One of the main features of perturbed microscopic particles (solitons)is the appearance of oscillatory tails on their back or front. Structural stability ofother microscopic particles (kink solitons) was studied by Fogel et al. and Currie etal., who worked on the initial value problem of the following Sine-Gordon equation,

a(x,O) = f(x).

Here the x-dependence of the eigenfrequency u>0 of the microscopic particles can beused to simulate the presence of impurities (non-uniformities) in the system. Thethird term in (4.124) describes the influence of external actions (fields, currents, andso on). Analytical and numerical results obtained by Currie et al. show that for suf-ficiently small perturbations the Sine-Gordon kinks behave as Newtonian particleswith an internal structure in external fields, they may be accelerated (decelerated),may radiate, and change slightly their structure, which is accompanied by tran-sition radiation. However, they can also be trapped in some spatial region. Theradiation in the model described by (4.124) would strongly influence the structureand especially the dynamics of microscopic particles, for example, their formation,lifetimes, breaking up.

We now briefly discuss the structural stability of a perturbed nonlinearSchrodinger equation (3.2) with A(<p) = 0, which has the form

i&'+tfw+M20 = M<- (4-125)

It is extremely difficult to obtain the Lax representation and the Lagrangian ofthis equation. Makhankov et al. considered at first the plane-wave solutions andtheir stability. Multiplying (4.125) by 0*, then by <£*,, and combining the resultingequation with its complex-conjugate, we get

Pt' = -JX', |p| = H 2 , (4-126)

pt, = 40R£, + 2dx.[H - » 0 W * ' + 0**.<»')]. (4-127)

Ut, = -ppdx,{%<t>2) + dx.\pp - 9 0 8 ^ , - 2<f>*x,<j>x,x,), (4.128)

where

j = /3%<j>2 -p, p = 290V,. , W = |0x '« ' | 2 - | l0 l 4 - (4-129)

Integrating (4.126) - (4.128) over x', we found that for the entire set of integrals ofmotion of the non-perturbed nonlinear Schrodinger equation, only the first, Nt> = 0,survives, i.e., the particle number in the system (4.125) is conserved.

We assume that the solution of (4.125) is of the form

<j>(x',t') = D'e-i^t'+*°\ (4.130)

with

u> = 1 — D'2 and I9Q = constant.


Separating the real and imaginary parts of the solution, <f> = F - iF', equation(4.125) becomes two equations

Ff = F' + pF'x, - F'x,x, - (F2 + F'2)F,

F't, = -F + pFx, + Fx,x, + (F2 + F'2)F. (4.131)

In the zeroth-order approximation in D'2, we get the following solution of the lin-earized system

Fo = D'cosd',

•d1 = kx1 - cjot', (4.132)

F = D' (1±*1 sintf' - &— costf1^V w 0 oJo )

and the dispersion relation

u>2 = (l + k2)2+p2k2.

Corrections proportional to D'2 can be found by the expansion

SF = F - Fo = D' J2 [a(2n+1) cos(2n + l)i?' + a(2n+1> sin(2n + l)i?'] ,n

SF' = F' -F^D'^2 [42"+1) cos(2n + 1)& + ft(2n+1) sin(2n + l)i?'] , (4.133)

n

where a(£+1) « tgr" and C " + 1 ) « b ^ ^ • The result is

CJ2 = (1 + fc2)2 + pk2 - 2(1 + fc2)D'2. (4.134)

To look at the stability of the plane-wave solutions, we consider the solution(4.130) as D' -4- 1. Assuming

F = £»'cosi?0 [l + <5ieifca:'-im'] ,

F' = D' cosi9o [l + 62eikx'-int>] ,

(<5* = Si), and linearizing the system in (4.131) in S\ and S2, we can get

^ V = jfca-02-2-2i£cos0o,

or

n = ±Vâ(v\/^?+l-V1/^?"1)'where a = fc2(A;2 + /32 — 2), c = 2/3fccos2??o- Prom these formulae we see that thesolution (4.130) is unstable at large amplitudes (D' -*• 1). The growth rate of the


instability has a peak, 7max = 1 - 02/2, for perturbations with a wave numberk=l- p2/4.

The stability of solution of (4.131) can be examined similarly by expanding Fand F' in harmonics of frequency OJ, which, however, is much more tedious. Sincethe amplitudes are small, i.e., D' <C 1 and w -> 1, assuming again that ft < ui andk <gil, we can get the following after some lengthy but straightforward derivations

( £ ) ' = -2D'2 ( l + i/J2fc2) + fc2 ( l + i/32) + l-pkD'\

where a = k2(k2-/32k2/2-2D'2) and c = /3A;3£>'4/4, which means that this solutionis still unstable.

Therefore, the plane-wave solutions are unstable for both small and large am-plitudes. Such an instability is responsible for the breakup of initial packets intosolitons in the framework of the nonlinear Schrodinger equation. We note thatft2 = k2{2D'2 - k2) when /? = 0 in (4.134). Although the instability is slightlymodified at non-zero, but small 0, it still leads to the breakup of the initial packetinto soliton-like objects as in the nonlinear Schrodinger equation case. This resultshows that the plane wave (4.130) is not a solution of (4.125).

For all the examples considered the perturbation is small for small /?. It isinteresting to observe its influence on a soliton-type quasi-stationary solution. Wecan write (4.125) in terms of the amplitude D and phase 1? of in the following

Df + 2dx'Dx, - pDx, sin2i? - ptix,Dcos2d + Ddx>x. = 0,(4.135)

- D i V - D + Dx>x> - Dti2x, + -D3 + 0DX> cos2t? - p&x,Dsm2d = 0.

Multiplying the first equation by D, we get

dfD2 + dX'[{2tix, - Psin2i9)D2] = 0.

It is easy to find a solution for i?^ -+ 0. In that case,

Dv = PDX- sin2i9;

-Di?t< -D + Dx,x> +D3 + &DX, cos 2i? = 0.

Thus D is a function of

ft'z = x' +/3 sin2i?(r)dT

Jo

and a solution for $ may be found in the form of

i?t, = -u + Pf(z) cos 2tf.


Here the amplitude D satisfies the nonlinear Schrodinger equation

(w - 1)D + Dzz + D3 = 0,

if the condition f{z) = 92(lnD) is fulfilled. Makhankov et al. finally obtained thesolution

D(z) = /.osech O^j ,

where

tio = y/2(l-u),

or

4> = Mosech te ^ + cos 2t'^ 1 (4.136)

H-#4)''+f^(^)H}which holds when /?//o <C 1 and pt§ <C 1. This is a perturbed (oscillating) versionof the conventional nonlinear Schrodinger equation soliton. It shows that micro-scopic particles can still maintain the soliton feature in motion under structurealperturbation.

In addition to the above solution (4.136), equation (4.129) has a family of station-ary solutions which essentially differ from a soliton solution. To find the stationarysolutions, we assume df — 0 in (4.135) and integrate the first equation over x'

D2 (tix /3sin2i? J = const.

Because of the boundary conditions at infinity the constant must vanish. Therefore,the phase •d satifies the stationary Sine-Gordon equation

B2

dxixi = — sin4i9,

which has the solution

tf^tan-ife^'-^j+tfi,

where

tf+=(n+|)7r> d-=nn' (» = 0 ,±l , -" ) -

In this case the second equation in (4.135) is reduced to

Dx.x> -D + D3 - -/32D sin2 2d - $DX, cos 2t? = 0.4


Here the sign of the last term is determined by the phase behavior at -co. Fixingt?+ or i?_ as an "initial" phase, we can get

Dx,x, -D + D3 - -/32£>sin2<x ± /3DX, COS2CT = 0.

Hence for /? -C 1 we get

Dx.x. -D + D3 ±/3Dtanhfix' = 0. (4.137)

In the first case, the existence of a solution is proved. Equation (4.137) has acountable set of soliton-like solutions Dn, one of which, Do, is nodeless, and theother ones have an increasing number of nodes n.

The above results show that the influence of the small term /?</>*, can lead to in-teresting results. This means that the structural stability of the nonlinear quantummechanical systems is a rather delicate problem which must be solved either for aspecific system or for a restricted class of systems.

4.8 Demonstration on Stability of Microscopic Particles

In addition to the initial and strutural stabilities discussed above, we can alsodemonstrate the stability of the microscopic particles in nonlinear quantum mechan-ics by means of the minimum energy concept. A system of microscopic particles isstable, when the particles are located in a finite range which results in the lowestpotential energy. This stability principle is very effective, when the microscopicparticles are in an extenally applied field. As a matter of fact, since interactionbetween the microscopic particles is very complicated, it is not easy to define thebehavior of each one individually. We cannot use the same strategies as those usedin the discussions of initial and structural stabilities in this case. Instead, we ap-ply the fundamental work-energy theorem of classical physics: a mechanical systemin the state of minimal energy is said to be stable, and in order to change thisstate, external energy must be supplied. Pang applied this fundamental conceptto demonstrate the stability of microscopic particles described by the nonlinearquantum mechanics, which is outlined in the following.

Let (f>(x, t) represent the field of the particle, and assume that it has derivativesof all orders, and all integrations of it are convergent and finite. The Lagrangiandensity function corresponding to the nonlinear Schrodinger equation (3.2) withA(<j>) = 0 is given in (3.93). The momentum density of this field is defined asP = dC/d<f>. The Hamiltonian density of the field is given in (3.99). In the generalcase, the total energy of the field is a function of t', thus (4.4) is replaced by

Eiti)=IZ \^]\'l^+vix'M2 dx>- (4-i38)


However, in this case, b and V(x') are not functions of t'. So, the total energy ofthe system is a conservative quantity, i.e., E(t') = E = const., as given in (4.4).

We can demonstrate that when x' -> ±00, the solutions of (3.2) with A((p) = 0,and <j>(x',t') = 0 approach zero rapidly, i.e.

lim 4>{x',t')= lim | ^ = 0 .|z'|->oo |z|->oo dx'

Then/•OO

/ <t>*<j>dx' = const, (or a function of t'),J-00

lim * v £ * = lim 2£*V = 0.|i'|-t-oo dx' |x'|-».oo 9a;'

The average position or the center of mass of the field (p can be representedas (x1) = x'g, as given in (4.9). Making use of (4.13), the average velocity or thevelocity of the center of mass of the field can be written as (4.18).

However, for different solutions of the same nonlinear Schrodinger equation (3.2),JLQQ <fr*<(>dx'', (a;') and d(x')/dt' can have different values. Therefore, it is unreason-able to compare the energy of one particular solution to that of another solution.The comparison is only meaningful for many microscopic particle (soliton) systemsthat have the same values of / ^ (f>*<j)dx' = k, {x') = u and d(x')/dt' = u at thesame time t'o. Based on these, we can determine the stability of the soliton solutionof the nonlinear Schrodinger equation. Thus we assume that different solutionsof the nonlinear Schrodinger equation (3.2) with A{4>) = 0 satisfy the followingboundary conditions at time t'o

[°° <t>*4>dx'=K, (x%,=t,o=u(t'o), ^ =«(«{,). (4.139)J-00 a i t'=t'o

Now we assume that the solution of nonlinear Schrodinger equation (3.2) withA(<j>) = 0 is of the form

(j)(x',t') = tp(x',t')ei$ix''t'). (4.140)

Substituting (4.140) into (4.138), we obtain the energy of the system

E -11 [{&)'+^ ( £ ) ' - b v ' + " H *"• ("41>

Equations (4.139) then become

/

oo roo an

xVdx' 2 J^dx<1 <p-ax=n, -fk = «(f0), - ^ = u(t0). (4.142)

J-°° / <p2dx' / <^dx'J~ OO J— OO


Finding the extreme value of the functional (4.141) under the boundary condi-tions (4.142) by means of the Lagrange multiplier method, we obtain the followingEuler-Lagrange equation,

-0P = \v(X')+dtocâ*' - «(*{,)]

+C3(t'o) [ 2 ^ - «(f0)] + [^j y - 6<p3 = 0, (4.143)

_ ^ + 2 ^ + 2 C 3 ( i i ) ^ = 0 , (4,44)

where the Lagrange factors C\, Ci and C3 are all functions of t'. Let

C5(t'o) =-\u(t'o).

If

rift

we can get from (4.144)

d2e2 dip _ ~9^2

Integration of the above equation yields

* = d6 1 'ft? - 2 U ^

or

S« _ £(<o) , «(*o) f 4 14 5 )

a * * = * " ^2 2 'where g(t'o) is an integral constant. Thus,

6(x',t') = 9(t'o) [X% + ^-x' + M(t'o). (4.146)

Jo V z

Here M(t'o) is also an integral constant. Let again

C2G0) = ^(ô)- (4-147)


Substituting (4.145) - (4.147) into (4.143), we obtain

-bv3 + p-. (4.148)

Letting

Cl{Q = U^f^-^ + M(t'o) + p', (4.149)

where /3' is an undetermined constant, which is i'-independent, and assuming Z =x' — u(t'o), then

ay _ ayd(x>)2 ~ dz2

is only a function of Z. In order for the right-hand side of Eq. (4.149) to be afunction of Z, the coefficients of y>, ip3 and 1/ip3 must also be funciton of Z. Thus,9(t'o) = 9o = const., and

V(x>) + J^x' + M(t'0)-^. = V0(Z).


•0p = {VW - u(t'o)} + / ? > - bV3 + 9^-. (4.150)

Since V0(Z) = V0[x' - u(t'o)] = 0 in the present case, (4.150) becomes

g-.^-v+m. (4.I5I)Therefore, ip is the solution of (4.151) for /?' = constant and g(t'o) = constant. Forsufficiently large \Z\, we may assume that \ip\ < /3/|Z|1+A, where A is a smallconstant. However, in (4.151) we can only retain the solution <p(Z) correspondingto g(t'o) = 0 to essure that lim|f ^ d?<p/dZ2 = 0. Thus, equation (4.151) becomes

As a matter of fact, if dO/dt' = ii/2, then from (4.149) and (4.150), we can verifythat the solution given in (4.8) satisfies (4.152). In such a case, it is not difficult toshow that the energy corresponding to the solution (4.8) of (4.152) has a miminalvalue under the boundary conditions given in (4.142). Thus we can conclude thatthe soliton solution of nonlinear Schrodinger equation, or the microscopic particlein nonlinear quantum mechanics is stable in such a case.

(4.152)


4.9 Multi-Particle Collision and Stability in Nonlinear QuantumMechanics

One of the most remarkable properties of microscopic particles (solitons) in non-linear quantum mechanics is that elastic collision between them results only in aphase shift, and the phase shift due to the collision with several microscopic par-ticles (solitons) is the sum of partial phase shifts resulting from separate collisionsbetween a given particle and each other microscopic particle (soliton), as discussedin the previous sections. Their stabilities have a similar property. This property iscommonly referred to as absence of multisoliton (or "many-particle") effects in theintegrable models. A natural question arises: what about multi-microscopic parti-cle (soliton) collisions in the non-integrable nonlinear quantum mechanical models?It is commonly believed that the main difference between an integrable model anda non-integrable model is due to radiation emitted by the interacting microscopicparticles (solitons). Prauenkron et al. studied the properties of multi-microscopicparticle collisions and demonstrated the existence of nontrivial effects, which donot involve radiation and exist for any value of the perturbation parameter e inmulti-microscopic particle collisions, by extended numerical simulations based on asimple integration scheme. These effects are due to energy exchange between thecolliding microscopic particles (solitons) and excitation of internal soliton modeswhich distinguishes multi-microscopic particle collisions in the integrable and thenonintegrable models.

To understand the multi-particle effects in multi-microscopic particle colli-sions in nonlinear quantum mechanics systems, we consider a perturbed nonlinearSchrodinger equation with a small quintic nonlinearity,

«|£ + 0 + 2M2tf = <#|V (4-153)where e is the amplitude of the perturbation which is assumed to be small. Equation(4.153) can be used to describe evolution of the electric field of a light in an opticalwaveguide with an intensity-dependent refractive index nni (I = \cj>\2), which slightlydeviates from the Kerr dependence. Its solutions will be given in Chapter 8.

In the absence of the perturbation (e = 0), the nonlinear Schrodinger equation(4.153) is known to be exactly integrable and it supports propagation of an mi-croscopic particle (envelope soliton) with amplitude Ao and velocity v, as given in(4.8). Three elementary integrals are the norm N, field momentum P and energyE, represented here by

iVs = 2A0, Ps = Ao, Es = Âov2-Â3

Q (4.154)

respectively. In addition, equation (4.153) possesses an infinite number of integralsof motion. Unlike higher (non-elementary) integrals of motion, the above three basicquantities remain conserved in the case of perturbed nonlinear Schrodinger equation,


(4.153). As a matter of fact, the effect of conservative perturbation in (4.153) on themicroscopic particles in nonlinear quantum mechanics is trivial. This is consistentwith the fact that a perturbed equation has an exact solitary wave solution whichis a slightly modified nonlinear Schrodinger soliton. However, interaction of thesesolitary waves for (4.153) differs drastically from the interaction of two solitonsdescribed by the integrable nonlinear Schrodinger equation.

To study multi-microscopic particle collisions in nonlinear quantum mechanics,Frauenkron et al. integrated (4.153) using the fourth-order symplectic integra-tor. The symplectic numerical method is much better than traditional numericalschemes, because it allows conservation of the norm within a relative accuracy of10~n and conservation of the energy within 10~6 during the integration. They useda grid spacing of dx' = 0.1, with the total length L = [—800,800], and a time stepdt' = 0.005. They studied collisions of three microscopic particles (solitons) andtwo microscopic particles (solitons), respectively, in nonlinear quantum mechanics.In the case of three microscopic particles, a "fast" particle, with a relatively largevelocity, was taken as the exact solution of (4.153) to avoid initial oscillation of itsamplitude. The other two "slow" microscopic particles were modeled by an exacttwo-soliton solution of the unperturbed nonlinear Schrodinger equation to avoid ra-diation due to strong initial overlapping. The microscopic particles were put on thegrid at positions a;'- = —650 and ia;^ = ±45 with initial amplitudes Aj = l/\/2,As\ = 0.35 and velocities Vj = 3.0 and vs\ = ±0.2. These values were selected insuch a way that all three microscopic particles collide when the two slow particlesare overlapping significantly (see Fig. 4.10).

Fig. 4.10 Triple microscopic particle collision in nonlinear quantum mechanics. The fast micro-scopic particle with amplitude Aj and velocity Vj collides with two slow microscopic particlesof equal amplitudes Asi propagating towards each other with velocities ±vsi, so that the threemicroscopic particles significantly overlap at the moment of collision.

Frauenkron et al. first studied collision of two-microscopic particles in nonlinearquantum mechanics. Simulations of the collision were performed using the same


initial conditions as mentioned above, but with only one slow microscopic particle(with negative velocity). For e < 0.1, the changes in velocities of the microscopicparticles after the collision were so small that they could not be measured with asufficient resolution, and they were expected to be on the order of the numericalerror. To understand this, we note that for collision of two microscopic particles,inelastic effects may exist due to emission of radiation. The emitted energy Eradhas been calculated analytically in the limit of two symmetric microscopic particleswith equal amplitudes A\ = A2 — A and velocities i>i]2 = ±v, where v > A. Theresult is £racj = C'e2A7[l + F(v/A)}, where C" is a constant. Similarly, it was foundthat the radiation-induced change of the norm is iVra(i = {A/v2)ET&d. Because of thesymmetry, the total momentum of the two microscopic particles was not affectedby the radiation. Due to conservations of the norm and the total energy, we have

NS = N'S + JVrad and Es = E's + £ r a d ,

where N's and E's are given in (4.154). The left-hand sides of the above equationspertain to the microscopic particles before the collisions, whereas the right-handsides take into account the changes due to the radiation emitted. These two bal-ancing equations for the conserved quantities allow us to determine the changes inthe amplitudes AA and velocities Av of the microscopic particles, and they aregiven by

•I nA

AA0 = - j £ r a d , and Av = —£-Erad.

This shows that the changes in the microscopic particles' parameters are propor-tional to e2 and inversely proportional to v2. For collision between two microscopicparticles with large velocities v, the interaction time is small and therefore thechanges in the microscopic particles' parameters are smaller. For two different mi-croscopic particles these changes will be further reduced due to shorter time ofoverlapping between the microscopic particles.

Unlike the case of two microscopic particles in nonlinear quantum mechanics,Frauenkron et al. obtained nontrivial effects which are in the first order of e,for the collision of three microscopic particles. Figures 4.11 show the changes inthe velocities of the microscopic particles after the collision for different values ofthe perturbation amplitude e. It is clear that the change is linear in e. Afterthe collision small oscillations in the amplitudes of the microscopic particles wereobserved, which is more prominent for larger e. It is obvious that these effectsare different in collision of two microscopic particles and that of three microscopicparticles.

The results also show a nontrivial energy exchange in the first order of theperturbation amplitude e. To find analytical results, Frauenkron et al. considereda collision between a fast microscopic particle with amplitude Aj and velocity Vj,and a symmetric pair of "slow" microscopic particles with equal amplitudes Asi andopposite velocities ±vsi. Assuming Vj 3> vsi > Asi, they presented a three-soliton


Fig. 4.11 Velocities of the microscopic particles after collision between three microscopic particles,for different values of e: (a) the fast particle with Vj = 3.0; (b) one of the slow particle withDS1 = 0.2. Open squares and circles are numerical data. Dashed lines are fitted linear functions.

solution in the form of <j) = (/>j(vj) + <j>ai(vsi) + si(-vsi), and calculate the changesin the microscopic particles' parameters by means of a perturbation theory basedon the inverse scattering method, with the one-soliton Jost functions. If the initialamplitudes of the slow microscopic particles are equal, the microscopic particlesparameters after the collision are given by

v'j = VJ + AVJ and ± v'sl = ±vsl ^ Ausi,

where

At;,- = -192e^^-G(5) and Avsl = 96e^lG(6). (4.155)V] Vj

Here 6 = Asi(x'^2 ~ x'sii) is t n e separation between the slow microscopic particlesat the moment of collision with the third (fast) microscopic particle. In (4.155),

sinh 5 [ tanlr<5 J

which vanishes as 5 —* 0 and S —> oo, and is of the same order as the perturbation.There is no change to the amplitudes of the microscopic particles. Because theamplitudes of the microscopic particles do not change after the collision, they furtherfound that AN = J2i A-Aj = 0, and the total momentum is conserved up to theorder of v~2 due to the symmetry of the problem, whereas, the energy of themicroscopic particles changes by AE = (2AS1VSIAVSI+AJVJAVJ). Using the resultsof (4.155), they got AE = 0. This shows that the results given in (4.155) areconsistent with the conservation laws.

To analyze the dependence of the energy exchange between the microscopic par-ticles on the separation of the microscopic particles, Prauenkron et al. fixed the

(4.156)


perturbation amplitude at e = 0.01 and varied the initial separation between thetwo slow microscopic particles. This results in a variation of the average distancebetween the two slow microscopic particles at the moment of collision, i.e., effec-tively 8 in (4.156). To measure the separation, they considered simultaneously thecollision between the two slow microscopic particles under the same conditions butwithout the third microscopic particles. The initial distance between the slow mi-croscopic particles, x'D, was measured at the moment when the fast microscopicparticle passed the point x' = 0. The final velocities were measured by the linearregression analysis. The numerical results are summarized in Fig. 4.15, where therelative changes in the velocities of the microscopic particles, AVJ and Awsi, re-spectively, are shown as functions of x'D. Indeed, the changes in the velocities ofthe slow microscopic particles are due to the energy exchange during the collision.This effect strongly depends on the separation between the colliding microscopicparticles at the moment of collision and it vanishes for larger separation, which isdifferent from what has been predicted by previous theory. It is also noted thatthe energy exchange vanishes when the centers of the slow microscopic particles(solitons) approximately coincide.

Fig. 4.12 Changes in the velocities of the microscopic particles vs. the extrapolated initial dis-tance between the slow microscopic particles xD. The dotted line shows the change in the velocityof the fast microscopic particle, Auj, and the dash-dotted line shows that of the slow microscopicparticle, Avs\. Symbols indicate data obtained from direct numerical simulations at e = 0.01.

More detailed analysis indicates that the energy exchange is more complicatedand it involves the excitation of an internal mode of the fast microscopic particles.In fact, the internal mode appears as a nontrivial localized eigenmode of the linear


problem associated with the microscopic particle (soliton) of the perturbed nonlin-ear Schrodinger equation (4.153). This mode always exists if e > 0, and describeslong-lived oscillations of the amplitude of the microscopic particles. Therefore, in-ternal mode of solitons plays an important role in the energy exchange betweenmicroscopic particles during the three particle collisions.

In conclusion, the above discussion shows that, unlike collisions betweeen twomicroscopic particles, the collision involving three microscopic particles in nonlinearquantum mechanics is accompanied by a radiationless energy exchange among theparticles and excitation of internal modes of the colliding microscopic particles.Both effects lead to changes in the soliton velocities which are of first order in theperturbation amplitude e. This effect depends nontrivially on the relative distancebetwween the microscopic particles, and vanishes when the microscopic particlesare separated by a large distance.

4.10 Transport Properties and Diffusion of Microscopic Particlesin Viscous Environment

We have seen that a remarkable feature of microscopic particles in nonlinear quan-tum mechanics is their ability to maintain the shape, velocity and energy duringpropagation in free space. But what about propagation of microscopic particlesin viscous systems? Can the microscopic particles retain these properties in suchsystems at finite temperature? This will be the focus of this section. We will dis-cuss the dynamics of microscopic particles in such viscous systems. The motionof microscopic particles in viscous media can be described using two parameters,the damping and the diffusion coefficients. The former is related to the system-atic force applied to the microscopic particles by the environment, while the latterdescribes the effect of fluctuations in the interaction. We will show that the micro-scopic particles move in the form of low energy Brownian motion in such a nonlinearsystem due to the scattering of internal excitation. It also exhibits classical featuresof microscopic particles in nonlinear quantum mechanics.

As metioned above, in the nonlinear quantum mechanics the position of thecenter of a microscopic particle (soliton), being a function of time, is sufficient todescribe its dynamics in the classical nonlinear Schrodinger equation. In the classi-cal field theory, we may think that the microscopic particles are field configurationswhich move in space without changing their shape. In quantum field theory the cen-ter of mass of the microscopic particles plays also a special role. It can be viewed asa true quantum dynamical variable and the microscopic particle can be treated as aquasi-classical particle. Thus classical and translational invariant theories with thesoliton solutions described by dynamic equations in nonlinear quantum mechanicscan be quantized using the collective-coordinate method. The problem is that atfinite temperature not all the degrees of freedom contribute to the formation of thesoliton, and the soliton is never free to move as in the classical theory. In other


words, there is always a residual interaction due to quantum degrees of freedomwhich changes drastically the dynamical properties of the microscopic particles.This residual interaction results in a damped motion of the microscopic particle(soliton). The microscopic particle thus undergoes a Brownian motion. This prob-lem was studied by Neto et al. using a model of dissipation in nonlinear quantumfield theory. In the following, we will follow their approach in our discussion onthe transport properties of the microscopic particles described by the nonlinearKlein-Gordon equation in a viscous system.

Let's consider, according to (4.1) or (3.75) and (3.94), the following classicalstatic action in the 1+1 dimension for a nonlinear Klein-Gordon scalar field underthe rescaling <f> -> <f>/g,

where g is a coupling constant of the fields. The natural unit system is used in theabove equation. The extremum value of the action is given by SS/5<f)s = 0, namely,

Equation (4.158) is similar to (3.77). It is a relativistic dynamic equation of a micro-scopic particle in nonlinear quantum mechanics. Due to its translational invariance,equation (4.158) must have the soliton solution (j>s = (f>s(x — XQ). We can expand(4.157) near the extremum (4.158) in terms of the coupling constant g, accordingto (3.78), in the first order approximation,

(t>s{x,t) = <f>s(x - x0) + g6<j>{x,t), (4.159)

which gives

f f d2 f<PV\

S[6<t>} = S{4>s] + jd2x6<f>(z,t) - ^ 2 + ( ^ r J W*,t),

plus some higher-order terms in g. Using (4.158), we get

m = SM = l y > * ( ^ ) 2 , (4.160)which is the classical mass of the microscopic particles. Therefore, up to the leadingorder in g, the eigenmodes of the system are given by

-TT+CID ]*n(x-x0)=u£*n(x-xo). (4.161)dx2 Vd02/0.J

Using (4.158) in (4.161), it can be shown that the function d<f>s/dx is an eigen-function with an eigenvalue of zero. This zero mode can be understood as follows.

(4.157)

(4.158)


Suppose we displace the center of the microscopic particle (soliton), x0, by an in-finitesimal amount 8x0. Then up to the first order in 6xo, we have

<W« = 4>s{xo + Sx0) - <j>s(x0) = -z-Z-Sxo.oxo

However, since 4>s is a function of the difference x — x0, we have 60<j>s =-(d<f>s/dx)5x0. Therefore, the eigenmode d(j)s/dx is related to the movement ofthe localized solution or the motion of the microscopic particle. Thus,

where N' is a normalization constant. We can then get from (4.159)

00

84>{x,t) = n£ian(t)*n(x-xo),n=0

, . 00

4>(x,t) = 4>s{x - x0) + •fiiao(t)-£+g52an(t)&n(x - x0).n-0

It can be seen that the center of the microscopic particle is a true dynamical variable.We rewrite the above expansion as

00

<P(x,t) = cj>s[x - xo(t)] +gân{t)$[x - xo(t)], (4.162)n—l

where

a:oW=*o-^70o(t1)-

Equation (4.162) is known as the collective-coordinate method and is consistentwith (4.39) in the first order approximation. It has all the physics of this system.The microscopic particle whose motion is represented by that of its center of mass isa collective excitation (since it is a solution of the nonlinear field equation), but it isalso coupled to all other modes by the relative coordinate x - x0. The correspondingclassical Hamiltonian can be expressed as

( \ 20 0 \ °° / 2 2 2 \p-Y, G**»* + E ( f + ^ t ) ' (4-163)

n,»=l / n=l ^ '

where P is the momentum canonically conjugate to xo, qn and pn are also conjugateparts, and

Gin = -Jdxd^-*n(x)

couples the microscopic particle to the other modes (phonons) in the system.


This shows that the microscopic particle cannot move freely. Phonons are scat-tered by the microscopic particle, resulting in the microscopic particle moving like aBrownian particle. Although (4.163) is obtained using a perturbative method, thisexpression is general and captures the essential physics of this problem.

Making use of the commutation relations, [xo,P] — ih, [qn,Pi] = ih&ni, and inthe context of quantum dissipation, the quantum model for the Hamiltonian (4.163)can be expressed as

/ oo \ ooH = 2^ : K " ? h9inbtK : + £>„&+&„, (4.164)

y n,i=l J n—1

where : • • • : denotes the normal order, and

I (un\ /„ iPn\K = (2h) [qn + ~n ) 'with

[bn,b+) = Smn, [b+,b+)=O,

and

i r /wr [ur\r9ni = 7T: \\ h bin

is the new coupling constant of the system. Equation (4.164) describes the dynamicsof microscopic particles at low energy and can also be used to model other physicalsystems where microscopic particles are coupled with their environment.

Here we are interested only in the quantum statistical features of the microscopicparticles and phonons acting as a source of relaxation in the diffusion processes.Consider the density operator of the system consisting of microscopic particles andphonons, p(t). This operator evolves in time according to

p(t) = e-ifltlhp{ti)eitltlh, (4.165)

where H is given in (4.164), /5(0) is the density operator at t = 0, which is assumed tobe decoupled and is given by the product of the density operators of the microscopicparticles, ps(0), and the phonon, /3R(0), respectively, i.e., p(0) — PS(0)PR(Q).

Neto et al. studied this problem in which they assumed that the phonons are inthermal equilibrium at t = 0, that is,

Pfl(0) = z ,

where


and HR is the Hamiltonian of the free phonons which is given by the last term in(4.164). We define a reduced density operator ps(t) — trn[p(t)], which contains allthe information about the system when it is in thermal contact with a reservoir.Projecting p(t) in the coordinate representation of the microscopic particle systemxo\i) — Q\Q) and in the coherent state representation for the phonons bn\an) =otn\o-n)- We have

Ps(x,y,t)= Jdx" f dy'J(x,y,t;x",y',0)ps(x",y',0),

where J is the superpropagator of the microscopic particle (soliton) which can bewritten as

J= f dx [Vdyei(-s°W-s°WkF{x,y], (4.166)Jx" Jy'

where

W={r[5ffl] (4.167)is the classical action for the free microscopic particle. F is the influence functional,

F[x,y] = l ^ 0 J WpR0j.f^-W'-lflVÎ'/ax f" D2a /°'JD27e5il».«]+sri».7]| (4.168)

h h*where j8' denotes the vector (ft,/J2> • • • , JSJV) and S\ is a complex action related tothe reservoir and the interaction,

*[«,«] = I clt" [I ( a - f - «• • £ ) - j ( ^ - x.0] , (4-169)with

oo oo

HR = ^J/ia;na*an, /ii = 2 J ^9nmO*man-n=l n,m=l

Now we expand the action (4.169) around the classical solution of its Euler-Lagrange equation. After integrating (4.168), we can get

00

F[x,y}='[[(l-rnn[x,y]nn)-1,

n=l

where

?nm = W*nm[y\ + Wmn[x] + J2 WL[y}Wln[x],1=1


oo - r

Wnm[r] = 6mn+J2 / dt"Knn,(t")Wnlm(t"),

n'=l J°

Knm([x],T) = ignmXir'y^-^',

nn = ( e ^ f t « - - l ) - 1 .

Here iiTnm is the kernel of the integral equation, Wnm[r] is the scattering amplitudefrom mode k to mode j . The terms that appear in the summation represent thevirtual transitions between these modes. If the velocity of the microscopic particleis small, we can use the Born approximation. In matrix form, this can be writtenas

W = (1 - W0)-^0 &W° + W°W°.

In such a case, the terms Tnm are small. Then we can write approximately

F[x, y] « exp J ] T Tnm[x, y}nn \ . (4.170)ln=l J

If the interaction is turned off (F -»• 0) at T = 0, the functional (4.170) is unity,and, as we would expect, the microscopic particle moves like a free classical particle.

Using the Born approximation for W in (4.170), Neto et al. obtained the fol-lowing.

J = J*Dx j " Dy exp j l-S[x, y) + \G\X, y] J , (4.171)

where

§ = j T * " { x [*2(O - f(t")] + [±(*") - y(t"))

= J df'T^t" - t'")[x(t") + y(t'")}V (4.172)

and

G= f dt" f dt1" {[TR(t" - t'")][x{t") - y(t")][x(t'") - y(t'")}} , (4.173)Jo Jo

oo

TR(t) = H&(t) ^ Olm^n COs(wn - Um)t,n,n=l

oo

rl(t) = m'(t) Y^ 9lmnnsin(u;n - ujm)t.n,n=l


Here ®'(t) is the theta function defined as

, r i , i f*>o( j ~ \ 0 , if t < 0 •

Introducing the following new variables

X = Z±*, Y = x-y,

the equations of motion for the action (4.172) can be written as

X{T) + 2 / dt"7(T - t')X{t") = 0, (4.174)Jor*

Y(T) - 2 / dt"j(t" - T)Y(t") = 0,Jo

where

77lQ Ut

or

h®'(t) °°l{t) = — Y\ glmnn(u!n - wm) cos(wn - ojm)t (4.175)

n,m=l

which is the damping function. The two equations in (4.174) have the same formas that obtained in the case of quantum Brownian motion by Calderia and Leggett.In the limit of the time scale of interest being much greater than the correlationtime of the phonon variables, we can write *y(t) = j(T)S(t), where 7(T) is a damp-ing parameter which is temperature-dependent and 6{t) is the Dirac delta function.The above form of *f{t) is the same as that in the Markovian approximation in thesesystems. If we use (4.174) and expand the phase of (4.171) around this classical so-lution, we can get the well-known result for the quantum Brownian motion providedthat the damping parameter 7 (temperature independent) is replaced by j(T) andthe diffusive part is replaced by (4.173). Neto et al. gave the diffusion parameterin momentum space

J2T-> °°

D(t) = h-^-f = -H2&(t) J2 92nmnn(Un - um)2 cosK - um)t.

n,m=l

As discussed before, in the Markovian limit, we can write D(t) in the Marko-vian form, D(t) = D(T)S(t), where D{T) and j(T) obey the classical fluctuation-dissipation theorem at low temperature. If we define the scattering function as

00

S(w - J) - S(CJ' - u) = ^2 glm6(LJ - um)2 cos(u' - w m ) ,n,m=l


D(t) can be written as

h2 f°° r°°D(t) = —z-6(t) / dcu / dJS(u - J){J - u)2{n(ujn) + n(wm)] COS{LJ - u')t.

* Jo Jo

In the above discussion, we established that the Hamiltonian (4.164) leads toBrownian dynamics. That is, a microscopic particle in nonlinear quantum mechan-ics moves like a classical particle at low energy in a viscous environment where itsrelaxation and diffusion are due to scattering by the phonons. These phonons arethe residual excitations created by the presence of the microscopic particle. Weshowed that the damping and the diffusion coefficients of the microscopic parti-cle depend on temperature, since phonons must be thermally activated in orderto scatter off the microscopic particle. Therefore at absolute zero temperature themicroscopic particle moves freely, but its mobility decreases as the temperature in-creases. This shows again the classical feature of microscopic particles in nonlinearquantum mechanics.

By expanding the perturbation in powers of temperature in the systems, Dziar-maga et al. developed a perturbative method to obtain an explicit expression forthe diffusion coefficient of microscopic particles (solitons) moving in a dissipationsystems with thermal noise. For a microscopic particle described by the (jA-equationin one spatial dimension, we have (in natural units)

r4H = 4>zx+2(l-<t?)(l> + Ti{t,x), (4.176)

where F is the dissipation coefficient and r)(t, x) is a Gaussian white noise withcorrelation (r)(t,x)) = 0, and

(r1(t1,x1)ri(t2,X2)) = 2KBTTS(t1 - t2)S(Xl - x2). (4.177)

The system is coupled to an ideal heat bath at temperature T. At a finitetemperature the field 4>{x,t) performs random walk in its configuration space. Ifthe noise is absent at T = 0, equation (4.176) admits a static kink solution (p(t, x) =4>s(x) = tanh(a;), and an antikink is given by (j>s{—x). Small perturbations aroundthe kink take the form 4> — e~7t/rcp(x). Linearization of (4.176) with respect totp{x), for r](t, x) = 0, gives

axz [ cosh (x)\

Its eigenvalues and eigenstates are

7 = 0, ip(x) = cj>sx(x) = - j — ,cosh {x)

7 = 3, <fi(x)=<po(x) = ^ & - , (4.179)cosh (x)

(4.178)


A , ;2 , \ , \ ikx\-i 3ifcmtanh(a:) - 3tanh2(x)l7 = 4 +ft2, <p(x) =(pk(x) ~elkx 1 + ^ p i-i I

where k is the momentum of the microscopic particle. The zero mode (7 = 0)is separated by a gap from the first excited state (breather mode, 7 = 3). Thecontinuum states are normalized so that

/ dx<pl(x)<pk,(x) = 27r^!<5(ft - ft') = N(k)8{k - k1).J—00 1 + ft

The field in the one kink sector for the theory (4.176) can be expanded usingthe complete set (4.179) as

4>{t, x) = 4>,[x- m\ + £ An(t)yn[x - mi- (4-180)

Inserting (4.180) into (4.176) and projecting on the orthogonal basis (4.179)gives a set of stochastic nonlinear differential equations,

TNoi(t) - r&) £ An(t)Mn0 + J2 Ai(t)Aj(t)pHo+

£ Ai(t)Aj(t)Ak(t)Rijk0=ilQ(t),

TNnAn(t) + lnNnAn{t) - Ti{t)Y,Ai{t)Min + £ Ai(t)Aj(t)Pikn+

£ Ai(t)Aj(t)Ak(t)Rijkn = Vn(t), (4.181)»,j,t#o

where the coefficients are/•oo /•oo

Nn= dx(pn(x)(p*n(x), Mni= dxtpnx(x)(p*(x),J—oo J—oo

/>OO /-CX)

P m j = 6 / dx<t>s(x)ipn(x)ip*(x), Pnijk = 2 / dxipn(x)ipi(x)ipj(x)ip*k(x),J—oo J—oo

Vn(t)= f dxr){t,x)ip*n{x).J—oo

Applying (4.177) and the orthogonality of the basis (4.179), we get

(Vn(t)) = 0, « ( t i )»?*(«)> = 2KBTTNn5ni5{t1 -t2). (4.182)

The natural parameter of expansion at low temperature is \/KBT. The pro-jected noises, (4.182), are of the same order. Dziarmaga et al. introduced therescaling y/KsT —> E\/KBT. Terms in (4.181) can then be expanded in powers ofe and a power series solution of the equation can be obtained, e was set back to1 at the end of the calculation. With the expansion of the collective coordinates,


£(t) = e£tt) + £(2)^(2) + • • • and An = e^1* + £(2Ul2) + • • •, equations in (4.181)become, to the leading order in e,

IW0£(1) W = ijb(t), TNnAgHt) + lnNnA^(t) = r,n{t). (4.183)

In this approximation the zero mode £^(t) and the excited modes An (t) are un-correlated and stochastic process driven by their mutually uncorrelated projectednoises. f^(*) represents a Markovian-Wiener process whose only nonvanishing sin-gle connected correlation function is <fW(t)fW(t")) = (2KBT/rN0)5{t-t"), whichis singular at t -> t". According to the first equation in (4.183), the probability,p(t, £(£)), for the kink random walks to £(i) at time t satisfies the diffusion equation

dp_ avat de'

where the diffusion coefficient is given byKBT = 3KBTFN0 4r '

Other terms in the expansion of collective coordinates can be recursively workedout and they are

42)(*) = E fdre-^-î^{r)Af\r) -

E^r^-^^M^w^w,»,ô l i V n -70

e(3)W = £K ( 1 ) (*)4a)W + (i«- 2 ) ] ^ -

^[^(^fw+do^î-

E ^w^w^w^.The equilibrium correlations of ^(t), up to one order higher than the leading order,are

(mm)«£2(^(1)(*K(1)(i))+£4(^(2)(^(2)©)+e4[<4(1)(*)^(3)(*)> + a «• 3)]

= <^>w^>(«)> i + ^ E ^ + E ^ ^ M ) + . . . .


The integrals over the continuous part of the spectrum and the summation overthe breather mode yield

(i(t)£®) = f p^C - *)[1 + 1.8164**71 + O[(KBTf].

Finally, Dziarmaga et al. gave the diffusion coefficient in the equilibrium state,which is given by the average over time much longer than the relaxation time of thebreather mode. It is given by

£>(oo) = ^ ^ ( 1 + I&VMKBT) + 0[{KBT)3]. (4.184)

Dziarmaga et al. performed numerical simulation of (4.181) with T — 1 fora range of temperature values. The numerical results were consistent with theprediction of the perturbative theory, (4.184).

4.11 Microscopic Particles in Nonlinear Quantum Mechanics ver-sus Macroscopic Point Particles

Prom previous sections of this Chapter, we understand that a microscopic parti-cle in nonlinear quantum mechanics is a soliton which exhibits many features of amacroparticle. However, the microscopic particle cannot be regarded as a macro-point particle. We already know from sections 4.9 and 4.10 that the excitationsof the internal mode of soliton - a long-lived oscillation of the soliton amplitude -participates in inelastic and radiationless energy exchange among particles duringthe three-soli ton collisions, and the residual excitations (phonons) created by thepresence of the microscopic particles result in damping and diffusion of the particlesin viscous systems. These phenomena do not occur in system of macro-point parti-cles. In this section, we will further explore these phenomena and address the pointparticle limit of microscopic particles in nonlinear quantum mechanics, followingthe work of Kalbermann. A simple coupling between the center of the microscopicparticle (soliton) and a potential will be introduced, which is useful in the discussionof differences and similarities between a microscopic particle in nonlinear quantummechanics and a point particle in classical physics, as well as the pointlike limit ofthe microscopic particles.

Kalbermann considered the center of mass of a microscopic particle (soliton) asa candidate for the location of the particle, and coupled it to an external potential.He took a kink mode supplemented by a potential and a Lagrange multiplier toforce the interaction with the center of mass. This procedure treats the centerof the microscopic particle (soliton) as the collective coordinate, as it is usuallydone in the case of topological defects such as those by Kivshar et al. and Neto etal, given in Section 4.10. In this approach, certain parameters of the microscopicparticle, related to the symmetries or zero modes, are identified and coupled toexternal sources. In this case, translational invariance yields the corresponding


symmetry. In other cases, internal symmetries can be identified and the generatorsof the symmetries can be treated as the collective coordinates of the microscopicparticles as a whole. This enables the separation of the bulk characteristics of themicroscopic particle from its internal excitation. In this way, the coupling betweenthe center of the microscopic particle and the collective coordinate correspondingto translational invariance breaks the symmetry, which allows a clear distinctionbetween internal excitations and bulk behavior of the microscopic particles as awhole. As our aim is to demonstrate the importance of the extended character ofthe microscopic particles on its interaction with external sources, the above methodseems to be the most straightforward approach.

Kalbermann used the following Lagrangian in his study of the problem

C = \d^dv<$> + Â ( V - ?p} - V(x0) + x(s, t)4>S(x - x0), (4.185)

where x{x-> i) is the Lagrange multiplier that enforces the coupling between thecenter of the microscopic particle, XQ, and the external potential, U(XQ). For kinkswith topological charge =±1 for which <fi(xo) = 0, the Lagrange multiplier selectsthe value of xo at (/> = 0. To couple the potential to a point at which the microscopicparticle vanishes, Kalbermarnn chose

U{xo) = \ j dxV{x)\U2-T^-\ , (4.186)

which represents interaction between a microscopic particle and an impurity. Theintegrand is zero except in the region around the center of the microscopic particlewhere 4> « 0. If V(x) is a smooth potential (in contrast to the 6 spike impurity), thento the lowest order of the potential, the above equation becomes U(x0) = C'V(XQ),

where c' is a constant depending on the initial parameters of the microscopic parti-cle. The choice for the interaction corresponds to the case in which the microscopicparticle propagates in a smooth medium and the impurity is similar to a backgroundmetric. If the microscopic particle can be regarded as topological soliton, then thepresent treatment would correspond to the classical propagation of a microscopicparticle in a background potential. In order to see clearly this correspondence,Kalbermarnn chose the interactions which consist of introducing a nontrivial met-ric for the relevant space time. The metric carries the information of the medium.

The Lagrangian in the scalar field theory for the (l-l-l)-dimensional systemimmersed in a background with the metric g^v is

C- = y/g \gw\d^dv<t> - U(4>)] , (4-187)

where g is the determinant of the metric, and U is the self-interaction that makesthe existence of the soliton possible. For a weak potential we have

g0Q « 1 + V(x), gu = - 1 , fl_u = (?!_! = 0, (4.188)


where V(x) is the external space-dependent potential. The equation of motion ofthe microscopic particle in the background becomes

This equation is identical, for slowly varying potentials, to the equation of motion ofa microscopic particle interacting with an impurity V(x). The constraint in (4.185)only insures the proper identification of the center of the microscopic particle in thepresent case. In the following, we will let A = m2 = 1 in (4.185). Variations of thefield 4> as well as the collective coordinate x0 in (4.185) yield the equation of motionof the microscopic particle constrained by the Lagrange multiplier,

where

B(x,t) = 8(x-xo)(j£) (4.191)

is the new source term that denotes the coupling between the center of the mi-croscopic particle and the external potential. The limit of a pointlike object isachieved, when the width of the microscopic particle is negligible as compared tothe characteristic scale of the potential. This method provides a clean separationbetween the particulate behavior of the microscopic particle and its wavelike ex-tended character in the nonlinear quantum mechanics. For the potential in (4.187),Kalbermann used a bell-shaped function,

U(x0) = A sech2 (^—^-) > (4-192)

where A is the height (depth) of the potential located at r, and a is related to thewidth, W, of the interaction potential by a = W/(2n).

Using a 5-function distribution for p(x),

6(x)=p(x) = \imp((x), pe(x) = eM~^/e), (4.193)

B(x,t) can be expressed as

B(x,t)= P(XZ^L w • (4-194)J p(x — xo)(dcp/dx)dx

In actual calculations the limit of e -* 0 is achieved when e « dx, the spatial step,is small enough compared to the size of the microscopic particle.

In solving the partial differential equation of motion, the spatial boundaries aretaken to be -25 < x < 25, with a grid of da; = 0.04 and a time lapse of dt = 0.02 upto a maximal time of t = 200 (10000 time steps). The upper time limit of t = 200

(4.189)

(4.190)


is sufficient for the resonances to decay, to define clearly the asymptotic behavior ofthe microscopic particle (soliton) while at the same time to avoid reflection from theboundaries. Kalbermann started with a free microscopic particle impinging fromthe left, at distance from the potential range, with an initial velocity v,

with xo(t — 0) = — 5. The total and kinetic energies of the free microscopic particleare

E(v) = ! ^ _ , Ek(v) = E{v) - E(0), (4.196)2v l - v

respectively.The value of A in (4.192) was chosen such that there are visible results for the

kink with m = 1. Kalbermann used A = 0.1 and A = —0.2. The former allowestransition through the barrier for a pointlike particle with velocity v w 0.43, and thelatter provides enough strength to trap it in a reasonable range of the initial velocity.The location of the potential was fixed at r=3. For the attractive potential, it wasfound that there were no islands of trapping between transmissions. The microscopicparticle either is trapped, when its velocity is below a certain threshold, or passesthrough the impurity. For a barrier depth of A = —0.2 and W = 1 which iscomparable to the size of the microscopic particle, the critical velocity is v — 0.583.At this initial velocity, the microscopic particle drifts extremely slowly through thepotential well, and lags behind the free propagation by almost an infinite time, but iteventually goes through. If the velocity is below this critical value, the microscopicparticle is always trapped. Contrary to the case of a point particle, there appear tobe bound states for velocities below the threshold for positive kinetic energies. Thesestates may be referred to as bound states in the continuum, extraneous to classicalparticulate behavior. This feature is due to the finite extent of the microscopicparticle (soliton) in nonlinear quantum mechanics.

One way to show that such a phenomenon is indeed due to space extension ofthe microscopic particle is by increasing the well width. In order to confirm thisassertion, Kalbermann selected an initial velocity of the microscopic particle whichwas well below the threshold transmission velocity, v — 0.2, and a well depth ofA = —0.2 as before. Starting from W = 1 he gradually increased the well widthand found that transmission occured when the well width became W = 5, whichwas much wider than the extent of the microscopic particles. When the velocitywas above the threshold velocity, the microscopic particle slowly drifted through thewell in an effectively infinite time. It remained within the barrier range for almostt — 200 and then started to emerge from the opposite side. Figure 4.13 shows theasymptotic velocity of the microscopic particle (soliton), plotted versus the potentialwidth W. In the case of the microscopic particle being trapped, the asymptotic

(4.195)


velocity vanishes. The microscopic particle oscillates with zero mean velocity insidethe well. It never emerges from it. Its amplitude decreases when the particle movesfarther from the center of the well during the oscillatory motion. The energy ofthe microscopic particle is conserved in all cases. When the microscopic particleis trapped, the initial kinetic energy transforms into deformation and oscillationenergies as well as radiation that damps the oscillations.

Fig. 4.13 Asymptotic velocity of the microscopic particle (soliton) as a function of the width ofthe attractive potential.

When the microscopic particles is transmitted, there is no radiation due to thetopological conservation of winding number that forces the microscopic particle(soliton) to be exactly the one that impinged onto the potential. This result wasborne out in the case of multi-soliton collisions in section 4.8 for which the domi-nant effect is the inelastic nonradiative exchange of energy between the microscopicparticles (solitons). In the present case, there is no possible recipient of such energy.Therefore, the microscopic particle (soliton) will emerge elastically. The asymptoticvelocities of the reflected and transmitted microscopic particle were calculated us-ing the actual motion of its center, and so were the theoretical expressions for thekinetic and total energies of the free microscopic particles. This comparison provedenergy conservation and served as a measure of numerical accuracy.

For the repulsive potential, the microscopic particle (soliton) is reflected even forinitial kinetic energies above the barrier height. When A = 0.1, the correspondinginitial velocity whose kinetic energy would yield classically to transmission is v ~0.43. However, this does not occur. For a barrier of width W = 1, the microscopicparticles goes through, only when the initial velocity is v = 0.52, corresponding toan initial kinetic energy of Ek = 0.16, well above the barrier height.

The particulate behavior will be restored only when the barrier width is largerthan the extent of the microscopic particle (soliton), such that it looks as if almostpointlike. In particular for v = 0.43, transmission starts for barrier widths greaterthan W = 19. Fig. 4.14 shows the asymptotic velocity of the microscopic particleas a function of the barrier width. The microscopic particle is repelled by sharpbarriers and transmits through wider barriers, while trapping never occurs.


Fig. 4.14 Asymptotic velocity of microscopic particle (soliton) as a function of the repulsivebarrier width.

Prom the above discussion we see that a microscopic particle in nonlinear quan-tum mechanics behaves as a particle only when the width of the potential is largecompared to the size of the particle so that the microscopic particles (soliton) ap-pears pointlike. In such a case the microscopic particle obviously has corpuscle fea-ture of a classical particle. In contrary cases, the coupling between the center of themicroscopic particle and a repulsive or an attractive potential can result in boundstates or trapped states in the continuum, which does not occur for point-particleobjects. Kalbermann simulated numerically the behaviors of the microscopic par-ticles in terms of collective coordinates. The simulation results showed also chaoticbehavior depending on the initial position of the center of the microscopic parti-cle. The initial states can transform a trapped state into a reflected or transmittedstate. This exhibits clearly the wave feature of the microscopic particle. Therefore,a microscopic particle in nonlinear quantum mechanics has not only corpuscle butalso wave features and it is necessary to study its wave feature which will be thetopics of discussion in the following sections.

4.12 Reflection and Transmission of Microscopic Particles at In-terfaces

As mentioned above, microscopic particles in nonlinear quantum mechanics repre-sented by (3.1) have also wave property, in addition to the corpuscle property. Thiswave feature can be conjectured from the following reasons.

(1) Equations (3.2) - (3.5) are wave equations and their solutions, (4.8), (4.56)and (4.195) are solitary waves having features of travelling waves. A solitary waveconsists of a carrier wave and an envelope wave, has certain amplitude, width, ve-locity, frequency, wavevector, and so on, and satisfies the principles of superpositionof waves, (3.30), (3.43) and (3.44), although the latter are different from classicalwaves or the de Broglie waves in linear quantum mechanics.

(2) The solitary waves have reflection, transmission, scattering, diffraction and


tunneling effects, just as that of classical waves or the de Broglie waves in linearquantum mechanics. Some of these properties of microscopic particles will be de-scribed in the following sections. In this section, we consider first the reflection andtransmission of microscopic particles at an interface.

The propagation of microscopic particles (solitons) in a nonlinear and nonuni-form medium is different from that in a uniform medium. The nonuniformity herecan be due to a physical confining structure or at the interface of two nonlinearmaterials. One could expect that a portion of a microscopic particles that was in-cident upon such an interface from one side would be reflected and a portion wouldbe transmitted into the other side. Lonngren et al. observed the reflection andtransmission of microscopic particles (solitons) in a plasma consisting of a positiveion and negative ion interface, and numerically simulated the phenomena at theinterface of two nonlinear materials. In order to illustrate the rules of reflection andtransmission of microscopic particles, we introduce here the work of Lonngren et al.

Lonngren et al. simulated numerically the behaviors of microscopic particle(soliton) described by the nonlinear Schrodinger equation. They found that thesignal had the property of soliton. These results are in agreement with numericalinvestigations of similar problems by Aceves et al., and Kaplan and Tomlinson. Asequence of pictures obtained by Lonngren et al. at uniform temporal increments ofthe spatial evolution of the signal are shown in Fig. 4.15. From this figure, we notethat the incident microscopic particles propagating toward the interface between thetwo nonlinear media splits into a reflected and a transmitted soliton at the interface.From the numerical values used in producing the figure, the relative amplitudes ofthe incident, the reflected and the transmitted solitons can be deduced.

Fig. 4.15 Simulation results showing the collision and scattering of an incident microscopic par-ticle (soliton) described by the nonlinear Schrodinger equation (top) onto an interface. The peaknonlinear refractive index change is 0.67% of the linear refractive index for the incident microscopicparticle, and the linear offset between the two regions is also 0.67%.

They assumed that the energy carried by the incident microscopic particle (soli-


ton) is all transferred to either the transmitted or the reflected solitons and none islost through radiation. Thus

•Sine = Eref + £ t r a n s . (4.197)

Lonngren et al. gave approximately the energy of a microscopic particle (soliton)

A*

&C

where the subscript j refers to the incident, reflected or transmitted solitons. Theamplitude of the soliton is Aj and its width is Wj. The characteristic impedanceof a material is given by Zc. Hence, (4.197) can be written as

^Winc = ^Wre{ + ^«Wtrans. (4.198)

Since AjWj = constant, for the microscopic particles (solitons) of the nonlinearSchrodinger equation (see (4.8) or (4.87) in which Bj is replaced by Aj), we ob-tain the following relation between the reflection coefficient R = Ave{/Ainc and thetransmission coefficient T = Atrans/Ainc

l = R+4±T> (4-199)

for microscopic particle (soliton) of the nonlinear Schrodinger equation.

Fig. 4.16 Sequence of signals detected as the probe is moved in 2 mm increments from 30 to 6 mmin front of the reflector. The incident and reflected KdV solitons coalesce at the point of reflection,which is approximately 16 mm in front of the reflector. A transmitted soliton is observed closerto the disc. The signals at 8 and 6 mm are enlarged by a factor of 2.


Lonngren et al. conducted an experiment to verify this idea. In the experiment,they found that the detected signal had the characteristics of a KdV soliton. Asequence of pictures as shown in Fig. 4.16 was taken by Lonngren et al. using asmall probe at equal spatial increments, starting initially in a homogeneous plasmaregion and then progressing into an inhomogeneous plasma sheath adjacent to aperturbing biased object. In Fig. 4.16, we can clearly see that the probe firstdetects the incident soliton, and some time later the reflected soliton. These signalsare observed, as expected, to coalesce together as the probe passed through the pointwhere the microscopic particle (soliton) was reflected. Beyond this point where thedensity started to decrease in the steady-state sheath, a transmitted soliton wasobserved. From Fig. 4.16, the relative amplitudes of the incident, the reflected andthe transmitted solitons can be deduced.

For the KdV solitons, there is also AjWf = constant (see (4.92)). Thus,

! = #3/2 + | c L r 3 /2 >

Zcn

for the KdV soliton.The relations between the reflection and the transmission coefficients for both

types of solitons are shown in Fig. 4.17, with the ratio of characteristic impedancesset to one. The experimental results on KdV solitons and results of the numericalsimulation of microscopic particles depicted by the nonlinear Schrodinger equationare also given in the figure. Good agreement between the analytic results andsimulation results can be seen. The oscillatory deviation from the analytic resultis due to the presence of radiation modes in addition to the soliton modes. Theinterference between these two types of modes results in the oscillation in the solitonamplitude. In the asymptotic limit, the radiation will spread, damp the oscillation,and result in the reflection-transmission coefficient curver falling on the analyticcurve.

The above rule of propagation of the microscopic particles in nonlinear quantummechanics is different from that of linear waves in classical physics. Lonngren et al.found that a liner wave obeys the following relation

1 = R2 + -f^T2. (4.200)

This can also be derived from (4.198) by assuming linear waves. The widths of theincident, reflected and transmited pulses Wj are the same. For linear waves

n Zcn — Zc\ 2ZcnR=~5 T~5~' a n d T=~5—T^~'

and equation (4.200) is satisfied. Obviously, equation (4.200) is different from(4.199). This shows clearly that the microscopic particles in nonlinear quantummechanics have wave feature, but it is different from that of linear classical wavesand the de Broglie waves in the linear quantum mechanics.


Fig. 4.17 Relationship between the reflection and transmission coefficients of a microscopic parti-cle (soliton), Eq.(4.199). The solid circles are experimental results on KdV solitons, and the opencircles represent results of Nishida. The solid triangles indicate numerical results of Lonngren etal. for microscopic particles(solitons) described by the nonlinear Schrddinger equation.

However, the above results are based on two assumptions: (1) no radiationmodes are excited through the interaction of the incident soliton with the interface;(2) the product of the interaction of the incident soliton with the interface is a singletransmitted and a single reflected soliton. If these conditions are not satisfied, theabove relation will not hold.

As a matter of fact, the reflection of a microscopic particle (soliton) in nonlinearquantum mechanics is complicated. Alonso discovered a shift in the positions ofmicroscopic particles which occurs in the case of nonzero reflection coefficient. Heused the inverse scattering method to obtain the phase shifts of the microscopicparticles in (4.40) with 6 = 2. As it is known, the inverse scattering method forsolving the nonlinear Schrodinger equation is based on the resolution of the integralequations of the Marchenko type (4.51), which can be written as

Fl{t',x',y)- l°° K*(t',x' + y + z)FZ(t',x',z)dz = K*(t',x' + y),Jo

Ft(t',x',y)+ fO°K(t',x'+y + z)F1(t',x',z)dz^0, (4.201)Jo

where the kernel K is determined from the set of scattering data

S(t') = {0 = & + irij, btf) = bt exp(4iC^'), j = 1, • • • , TV;

in the form of

1 f°°K(t',x') = 2^2bj(t')e

2i^x> + - c(t',C)e2iCx'dC. (4.202)j "^ J-OO

Solution of (4.40) is given by 4>{t',x') = -F1{t',x',+0). As t' -> ±oo, cj>{t',x')


evolves into a superposition of N freely moving solitons with velocities Vj = 4£,-,and a radiation component which decays as I*'!"1/2. The parameters q^ whichcharacterize the asymptotic trajectories qf(?) « qf + Vjt' for the solitons as t' -4±00 remain to be determined.

It is known from Section 4.1 that the nonlinear Schrodinger equation is aGalilean-invariant Hamiltonian system and its corresponding generator for the pureGalilean transformation can be written as

" J—oo

= - £ > M<a(O)| - ^ J™ In |o(C)|0c argF(C)dC, (4.203)

where

J VW l l c - C ; P L2vr 7 . ^ g - C - iO q\' ' (4.204)

[ p(Q = ln(l + |c(C)|2), F(0 = c(Oo(C), Qv = 0.

From (4.203) and (4.204), the following alternative expression for G can beobtained

°-E[-^5>|&|-£/" i^P«+5r£"«)a< h^+s'jC^*]* (4205)

where P denotes the principle value. We observe that for a pure one-soliton so-lution (j>so\, the position of its center is q(t) = (2r])~1 ln[\b(t)\/2rj\, and therefore,G[&oi(*)] = -2nq(t).

As a consequence of the Galilean invariance of the nonlinear Schrodinger equa-tion, we can define the mass and momentum functionals which have the followingforms in terms of scattering data

M = - / |0|W = 2 j > - — / p{QdC,* J-00 j Z 7 r J-00

P = -i Pfa'dx' = -8^2^ + - / OKCR-^-oo j n J-00

According to these expressions, one may think that the field of microscopic parti-cle depicted by the nonlinear Schrodinger equation appears as a Galilean systemcomposed of N particles with masses 2TJJ and velocities Vj = 4fj (solitons), and ahas continuous mass distribution with density —p(()/2n and velocity vg(Q = 4£


(radiation). This velocity spectrum associated with the scattering data can be un-derstood through the asymptotic analysis of the nonlinear Schrodinger equationfield as follows.

Prom (4.201) we know that the nonlinear Schrodinger equation field 0 at agiven point (t',x'o) depends only on the restriction of the kernel K(t',x') to theinterval (x'0,+oo). If the evolution of the scattering data is inserted into (4.202),the modulus of the jth term in the summation propagates with velocity Vj — 4£,,while the group velocity of the Fourier modes in the integral term coincides withvg(C) = 4(. Hence, given arbitrary values x'o and v0, as t' ->• ±oo the restriction ofK to intervals of the form /±(t ' ) = [x'o + (v0 ±e)t',+oo] where € is an arbitrarypositive number, depends only on those scattering data with velocity v such that±(v - v0) > 0. In fact, it can be shown that the contribution to K due to theremaining scattering data has a L2 norm on /±(t) which vanishes asymptoticallyas t' -> ±oo.

We now consider the motion of the Ith soliton and let us denote by </>± the partsof the nonlinear Schrodinger equation field propagating to the right of the soliton ast' —> ±oo. Then the relevant kernels for characterizing <p± through the Marchenkoequations (4.201) are

K±(t', x') = 2^20[±(Vj - vftbiW**'

+ - r 9{±[vg(Q -vtMt'^y^'dC,

where 6 is the step function. The sets of scattering data related to <j>± are

S±(t') = {Q,bj(t'),j such that ± (Vj - vi) > 0\B(±[v9(Q - vt})c(t\(),teR}.

The scattering data S±(t'){Q, h(t')} correspond to the parts <j>'± of the nonlinearSchrodinger equation moving with velocity v such that ±(v — vi) > 0 as t' -> ±oo.In other words, <j>'± result from the addition of the Ith soliton to <j>±. Hence, asa result of the dispersive character of the radiation component and the localizedform of the soliton, it is clear that as t' ->• ±oo the difference between the values ofthe functional Gs for <f>'± and <f>±, respectively, must be -2r)eqf(t') of G at the Ithsoliton. Using this and the identity (4.205), Alonso obtained

The phase shift of the Ith microscopic particle (soliton) as it interacts with both


the other microscopic particles and the radiation component is given by

qf ~ 9f = - S s i S n ^ ~ v^ln TZ^

Prom these results we see that the microscopic particles experience a phase shiftwhen they are reflected due to interaction with other microscopic particles andwith the radiation component. It shows again the wave feature of the microscopicparticles.

4.13 Scattering of Microscopic Particles by Impurities

A wave can be scattered by an obstruction, and scattering is an essential featureof waves. In this section we discuss scattering of microscopic particles by impu-rities, and propagation of microscopic particles in nonlinear disordered media inthe framework of nonlinear quantum mechanics. It is well known that in linearsystems, disorder generally creates Anderson localization, which means that thetransmission coefficient of a plane wave decays exponentially with the length of thesystem. However, the nonlinearity may drastically modify properties of propagationof microscopic particles in disorder systems. We will find that besides reflection andtransmission, there are also excitations of internal modes and impurity modes inthe systems, which again reflects clearly the wave nature of microscopic particles.These properties of microscopic particles depend on the nature of the impurity andits velocity. As the first step towards a full understanding of these phenomena, westart by discussing scattering of a microscopic particle by a single impurity, whichwas studied by Kivshar et al, Malomed, Zhang Fei, et al, and others.

In such a case, the dynamics of the perturbed microscopic particle in nonlinearquantum mechanics is described by the following nonlinear Schrodinger equation

i<k + 4>x'X' + 2\4>\2t - ef&)4>, (4-206)

where e is a real, small parameter, f(x') is a localized potential function arising fromthe impurity in an otherwise homogeneous medium, which satisfies f{x' -> oo) = 0.If e = 0, equation (4.206) has a soliton solution which is given by (4.56), with theamplitude and velocity of the soliton given by 2r\ and +4£, respectively. Equation(4.56) can be interpreted as a bound state of N = 4TJ quasi-particles with a bindingenergy Eb = Ar)2.

We now consider the scattering of the microscopic particle (soliton) by the im-purity described by (4.206) using the Born approximation. We assume that thevelocity of the microscopic particle does not change during the scattering under thecondition |e|7j <C £2- The spectral density of the "quasi-particles", n(A), due to the


scattered microscopic particle, can be calculated following the inverse scatteringmethod of Zakharov et al., and the result is given in Section 4.3, n(A) « TT""1 |6(A)|2,where 6(A)(|6(A)|2 <C 1) is the Jost coefficient used in the inverse scattering method,and A is a real spectral parameter that determines the wave number k(X) = 2A andthe frequency CJ(X) = fc2A = 4A2 of the emitted linear waves. The perturbation-induced evolution of the Jost coefficient 6(A) was obtained by Karpman as

^P- = 4i\2b(\,t>) + £ | ° ° dx'f(x') [0(i',O*i(^t;, A]^V,f, A)

- 0V,t/)¥/2(s'.t/MV,t/>A')], (4-207)

where *i,2(x', t', A) and ^>i,2(a:', t', A) are the components of the Jost functions. Forthe one-soliton solution (4.56), these functions were obtained by Karpman.

Kivshar et al. assumed that before the scattering, i.e. at t' ->• — oo, all the"quasi-particles" were in the bound state, or the soliton state (4.56). This meansthat the initial condition for (4.207) should be in the form of 6(A, t' = -oo) = 0.Then, the total density rcraci(A) of the linear waves emitted by the microscopicparticles during the scattering is

nrad(A) = i|&(A,t' = +oo)|2

- e2*£i(A) [°° dxi [°° dx'f{xl)f{x')ei^^-x'\ (4.208)J— oo J— oo

where

0(X) = l ^ ' V . (4.209)

The number of "quasi-particles", N, emitted by the microscopic particle (soliton)in the backward direction (i.e. the reflected "quasi-particles") can be calculatedfrom

r°°Nr= nrad(A)dA.

J-oo

The reflection coefficient R of the microscopic particle can be defined as the ratioof Nr to the total number of "quasi-particles", N = 47?

R=±- [ nrad(-A)dA. (4.210)

Kivshar et al. studied the scattering of a microscopic particle by an isolatedpoint impurity in which f(x') is a delta-function. Prom (4.208), we get the emitted


spectral density nren = e2n^(A). The reflection coefficient can then be written as

fl(D = _![£L r ( * ' + D 2 + * 2 dx, ( 4 2 1 1 )

where a = 77/ . At a = 0, i^1' reduces to that of the linear wave packet, i.e.,R{o] « £2/4feo = £2/16£2. For small a, R^X^/R^ first increases slowly to 1.004(at a = ac sa 0.178) and then rapidly decreases to zero, so that for a » 1 thereflection coefficient approaches zero exponentially.

7/2R{1)" i ^ ^ 1 ) e " W 2 ) (a>>1)- (4-212)The fact that R^ differs from i?g shows that the wave nature of the microscopicparticles in nonlinear quantum mechanics differs from linear wave packet.

The number of transmitted "quasi-particles" Nt, i.e., those emitted by the mi-croscopic particle (soliton) in the forward direction, may be defined by

Nt= /°°nrad(A)dA.Jo

Note that for a C 1 this quantity is essentially less than Nr, since it has theasymptoic expansion Nt w e2a4/120£2iV. However, for a > 1, Nt asymptoticallyapproaches iVr.

We define the density, erad(A) = 4A2nrad(A), of the emitted energy

£rad= r' dx>(\4>*\2-\4>\*)J—oo

by means of the inverse scattering method. The total energy emitted by the mi-croscopic particles in the forward (Et) and backward (Er) directions can be foundby

r°°Et,T = / dAerad(±A).

Jo

For the microscopic particles described by the nonlinear Schrodinger equation(4.206), Kivshar et al. found that

Er = e2r, (l - l-a2 - ~aA V Et = ê2r,a\ for a « 1,

2 9/2Ert*Et = £ m

n e~na/2, for a > 1.

Kivshar et al. also considered scattering of a microscopic particle (soliton) bytwo point impurities separated by a distance a. The perturbation ef(x') take the


form of ef{x') = e\S(x') + S2S(x' — a). It can be shown that

nrad(A) = n£i | (£i - e2)2 + 4£le2 cos2 [â/3(A)J | ,

where n ^ and fi(\) have been denned in (4.209). In this case the reflection coeffi-cient of the microscopic particle, R^2\ is determined by two factors, correspondingto the soliton scattering by an isolated point impurity with the effective intensity(ei - £2), and to the resonant effect during the scattering, respectively, and it isgiven by

28a^2 70 cosh2[7r(a;2+a2-l)/4a] [4 J

where d = 2a^. When a < 1 and a2d < 1, equation (4.213) reduces to

RW = 2RP [l + . l^ ,, cos(2d)l . (4.214)u L sinh(2ad) J

Fig. 4.18 The reflection coefficient of the microscopic particle described by the nonlinearSchrodinger equation scattered by two identical point impurties, as a function of d = 2a£. (a)a < 1; (b) a > 1; R^ is the reflection coefficient for scattering by a single isolated impurity.

The dependence of i?(2) on d for small a is shown in Fig. 4.18(a). If the sizeof microscopic particle (« if"1) exceeds considerably the distance between the twoimpurities (i.e. ad w or) < 1), we can distinguish the interference phenomena whichmanifest themselves as oscillation of i?(2) against the parameter d. Different fromthe linear approximation, i?(2) does not approach zero, because x' < sinhx' for x' ^0. When d = dmin = {I~2a2/3)d°n, where dPn = ir/2+mr (n = 0,1, • • •), R{2) has theminimum values: R{^m = 2(admin)2/3 « (2a2/3)O/2+n7r)2. The maximum values,R&lx = 4i?[)

1)[(l - na7r/3)2], which occur at d = rfmax = (1 - 2a2/3)n7r, decreaseas d increases. When the size of the microscopic particle becomes comparable to

(4.213)


the separation of the impurities, a (i.e. ad m 1), the interference decays and forad » 1, RW approaches the value 2R§' (see Fig. 4.18(a)). That is, the scatteringby each impurity is independent of the other. Thus, the microscopic particle showsboth wave (ad <C 1) and corpusclar (ad 3> 1) properties during the scattering bythe impurities.

In the nonlinear case (a 3>1), we can obtain from (4.213)

RP) = 2ij(i) (a) fl + (1 + c')1/4 cos (\o?d + y'\ 1 , (4.215)

where d = 2ar)/-K, y' = tan~1(c')/2, and R^(a) is given in (4.212). The depen-dence of RW on d for a 3> 1 is shown in Fig. 4.18(b). For large values of d (whenad sa d ~S> 1), the reflection coefficient becomes equal to 2Rl-1\ which character-izes the non-resonant scattering of the microscopic particle in which the scatteringintensities by separate impurities are simply added. For da < 1 (d < 1), the re-flection coefficient oscillates with increasing d and the period of the oscillation isapproximately a~2 •C 1. The frequency of the internal oscillation of the micro-scopic particle for a » 1 is then « 7j2, and the resonant condition is a2d ss 1.For a -C 1, the period of the internal oscillations is of the order of £~2, and theresonant condition is a£ « d m 1. However, in the case of e\ = Zi = s, scattering oflinear wave is known to be resonant and the reflection coefficient RQ in the Bornapproximation oscillates as a function of d = afeg = 2a£ as R^' = 4R^' cos2 d,where R{

01] = £2/16£2. When ffl = 7r/2 + n7r (n = 0,1,- • •), R^ = 0, which corre-

sponds to the situation where the energy of the scattered "quasi-particle", fcg = 4£2,coincides with the energy of the resonant state between the two delta-function po-tentials. The maximum value, 4/?Q of RQ , corresponds to scattering of a linearwave by a single effective impurity with twice the intensity (e -> 2e). Therefore, wesee that scattering of microscopic particles described by the nonlinear Schrodingerequation is different from that of linear waves.

The above results can be easily generalized to the case of a periodic system ofN point impurities, with separation a between adjacent impurities. Tedious butstraightforward calculation leads to the following when Na2d <IC 1,

, (N-l)/2 fc_l

RW = J#> (N + 2 £ C2N

k+1C%+1 £ (_i)-cfc"L12-2(7V-m-1'^ fc,/#0 m=0

xsump=1 C2{JV_m_1) sinh{2adp) cos(Zpd)J ,

where C§ are the binomial coefficients and ( ) stands for the integer part. In thecase of aNd 2> 1, the microscopic particle is scattered as a corpuscle and it can beshown that RW « NRg\

However, transmission always occurs in the scattering process of the microscopicparticles by the impurities except in the case of complete reflection, the condition


for which is given above. The transmission coefficients are defined by T^ =Et/Ei for the energy and T ^ = Nt/Ni for the excitation, where £* and Nt areincident energy and number of microscopic particles, respectively, Et and Nt arethe transmitted energy and number, respectively. Because Ef = Et + Er andNi = Nt + Nr, then T(-E<N'> = 1 - R(E'Nl Thus the transmission coefficients canbe easily calculated from the reflection coefficient given above

When the impurity concentration p is low, the average distance between twoimpurities is larger than the size of microscopic particles. In this case, the scatteringby many impurities can be considered independent, i.e., T = FT • Tj, where Tj is thetransmission coefficient of the jth impurity. Because the transmitted microscopicparticle from scattering by the jth impurity is the incident microscopic particle forscattering by the j + lth impurity, we have

Ej+ÊjTîE^Nj),

NJ+1=NiTJIf){EJ,Nj)

and

AEj+1 = EJ+1 -Et = -EjRW (EJ ,Nj), (4.216)

ANj+1 = Nj+1 - Nd = -NjRfHEjNj).

When a < 1,

1 {X}~ N(0) ~ E(Q) ~

where

iee2(o) _ iAo - ~ ^ ~ - ^RW '

RW is the reflection coefficient due to a single impurity. This shows that thetransmission coefficent decays exponentially.

When a = T]/£ > 1, Kivshar et al. found that the asymptotic change inrp(N,E)f^z _ x'/x'o), where x'o = 64/npe2, depends essentially on the parametera(0) = »?(0)/£(0) which is related to the nonlinearity of the incoming microscopicparticle. The greater a is, the larger the number of excitations in the microscopicparticle (soliton) becomes, and the smaller its spatial extension. On the contrary,if a is small, the microscopic particle looks very similar to a linear wave packet.Therefore, for initial conditions a(0) <C ac = 1.28505, the system evolves to a finalstate in which N decreases exponentially to zero while the speed of the microscopicparticle, v, reaches a constant positive value. If ac > a ta 1, the decay consistsof an initial slow transient which followed by a fast exponential behavior. Finally,the initial condition a(0) > ac leads to a situation in which both JV and v becomepractically constants.


These results show that strong nonlinearity can completely inhibit the localiza-tion effect stimulated by the disorder. This effect appears over a threshold nonlin-earity. Below this threshold, the transmission coefficient decreases to zero as thesize of the system increases, either exponentially throughout or exponentially after ashort transient. Above the threshold value this model shows undistorted motion ofthe microscopic particle in the disordered systems, i.e., the transmission coefficientdoes not decay and localization does not occur because of the small width of themicroscopic particle compared to the large values of a.

In the above discussion, the only effect of the impurity is to give rise to aneffective potential on the microscopic particle. In particular, a microscopic particle(soliton) may be trapped by an attractive potential due to the impurity as a resultof loss of its kinetic energy through radiation. However, the impurity is not a "hard"object, and it may support a localized oscillating state, or the so-called impuritymode. As a consequence, a microscopic particle may be able to transfer part of itskinetic energy to the impurity and thus excite the impurity mode during an inelasticinteraction. Hence, there is a different type of interaction between a microscopicparticle and an impurity when the impurity supports a localized impurity mode.The microscopic particle can be totally reflected in the case of an attractive impurityif its initial velocity lies in certain resonance "windows". We now discuss theseproblems.

Let us considered the Sine-Gordon model in the 1 + 1 dimensional case, includinga local impurity

4>tt - <j>xx + s in (j> = eS(x) s in cf>, (4.217)

where 8(x) is the Dirac <5-function. The natural unit system is assumed in (4.217).It is known that the Sine-Gordon model supports a topological soliton (kink) ate = 0, given by

^=4tan-1|exp[^^]|, (4.218)

where X(t) = vt is the kink coordinate and v = X is its velocity. For e > 0,the impurity in (4.217) creates an effective attractive potential well for the micro-scopic particle (kink). Kivshar et al. studied the scattering of a microscopic particle(kink) by a pointlike impurity, and integrated (4.217) using a conservative numeri-cal scheme. Simulations were carried out in the spatial interval (-40,40) which wasdiscretized by a step size of Ax — 2Ai = 0.04. The Dirac <5-function was approx-imated by 1/Ax at x — 0, and zero elsewhere. The initial conditions were chosensuch that the microscopic particle (kink) always centers at X = — 6, moving towardthe impurity with an initial velocity v, > 0. Fixed boundary conditions, </>(-40) = 0and </>(40) = 2TT, were used in the simulation. Only the atractive interaction by theimpurity, i.e., e > 0 were considered in the work of Kivshar et al.

Depending on whether the microscopic particle passes through the impurity, is


captured by the impurity, or is reflected from the impurity, there are three differentwindows of the initial velocity of the incoming microscopic particle, and a criticalvelocity vc. If the initial velocity of the microscopic particle is greater than vc, itwill pass through the impurity inelastically and escape to the positive direction,losing part of its kinetic energy through radiation and excitation of an impuritymode during the process. In this case, there is a linear relationship between thesquares of the initial velocity Vi and the final velocity ve of the microscopic particle,i.e., ve = a'(u? - vl), where a' is a constant. For e = 0.7, the critical velocity isapproximately 0.2678, and a' « 0.887.

Fig. 4.19 The kink coordinate X(t) versus time for initial velocity, Vi, of the microscopic particlein different windows: passing through (solid line, Vi = 0.268), being captured (dotted line, Vi —0.257), and being reflected (dashed line, Vi = 0.255).

If the initial velocity of the incoming microscopic particle is smaller than vc,the microscopic particle cannot escape to infinity from the impurity after the firstinteraction, but will stop at a certain distance and return, due to the attractive forceof the impurity, and to interact with the impurity again. For most of the velocityvalues, the microscopic particle will lose energy again in the second interactionand eventually get trapped by the impurity (see Fig. 4.19). However, for certainvalues of the initial velocity, the microscopic particle may escape to the negativeinfinity after the second interaction, i.e., the microscopic particle may be totallyreflected by the impurity (see Figs. 4.19 and 4.20). This effect is very similar tothe resonance phenomena in the kink-antikink collision which was explained by theresonant energy exchange mechanism proposed by Campbell et al.. The reflectionof the particle (kink) is possible only if the initial velocity of the particle is withinsome resonance windows. By numerical simulation, Campbell et al. found elevensuch windows. The details are shown in Fig. 4.21.

Similar resonance phenomena for the microscopic particle (kink) - impurityinteraction can also be observed in the </>4-model,

4>tt-cf>xx[l-e5{x)\{<t>-<l>z)=O,

which is given in the natural unit system. The inelastic interaction of the micro-


Fig. 4.20 4>(0, t) versus time in the case of resonance (vi = 0.255). Note that between the twointeractions there are four small bumps which show the oscillation of the impurity mode, andafter the second interaction the energy of the impurity mode is resonantly transferred back to themicroscopic particle (kink).

Fig. 4.21 Final velocity of the microscopic particle (kink) as a function of the initial velocity fore = 0.7. Zero final velocity means that the microscopic particle is captured by the impurity.

scopic particle with an impurity was first studied by Belova et al. However, theyignored the impurity mode, and attempted to explain the resonance effects basedon energy exchange between the translational mode of the microscopic particle andits internal mode. Later, Kivshar et al. studied again the (/>4-kink - impurity inter-action by intensive numerical simuiation and found that both the internal mode andthe impurity mode take part in the resonant interaction. For example, at e = 0.5,they found six resonance windows below the critical velocity vc w 0.185. The reso-nance structure in the </>4-kink - impurity system has an internal mode which alsocan be considered as an effective oscillator. These problems can be solved using thecollective-coordinate approach taking the three dynamical variables into account.

To summarize, we described here a new type of microscopic particle - impu-rity interaction when the impurity supports a localized mode. In particular, wedemonstrated that a microscopic particle can be totally reflected by an attractiveimpurity if its initial velocity is in a certain resonance window, which shows thatthe microscopic particle has similar properties as a classical particle. These reso-nance phenomena can be explained by the mechanism of resonant energy exchangebetween the translational mode of the microscopic particle and the impurity mode.


Other phenomena, for example, the transmission, the soliton modes and the impu-rity modes which occur in the interactions, can be understood by the wave featureof the microscopic particle. Therefore, the scattering of microscopic particles innonlinear quantum mechanics by impurities also exhibit wave-corpuscle duality.

4.14 Tunneling and Fraunhofer Diffraction

In the last section we discussed the resonant behaviors of microscopic particlesthrough trapping, reflection and excitation of impurity modes, when an microscopicparticle is scattered by impurities with an attractive ^-function potential well. Theuse of the <5-function impurity potential allowed us to investigate the dependenceof reflection and capture processes by the impurities on the initial velocity of theincoming microscopic particle. In these discussions, however, we did not considerthe effects of finite width of the potential well or barrier. If this is taken into con-sideration, tunneling of the microscopic particle through the well or barrier wouldoccur due to its wave property in the nonlinear quantum mechanics. Kalbermannstudied numerically the interaction of an microscopic particle (kink) in the Sine-Gordon model with an impurity of finite width, and attractive, repulsive or mixedinteractions, respectively. In these studies, the Lagrangian function of the systemwas represented by (in natural units)

£ = d ^ 9 ^ + i A U 2 - — J , (4.219)

where g is a constant, A = g + U(x), and U(x) is the impurity potential which isgiven by

U(x) = h! cosh'2 f ^ 1 ) + ht cosh"2 i^1—) • (4-220)

Here hi, h2 and xit x2 are the strengths and positions of the two impurities, re-spectively, a\ and a2 are some constants. If hi > 0 and h2 < 0, this potentialdescribes a repulsive impurity and an attractive impurity. The equations of motionwere solved by Kalbermann using a finite difference method. A microscopic parti-cle (soliton) was incident from x — — 3 with an initial velocity v0, on an impuritylocated at x = 3. The spatial boundaries were taken to be -40 < x < 40, witha grid of dx = 0.04. A time step of dt = 0.02 was used and the simulation wascarried out for a total elapsed time of t = 200 (or 10000 time steps), which wassufficient to allow for resonant tunneling to decay, permitting a clear definition ofthe asymptotic behavior of the microscopic particle. The asymptotic velocities forthe reflected and transmitted cases were calculated using the actual motion of thecenter of the microscopic particle.

In his calculation, Kalbermann used g = m2 in (4.219), and 0.7, 1.0, and 1.5respectively for m, which correspond to the cases where the size of the microscopic


particle, « 1/ro, is larger, comparable, and smaller, respectively than the barrierwidth, « a/6, where a is the parameter in the argument of U{x) in (4.220). Awidth of ai = 1 was assumed for the repulsive barrier while the attractive barrierhas a width of «2 = 0-3- These values were chosen to illustrate all effects in areasonable range of velocity values around v0 « 0.25. These considerations, plustrial and error, led to the choices of hi = 1 and h2 = -6 . Fig. 4.22 shows theimpinging microscopic particle (soliton) as well as the barriers. Figs. 4.23 - 4.25show the final velocity v' as a function of the initial velocity v0 for the repulsive{hi = 1 and h2 =0 ) , attractive {hi = 0 and h2 = -6), and attractive-repulsivecases, respectively.

Fig. 4.22 From top to bottom: Kink with m = 1 impinging from left onto a repulsive barrier;Kink with m = 1 impinging from left onto an attractive impurity; Kink with m = 0.7 impingingfrom left onto an attractive-repulsive system; Kink with m = 1.5 impinging from left onto arepulsive-attractive arrangement.

In the repulsive case (Fig. 4.23), the microscopic particle is reflected if the finalvelocity of the particle v' < 0, up to a certain value of the initial speed for whichthe effective barrier height becomes comparable to the kinetic energy, and then asudden jump to transmission occurs. In all three cases shown in Fig. 4.23, thetransmission starts at the same kinetic energy, with minor differences due to theeffective barrier.

In the attractive case there are islands of reflection between trappings and res-onant behavior in which the microscopic particle remains inside the impurity andoscillates, exciting the impurity mode as mentioned in the previoius section. Againthe higher the mass, the smaller the critical velocity for which transmission starts.


Fig. 4.23 Final velocity v' versus the initial velocity vo for rn = 0.7 (upper panel), m = 1 (middlepanel) and m = 1.5 (lower panel) for the repulsive barrier.

The details of the reflection islands depend strongly on the parameters, but thegeneral trend is similar for all three cases of different masses.

In the case of combination of attractive and repulsive impurities shown inFig. 4.25, the reflection dominates at low velocities which is induced by the re-pulsive impurity. Trapping and resonant behavior occur with islands of reflection.At high velocities, the transmission occurs which is essentially dictated by the sameimpurity. The repulsive-attractive case is similar to the repulsive case for velocitybelow transimission, and the critical speed here is determined mainly by the attrac-tive impurity that can drag back the microscopic particle (soliton) after it passesthrough the barrier. It appears that the larger the mass (the thinner the microscopicparticle), the more the attractive impurity is capable of trapping, thereby produc-ing a somewhat counterintuitive behavior in which a massive microscopic particleneeds a higher initial velocity in order to traverse them. Concerning the durationof the microscopic particle inside the barrier, there is always a time delay in theimpurities, in contrast to the quantum-mechanical Hartmann effect. Furthermore,the energy of the microscopic particle is conserved in the scattering.

The above discussion addressed the tunneling effect of microscopic particles inthe one-dimensional case. It shows again the wave property of microscopic particlesin the nonlinear quantum mechanics. However, in order to relate more closely toactual tunneling of microscopic particles in nonlinear quantum mechanics, one has toconsider higher dimensions, such as the 0(3) two-dimensional case, and eventuallyincluding rotations of the microscopic particle (soliton), and other effects such as


Fig. 4.24 Same as Fig. 4.23 but for the attractive case.

fluctuations. Moreover, actual barriers can be dynamic. There is then a need toallow for more flexibility in the modeling of impurities as well as the possibility ofenergy dissipation.

In the following, we consider the Praunhofer diffraction of microscopic particlesdescribed by the nonlinear Schrodinger equation (4.40) with a small scale initialphase modulation and a rather large initial intensity. These conditions allow usto use a perturbution approach based on multiscale expansions to investigate suchproblems, as was done by Konotop.

Following the approach of Lax and the inverse scattering method, the lin-ear spectral problem corresponding to the nonlinear Schrodinger equation sat-ifies the Zakharov-Shabat equation (4.42), where A is the spectral parameter,<po(x') = cj)(x',t' = 0) being the initial condition for the nonlinear Schrodingerequation (4.40). Konotop assumed the following for for <j)o(x'),

4>Q{x') = b'g(x')f(x'), (4.221)

here g(x') is a random and statistically homogeneous function

<<?(*')> = <?>(g(x')g*(x'1)) = Qdl(x'-x'1), (4.222)

(9(x')9(x'i)) = Gd2(x'- x'i),

where (• • •} denotes the average over all realizations of the random function g{x'),di and d% are correlation radii (from now on, we assume d\ = d2 = d for simplicity),


Fig. 4.25 Same as Fig. 4.23 but for the attractive-repulsive case.

f(x') is a regular function varying on a unit scale, \f(x')\ ~ |g(x')l ~ 1> &' representsthe amplitude of the initial pulse and is a nonlinearity parameter.

The solution of the Zakharov-Shabat equation is known to be of the form

* = (t1) = C(i')î(x',A) + D{x>)th&,V, (4-223)

with

/piXx'\ f 0 \Vi(x',A)=ê

0 y M^)=(e-ixx>)- (4-224)

C(x') and D{x') in (4.223) satisfy the "initial" conditions: C(x' = -oo) = Co,D(x' = —oo) = 0. The reflection coefficient R(\) was determined by Konotop,which is given by

R(X)= lim r{x')e-2iXx\ r(x') = ?^-e2iXx'. (4.225)

Prom (4.221), r(x') satisfies the Riccati differential equation

- ^ = 2i\r + i(j>o(x')b' - i^x'y"2. (4.226)

Konotop restricted the "potential" <f>o(x) in the interval [0, L], that is, CJ)Q(X') = 0for x' < 0 and x' > L. In this case, the initial conditions for (4.226) are r(0) = 0,


and r(x' = oo) = r(L). Equation (4.226) then becomes

^ = it [Ar + r(dZ')g'*(O - fidt'WiOr2} , (4.227)

where

A = A , e = db>, g(e) = g(x'), £' = £

For small correlation radius d <C k~l (e >C 1), r ( f ' ) may be expressed as a series

r ( O = r ( 0 ) ( r , 6 , - - - ) + e r ^ ^ 1 , . • • , ) + ••• (4.228)

where & = e l£' and correspondingly

i = W + €Wi+"" (4"229)

In the case of f(d£) = f(xi) (i.e. V = 1 and e = d), from (4.227) - (4.229),Konotop obtained

0r(o)- ^ T = 0, (4.230)

and

^ T = ~^T +«Ar(0) +i/*^)5"(e') - i / ^ iV^)^] 2 , (4-231)

where ^', i (i = 1,2, • • •) are independent variables. Equation (4.230) implies that

r(o) = r(°)(£i). An expression for r^ follows directly from (4.231)

r(1) = -e^- + iM'r™ + it'nWit') - t£7(&)F(£')[r(0)]3. (4.232)

Here F(£') represents the "average"

F(O = J, t dx'9{x'),? Jo

introduced by Konotop. r^ (î) is then determined by requiring the secular termson the right hand side of (4.232) to vanish. That is

^ ~ = iArW + ir&)F*(L) - if(^)F(L)[r^}2 (4.233)

where L = L/e. In the case under consideration the average reflection coefficient isgiven by r{L) = r^(L) + O(e2).

Now let

G{ti,x')=ir(Z1)F*{x')-if(ti)F{x>)[rM]2,


where

\ 3 G ( M = G 2 ) a n d | c j r ( i ) = r W .

Then, (4.232) may be rewritten in the following form

r(i1)(O = Z'[Gi(Si,O-Gi(S1,L)}.

In the case of stochastic initial conditions, i.e., g(x') is a random function, thereflection coefficient r^ (£') is a random function too. Hence, there is

e | ( r | 1 ) ( e ' , 6 ) ) | « l , (4.234)

and

eV/iy[rj1)(e',Ci)]«l, (4.235)

where W[r] = (r2) — (r)2 is the dispersion, and it is given by

W [rf >(£')] =W[eGi(t1,?)] + iw [LGitfuL)] (4.236)

~T [(S'Gi{Si,aLGifo,L)) - itCMi,?)) (LGi(ti,L))] .Since our discussion is limited to the statistically homogeneous function g(x') withfinite dispersion \Q(x')\ » |W(a;')| « 1, an estimate for W[r] follows from (4.236),W[r] « L. Therefore, the inequalities (4.234) and (4.235) may be rewritten in theform of L <$i e~l. If we further assume that g{x) is an ergodic process, the functionF(L) reduces to (g(L)) + 0{eL~l) and <rW(L)) = 0 with an uncertainty of cL'1.Konotop finally obtained, from (4.233), the following in the interval e « L « e"1,

^ ~ = iArW + iffoW - if(Zi)9(x')[rM}2. (4.237)

Konotop also studied Praunhofer diffraction of an microscopic particle (soliton)which has features of a phase modulated noncoherent wave through a slit. Thediffraction pattern detected on the screen is also affected by the nonlinearity ofthe medium behind the screen which was taken into consideration by Konotop.The corresponding initial condition for the nonlinear Schrbdinger equation in thisdiffraction is taken to be a square potential with a random phase,

, , M _ / cj>oeie^')+ie", for 0 < x1 < L,

0 o l a : J ~ \ O , for *' < 0 or x' > L.

where 0(x') is a Gaussian noise with statistical characteristics {0(x)} — 0,(0(x)6(x')) = Q{x' - ar'j), Q(0) = a2. For 0 < x' < L, it is easy to show that

(4>o(z')> = foe-'212 and (0o(O<M*i)) « Qtf - A)- (4-238)


If Q{x') -> 0 as x' ->• oo, the process <fo(z') is ergodic, i.e., one can use (4.237)to determine the reflection coefficient

_ (-j)(*+D/ôtanh [JV+WL&B] e-ie°-2iXL

(-*)(<!+1)/2Atanh[i('5+1)/2LAi]+jA5 ' ( 4 ' 2 3 9 )

where

A , = ^4>2oe-s2 +8X2, (<5 = ± 1 ) . (4.240)

We can also determine the property of the Fraunhofer diffraction in such a case.In fact, in the diffraction problem, 6 = 1 and 5 = —1 correspond to focusing anddefocusing medium, respectively. The diffraction picture in the Fraunhofer zone isdetermined by the Manakov's formula

l<Hz',t')|2 = i ^ l n | l + air(L)|2|'5. (4.241)

Prom (4.238), (4.240) and (4.241), it follows that the diffraction picture of thephase modulated microscopic particle is similar to that of the regular wave witha smaller amplitude. As far as the phase modulations are concerned, there areno limitations on the phase fluctuation dispersion, i.e., (el\-e(x')+0(x'i)'\) < 1 for anyQ(x'-x[).

Similar results may be obtained for a wave with an amplitude modulation of thetype

j . , ,, f <t>oe9{x>), forO<x' <L,MX)-\O, for z ' < 0 or z ' > L.

The solution takes the form of (4.238), with — a2 substituted in the place of a2.But for the case of a focusing medium one must take the possibility of creation ofa microscopic particle (soliton) into account. This leads to an upper bound for a2

for applicability of (4.241) in the case of 5 = 1.Equation (4.233) for r^ does not contain L explicitly. However, if we extend

the above results to infinite initial conditions with high decreasing speed, the timeinterval in which the initial condition cf>o(z') is essentially nonzero can be used asparameter L.

From the above discussion we see that the Fraunhofer diffraction phenomenontakes place when a microscopic particle described by the nonlinear Schrodingerequation moves in a perturbed systems with random initial condition and a smallcorrelation radius. This phenomenon also occurs for pulse waves with slowly varyinginitial phase modulations (amplitude and phase modulations) which propagate in adispersive medium. Once more, this shows the wave nature of microscopic particlesdescribed by the nonlinear Schrodinger equation.

Fraunhofer diffraction of microscopic particles, described by (4.40), from a beltin a nonlinear defocusing medium, was also studied by Zakharov and Shabat, under


the initial conditions

[ 1, for |a:'| > a'. y '

Here t' and x' are regarded as the longitudinal and transverse coordinates, and a' isthe half-width of the belt required for the diffraction, the amplitude of the infinitewave is set equal to unity.

In accordance with the results of Section 4.3, it is necessary to solve the eigen-value problem (3.21) and (4.42) with the function q = 4>\V=Q. We have

Cl[j]e-^'+c2[°]e^', (for \x'\ < a'),* = < c2

f ( 1 - A 2i )

1 / 2 - A L - v T r F x ' ) (fsxaj>a,)t (4.243)

Cl i ( l - A a ) 1 / a ~ A e~VT=JSX'> (for i ' < - a ' ) ,

where Ci and ci are the Jost coefficients. Requiring the solution (4.243) to becontinuous at x' = ±a, we obtain the following eigenvalue equation after someelementary transformations

cos2Aa' = A. (4.244)

Equation (4.244) has a set of zeros, ±An, which are symmetric and approachingzero, and |An| < 1, as expected. For sufficiently small a', there is only one pairof zeros. For large a', the number of pairs of zeros, N, can be estimated fromN « 2a'/n.

Local darker bands in the diffraction pattern, given by the zeros of (4.244), canbe found which correspond microscopic particles (solitons) that propagate alongstraight lines in the (x',t') plane. The microscopic particle (soliton) with the eigen-value An moves in a direction that makes an angle 0n = tan-1(2An) with the direc-tion of propagation of the wave. The minimum number of such bands is two. Thus,the diffraction of a microscopic particles by a belt in a nonlinear medium differs inprinciple from the diffraction in a linear medium in that it can be observed at anarbitrarily large distance from the belt, whereas in a linear medium the diffractionpicture becomes "smeared out" at large distances.

As a matter of fact, the Fraunhofer diffraction for superconducting electron (soli-ton) was experimentally observed in the superconducting junctions (See Fig. 2.3).Therefore, it is confirmed beyond doubt that Fraunhofer diffraction can occur formicroscopic particles in nonlinear quantum mechanics.


4.15 Squeezing Effects of Microscopic Particles Propagating inNonlinear Media

Interferometer experiments of Wu et al. showed that propagation of a laser fieldalong an optical fiber produces a squeezing effect, which refers to the reduction ofnarrow-band quantum fluctuation in one of the quadrature phases of the field to alevel below the usual limit for a coherent state. The squeezing of a laser field canbe induced by four-wave mixing, resulting from quantum noise. Calculation on thequantum fluctuations in a dispersionless medium predicted the squeezing effect inone of the quantum phases of a propagating continuous wave (CW) field that isconsistent with experimental results.

In nonlinear quantum mechanics such squeezing phenomena also occur for quan-tum microscopic particles described by the nonlinear Schrodinger equation (3.62),i.e., propagation of a microscopic particle in a dispersive nonlinear medium canlead to a wide-band squeezing of quantum fluctuations in the vicinity of the micro-scopic particle (soliton). This problem was studied by Carter et al.. In their work,source of quantum noise was added in the original nonlinear Schrodinger equationas a stochastic field, and the nonlinear Schrodinger equation becomes the followingstochastic form

iW£Jl = 1 \i ± ^ _ j ftfj) _ 101*0( ,0 +i][Ir1(x',t')ct>(x',t'), (4.245)

where n — k" /bt'Q is the average number of excitation of the particle field, or photonnumber in the CW field, k" = d2k/duj2\ko is the dispersion of overall medium groupvelocity, t'o = ^/w9|A;"|/(2Aw) is the time scale, b is the effective nonlinearity, vg isthe group speed of the microscopic particle (soliton). The real and stochastic fieldr\ and rj+ are introduced here with

(r,(xi,*i)r,(x'2:t'2)) = (V+ (x[,t[) V+ (x'2,t'2)) =S(x[- x>2)8(t[ - t'2),

(4.246)(v+(x'1,t

l1)v(x'2,t'2))=0,

where the (+, - ) signs in (4.245) correpond to the cases of normal (k" > 0) andanormalous (k" < 0) dispersions, respectively. Equation (4.245) can be derived fromquantum theory of propagation in dispersive nonlinear media in which quantumfluctuations are handled via the coherent-state positive-P representation of Carteret al. In (4.245), the stochastic part provides the quantum fluctuations in theproblem, and is the source of inherently squeezed quantum noise. This term iseasily modified to include thermal phase-modulation noise.

Carter et al. first studied the squeezing effect of a CW field. They first definedthe spectrum of quadrature fluctuations in the CW field at location x' with a phase


angle 8, following the approach of Gardiner et al,

S(u,,x',6) = ^ ° 5 K \e-2ie(A^D,x')A^(-u,,x'))

+ {AJ>(Q,x')A4>+{-u,x'))] • (4-247)

A similar equation for A<£+ can be obtained

1 rT/2t'o&<j>(Q,x') = - = / A^VK*' dt',

y/2-K J-T/2t'oA<t>(t',x') = cl>(t',x')-(4>(t',x')).

Squeezing occurs when 5 < 0. In (4.247), Tp represents a pulse duration definedfor normalization purpose. In the CW case, Tp = T, which is the total observationtime. The quantum limit of squeezing is given by 5 > — 1.

When one deals with relatively small quantum fluctuations, the results obtainedfrom a linearized fluctuation equation are good approximations. Hence we write (j>as 0o + A</>. Here cj>o = (<j>) is a first order approximate classical solution to thenonlinear Schrodinger equation. It corresponds to a coherent-state input of (f>o {T, 0)at x' = 0. Linearization requires n ~3> 1, although squeezing does not depend onn initially. For large propagation distances (x' 3> 1), linearization can break downeven for n > 1, if there is exponential growth in fluctuations.

Considering now the CW field, the constant solution is <f>l = 1/2. This impliesan input power of hu>on/2to, where u>o is a renormalized frequency. The constantsolution is therefore applicable to any CW of input power W provided that Aw =vgWb/(ftijj0), with vg = (duj/dk)\k0 being the group velocity of the moving frame.The resultant squeezing at the phase for the maximum fluctuation reduction is

1 - cos ts ' ) ism(jx) , /?[cos(73/) - 1 ] , .M w , z ) m a x = 5 ' 2 ' (4.248)

where /? = l±w2; 7 = wVw2 ± 2. This expression can also be derived using operatortechniques, and in the nondispersive limit it gives a rigorous basis. The spectrum,in the anomalous-dispersion case, is shown in Fig. 4.26. It has an exponentialfluctuation reduction at finite frequency (Q2 < 2), with perfect squeezing in the limitof a;' -> 00. This implies exponential growth in the complementary quadratures,so that linearization will not be valid for sufficiently large x'. Similar modulationinstabilities are known to exist for propagation of a classical CW.

In the case of a microscopic particle (soliton) in nonlinear quantum mechanics,an identical computational technique can be used. Carter et al. extended the abovestudy to the case of a known classical soliton solution. Here the fluctuations aredefined relative to a time-dependent solution of <j>o{t') = sech(i')- Aw is chosento give a characteristic time scale, t0, corresponding to the size of the microscopicparticle. The peak power is hu>o\k"\/btlQ. The linearized equations are similar to


Fig. 4.26 Graph of the logarithmic spectrum ln[l 4- S(x',u>)ma.x] of maximum squeezing in thecase of anomalous-dispersion.

those given above in the case of CW, except for the time-dependent four-wavemixing and stochastic terms. These can be treated numerically as a set of coupledlinear differential equations in x'. The maximum-squeezing curves are shown inFig. 4.27. The results clearly demonstrate a wide-band squeezing over the spectralwidth of the input microscopic particle (soliton). The phase angle of the largestsqueezing is approximately TT/4.

Fig. 4.27 Graph of the spectrum of maximum squeezing in the case of anomalous-dispersionmicroscopic particle (soliton).

Note that quantum limits in the CW and microscopic particle in nonlinear quan-tum mechanics are somewhat different. The fluctuations are now localized over atime interval ss 2to near the mass center of the microscopic particle. This corre-ponds to squeezing in a localized mode of the radiation field copropagating with themicroscopic particle. Hence the pulse duration used here to normalize the spectrumis Tp = 4£Q, which is approximately the size of a classical microscopic particle (soli-


ton). This is necessary in order to obtain a finite result for a pulsed field. Squeezingcan also be defined relative to more general mode functions.

In summary, the quantum noise source is added into the general nonlinearSchrodinger equation as a stochastic field, which reduces the quadrature fluctuationsin one quantum at a phase of TT/4 relative to the carrier. This intrinsic squeezinginitially increases linearly with distance. At high relative frequencies, the effectsof linear dispersion cause a phase rotation of the squeezing generated at differentspatial locations, which causes interference in the spectrum, and reduced squeezing.For long propagation distances the quantum noise undergoes further modificationsdue to four-wave mixing and nonlinear dispersion. In the normal-dispersion regime,this reduces the extent of the squeezing. In the anomalous-dispersion regime, thereis a range of frequencies where the linear dispersion counterbalances the nonlineardispersion. This results in enhanced squeezing and exponential noise reduction withdistance.

These general comments of the squeezing effect hold for both CW and the mi-croscopic particle (soliton) inputs. This squeezing effect of the solitons shows thata microscopic particle has wave property in nonlinear quantum mechanics, becauseonly wave is known to have the squeezing property. When the input is CW, the re-sults are well known. In the case of a coherent microscopic particle (soliton) input,the results were not obtained before. These demonstrate the squeezing of quantumfluctuations below the vacuum level in a microscopic particle (soliton). Therefore,this result suggestes that the squeezing is an universal property of propagation ofmicroscopic particles described by the nonlinear Schrodinger equation in quantumlimit.

4.16 Wave-corpuscle Duality of Microscopic Particles in aQuasiperiodic Perturbation Potential

The behaviors of microscopic particles subject to arbitrary perturbations are verycomplex in general. However, for certain types of perturbations, sufficiently simplebehavior can be expected even for relatively strong perturbations, which allow ananalytic description in terms of a small number of collective variables. In nonlinearsystems there always exist inhomogeneities or disorder, and the interactions involvedwill certainly have an anharmonic component. These can give rise to interconversionof excitations. Thus the microscopic particle (soliton) can emit radiation whichcan self-focus to form coherently localized excitations. Meanwhile the microscopicparticles can be scattered inelastically or be broken up. In this section, we examinethe behaviors of microscopic particles in nonlinear quantum mechanics subject to aspatially quasiperiodic perturbation potential in nonlinear systems. This problemwas studied by Scharf et al.

The perturbed nonlinear Schrodinger equation (3.2) in such a case can be written


as

i<t>v - 4>x>x> + 2 | ^ | V = eV(x')<t>, (4.249)

where V(x') = cos(kx') and e is a small parameter. As mentioned above, for a singlelocalization of a microscopic particle moving in the presence of the perturbation,we make a collective variable ansatz

<l>(x',t') = 27]ei«I'/2-i9sech[2)7(x' - q)}, (4.250)

where q is the location of the microscopic particle (soliton). Equation (4.249) pos-sesses two integrals of motion, the norm N and energy E. For <p(x, t) given in(4.250), they are given by

r+oo

N= \4>\2dx' = 4T?, (4.251)J — oo

and/•+OO

E= / [|<M2 - M4 + e|</>|2 cos(fcz')] & ' (4.252)

= 7 T " T 1 7 J + sinh(fe7r/4,) C ° S ( f c g ) i

respectively.The conservation of the norm of 0 leads to 77 = const. + O(e2). The time

dependence of the phase 6 is decoupled from the time dependence of the positionq of the microscopic particle, which is the relevant dynamic variable. The totalenergy given above depends only on the position q(t) of the microscopic particle.Conservation of the total energy E leads to an equation of motion for q which canbe derived from the following effective single-particle Hamiltonian

Hi. = — + ,£*7r , . cos{kq). (4.253)277 2 sinh(ifc7r/4r?) v ; v

The effective potential of the particle is

Veff(q) = ekir csch I — j cos(kq), (4.254)

which is derived fromVeS(q) = hfe j sech[27](a; - q)]V(x).

J—00

Scharf et al. found the following time-dependent equation for the collective coordi-nate

q(t) = -~V:Aq), (4-255)

where Ve'ff = dV^/dt.


Equation (4.255) resembles (4.17) and (4.25). It shows that the microscopicparticle satisfies the classical equation of motion, i.e., the microscopic particle hascorpuscle property just as classical particles, and the mass of the particle is 2TJin such a case. Equation (4.254) shows also that a periodic potential with shortwavelength becomes an effective potential which is exponentially decreasing for largek/rj in such a case.

We now neglect the perturbation (e = 0) and focus on a collision between twomicroscopic particles (solitons). When the two microscopic particles collide theircenters of inertia, q,, as well as their phases &i, suffer a shift as mentioned earlier. Asthe perturbed nonlinear Schrodinger equation (4.249) is U(l) invariant, we neglectthe dynamics of the phases. The shifts of the positions qi ("space shifts") of themicroscopic particles amount to an attractive interaction between the microscopicparticles. For example, the microscopic particle that was on the right hand side att -)• -co and on the left hand side at t ~> -t-oo has the form

lim \<f>\ = sech[2r?(x' - q) ± a].t'—nfcoo

For the other microscopic particle, the shift a should be replaced by —a. The shifta is given by

o = l l n [ 1 6 f a + ^ ) a + ^ - ^ a 1 , (4.256)2 [ (vi- v2)

2 Jwhere v\ and v2 are the asymptotic velocities of the two separated microscopic par-ticles and r]i and r\i their amplitude parameters [see (4.250)]. A simple calculationshows that the following two-particle Hamiltonian gives rise to the same space shiftsas (4.256)

* = £ + £- ***(* + m) -h2 p ^ - ^ l • (4-257)Combining the single-particle Hamiltonian Hi with the attractive two-

microscopic particles interaction in (4.257), we can find the following effective N-particle Hamiltonian:

l<i<j<N l " < 3 - l

This Hamiltonian is similar to that describing the motion of N anharmonicallycoupled nonlinear pendula which becomes nonintegrable for N = 2.

The microscopic particle (soliton) depicted by (4.250) has a spatial width andthereby selects a certain length scale. Thus two parameters, the size of the micro-scopic particle, Ls, and the width of the perturbation potential, Lp, respectively

(4.258)


can be introduced. Two situations, Ls > Lp and Ls < Lp, can be distinguished.In the former, the microscopic particle (soliton) covers many wiggles of the poten-tial, while in the latter the microscopic particle experiences only negligibe potentialdifferences over its size. In the intermediate case, Ls ss Lp, the behavior of themicroscopic particle is much more complicated and bears no resemblance to theunperturbed dynamics.

Scharf et al. carried out numerical simulation of (4.249) - (4.258) which gaverich and fascinating behaviors of the microscopic particle. In summary, if the widthof the microscopic particle (soliton) is small compared to the length of perturbationpotential, it is natural to neglect all degrees of freedom of the microscopic particlebesides its center of mass motion and therefore treat it as a particle. In contrastto the full dynamics this reduced dynamics will still be integrable for the nonlinearsystems. Its properties are governed by an effective Hamiltonian (4.258) or (4.253)and (4.257). If the width of a microscopic particle is large compared to the typicallengthscale of the perturbation, then one can expect the effect to be an averageover the fine spatial details of the potential. Thus the microscopic particle showsobvious wave property. This was verified by detailed investigations.

Starting from (4.249), with V(x') = cos(kx') and with a large microscopic parti-cle (77 <C k) of the form given by (4.250), Scharf calculated the correction to (4.250)due to the perturbation up to the second order in the small parameter ek~2. Hefound that

<t>e{z'J)=M*',t')[l + -tfX(kx')\,

with (f>o(x',t') given by (4.250) and the spatial modulation

- _ costs')-i(g/*)sin(fcc/)

The bare soliton <po(x',t') of the unperturbed equation acquires a dressing inthe presence of the perturbation. It can be shown that the dressed microscopicparticle (soliton) fulfills an unperturbed effective nonlinear Schrodinger equationwith renormalized parameters. The dressed microscopic particles behave like baresolitons when subject to additional long-wavelength perturbation. Again, theircenter of mass motion can be described by an effective single particle Hamiltonian.When two dressed microscopic particles collide they reemerge essentially unchangedthereby illustrating the near-integrability of the dynamics. As can be seen from(4.259) there are two types of dressed microscopic particles, namely the slow and thefast particles depending on whether q C k or q 3> k. The slow dressed microscopicparticles are spatially modulated, with the maxima appearing at minima of theperturbing potential, while the maxima of the fast dressed microscopic particlesoccur at the maxima of the potential. These two regimes are devided by a "phaseresonance" leading to a destruction of the microscopic particles.

This analysis has been extended to more general potentials with a quasiperiodic

(4.259)


part containing many short wavelengths and an arbitrary long wavelength part.The resulting solitary solutions are dressed microscopic particles (solitons) of morecomplicated forms moving like particles in a long wavelength effective potential.Lengthscale competition in general leads to complicated behavior in space and time.In the case of the nonlinear Schrodinger equation this can happen either throughthe "phase resonance" ( q sa k ) as mentioned above or a "shape resonance" {q » k).

The effective particle approximation for the microscopic particles in the nonlin-ear quantum mechanics shows that the nonintegrability of the perturbed nonlinearSchrodinger equation can manifest itself through "microscopic particle (soliton)chaos" induced by the long-wavelength part of the perturbation. The effective de-coupling of a few degrees of freedom leads to simple dynamics of the microscopicparticle through nonintegrable behavior in space and time. Inelastic effects canshow up in processes involving only few effective coordinates or many degrees offreedom. Larger curvature of the long-wavelength potential (i.e., |V"(a;')| ss TJ2|)can lead to a decay of the microscopic particles through radiation. The radiativepower can be calculated using perturbation theory.

Collisions of two microscopic particles with nearly vanishing relative velocity ina perturbing potential show effects no longer described by the effective two-particleHamiltonian He$ given in (4.258). The soliton parameters rn which played the roleof masses in the collective variable description are no longer constant, but becomedynamic variables themselves. As the amount of radiation generated might stillbe negligible an extended collective variable description seems to be feasible. Si-multaneous collision of three and more microscopic particles in the presence of aperturbation lead to effects which in general can no longer be treated by effectivetwo-particle interactions. The power of radiation generated by "dressed microscopicparticles" appears to be of higher order than r\jk, as concluded from numerical sim-ulation. Collisions of two "dressed microscopic particles" in the case of the nonlinearSchrodinger equation as well as kinks and antikinks for Sine-Gordon equation leadto a noticible increase in the power of radiation, probably to the order of 77/fc. Thesephenomena are due to the wave feature of the microscopic particles. Therefore, theabove discussion gave evidence to the wave-corpuscle duality of microscopic particlesin the nonlinear quantum mechanics.

In order to justify this statement, Scharf et al. carried out further studies onthe properties of microscopic particles described by the following perturbed Sine-Gordon equation (using the natural unit system)

<j>tt - <t>xx + [1 + £ cos(kx)} s in <j> = 0. (4.260)

This equation of motion is generated by the Hamiltonian

r+00 r 1 1 1H= dx\ -$ + -<g + [1 + ecos(fcB)](l - cos0) [ . (4.261)

J-00 [2 2 )

They used two different kinds of exact solutions of the unperturbed Sine-Gordon


equation (e = 0) as initial conditions, the breathers and the kink-antikink (K-K)solutions. The breather at rest has the form

«/>">,*) = 4tan-1 ( t a n MS i n « ; - ' ° ^ ) , (4.262)

[ cosh[(a;-xojsin^J J 'where fj, is the parameter governing the breather shape and size. As /x —j- 0, thebreather becomes shallower, its frequency, given by w&r = cosfi, grows, and it canbe effectively described by the nonlinear Schrodinger equation. On the other hand,when fi-^ n, the breather frequency goes to zero and it is actually very close to theK-K pair. As for the K-K solution, its expression at rest is given by

**-*(x,t) = 4tan- ( S i n h ; ] f - ^ 1 , (4.263)1 v cosh[7(x - x0)] J

with v1 < 1, 7 = l / \ / l — v2. Equation (4.263) can also be obtained from (4.262),by letting fi - ir/2 + ia', where ea> = 7(1 + v). Finally, both (4.262) and (4.263)can be derived using standard methods which will be discussed in Chapter 8.

The energy of an unperturbed breather at rest, as can be obtained from (4.260)

and (4.261), is EQT = 16 sin/i, whereas the same computation for an unperturbed

K-K solution with its center of mass at rest is E$~K = I67. In the absence of any

perturbation the energies for the breather and the K-K solutions at rest obviously

fulfill EQT < 16 < Eff'R. The perturbation may shift this borderline between the

breather and the K-K solutions. Of relevance to this analysis will be the potential

energy of an excitation in the absence of any perturbation

^ dx^-<j>l + l-cos<j>).

The amount by which the total energy is changed in the presence of the perturbationis given by

r+00

VeS= dx'ecos(kx'){l-cos<f>). (4.264)j-00

For a breather at rest at to = 0, Sharf et al. obtained

_n 8 tanh2 z 82 sin u(l - cot2 a sinh2 z)E° = 8 s i n M + — : ^ - -% '-,

p sin/* sinhzcosh3 z

and

trbri \ 47resinhzcos(fea;o) [sin(Kz) . , . 1VZZ(xo,z) = 5 —^— ' - + Kcos(Kz)smz\ ,

eff sin/xcosh2z sinh(/s:7r/2) L coshz J 'where K = fc/sin/z and z, denned by sinhz = tan fi sin(t cos n), is a measure of thedistance between the kink and the antikink bound in the breather.


For a K-K solution at rest at t0 = 0, they found correspondingly

0 _ , _ , , 8 7 z( l + i ;2S inh2/)

•^pot - °7 H — T5 ,sinh z cosh z

and

^rK-Ri \ 47resinh2cos(fcx0) \s\n( Kz) ,T N 1Kff ^ o , ^ = -2 . * ' , ; + # c o s # z sinz , 4.265

7CoslTz smh(Jff7r/2) L cosh 2 Jwhere if = fc/7 and 2, denned by sinh.z = v'1 sinhût), is a measure of thedistance between the kink and the antikink.

The effective potential Veff depends on the distance between the (virtual) kinkand antikink, z, and on the center of mass, XQ, of the K-K solution or the breather.For nonrelativistic velocities (7 « 1), Veff (xo,z) can be used to calculate the influ-ence of the potential on the motion of the center of the excitation as well as on therelative distance between the kink and the antikink in an adiabatic approximation,assuming that the parameter z can be considered as a second collective variable.

A breather is a bound state solution of the Sine-Gordon equation which oscillatesaround <j> = 0 with a period T — 2n/cos^x. For sufficiently strong perturbationsthe total potential energy J5pot = Eôt + Vea can have other minima besides 0 = 0around which the solution can oscillate.

Fig. 4.28 Schematic "phase diagram" for the microscopic particle in the perturbed Sine-Gordonequation.

Scharf et al. introduced two characteristic length scales, which are the breatherwidth (As) and the perturbation period (Ap) and carried out a number of numericalsimulations spanning large intervals of the parameters, namely, the length ratio(As/Ap) and the potential strength (e). They found that there are basically threepossible behaviors for the breather, largely independent of its amplitude. If AB < Ap

(see Fig. 4.28) the breather can indeed be considered as a particle in the externalperiodic potential, and if AB > Ap, this is still valid except that now the effectivepotential in which the particle moves is not the original but a renormalized one.On the contrary, if AB ~ Ap (competing lengths), this particle-like behavior ceasesto be true even for small e values, and the breather rapidly breaks up, either intoa kind-antikink (K-K) pair (if its amplitude is large enough) or into two or more


breathers, involving always a great amount of radiation. Interestingly, they alsofound that breather breakup happens also for noncompeting lengths when e isabove a certain threshold eT, which depends on A B / A P . They observed nonradiativesplitting of large-amplitude breathers into K-K pairs for large A B / A P , which seemsto have its origin in energetic considerations and not in the length competition.All these phenomena are smoothed out by dynamic effects, when the breathers aremoving, as observed in defail in the nonlinear Schrodinger equation. Therefore,the features shown in Fig. 4.28 indicate that microscopic particles described by theSine-Gordon equation and the nonlinear Schrodinger equation have wave-corpuscleduality because they have either corpuscle feature in one case, or wave feature inother case. Thus it gives us a satisfactory answer to the question of this chapter -whether microscopic particles have the wave-corpuscle duality in nonlinear quantummechanics.

Bibliography

Abdullaev, F. (1994). Theory of solitons in inhomogeneous media, Wiley and Sons, NewYork.

Abdullaev, F., Bishop, A. R. and Pnevmtikos, St. (1991). Nonliearity with disorder,Springer-Verlag, Berlin.

Abdullaev, F., Darmanyan, S. and Khabibullaev, P. (1990). Optical solitons, Springer,Berlin.

Ablowitz, M. J. and Segur, H. (1981). Solitons and the inverse scattering transform, SIAM,Philadelphia.

Ablowitz, M. J., Kruskal, M. D. and Ladik, J. F. (1979). SIAM. J. Appl. Math. 36 428.Abrowitz, M. J. and Kodama, Y. (1982). Study Appl. Math. 66 159.Aceves, A. B., Moloney, J. V. and Newell, A. C. (1989). Phys. Rev. A 39 1809 and 1839.Alonso, L. M. (1983). J. Math. Phys. 24 982 and 2652.Alonso, L. M. (1984). Phys. Rev. D 30 2595.Alonso, L. M. (1985). Phys. Rev. Lett. 54 499.Anandan, J. (1981). Phys. Rev. Lett. 47 463.Andenson, M. R.,et al. (1995). Science 269 198.Andersen, D. (1979). Phys. Scr. 20 345.Andersen, D. R. and Regan, J. J. (1989). J. Opt. Soc. Am. B6 1484.Andersen, D. R., Cuykendall, R. and Regan, J. J. (1988). Comput. Phys. Commun. 48

255.Anderson, D. and Lisak, M. (1985). Phys. Rev. A 32 2270.Anderson, D. and Lisak, M. (1986). Opt. Lett. 11 174.Aossey, D. W., Skinner, S. R., Cooney, J. T., Williams, J. E., Gavin, M. T., Amdersen,

D. R. and Lonngren, K. E. (1992). Phys. Rev. A 45 2606.Bass, F. G., Kirshar, Yu. S. and Konotop, V. V. (1987). Sov. Phys.-JETP 65 245.Berryman, J. (1976). Phys. Fluids 19 771.Bishop, A. R. (1978). Soliton and physical perturbations, in Solitons in actions, eds. by

K. Lonngren and A. C. Scott, Academic Press, New York, pp. 61-87.Boyd, J. P. (1989). Chebyshev and Fouier Spectral methods, Springer, New York, p.792.Campbell, D. K., Peyrard, M. and Sodano, P. (1986). Physica D19 165.


Campbell, D. K., Schonfied, J. E. and Wingate, C. A. (1983). Physica D9 1 and 33.Carter, S. J., Drummond, P. D., Reid, M. D. and Shelby, R. M. (1987). Phys. Rev. Lett.

58 1843.Castro Neto, A. H. and Caldeira, A. O. (1991). Phys. Rev. Lett. 67 1960.Castro Neto, A. H. and Caldeira, A. O. (1992). Phys. Rev. B 46 8858.Castro Neto, A. H. and Caldeira, A. O. (1993). Phys. Rev. E 48 4037.Chen, H. H. (1978). Phys. Fluids 21 377.Chen, H. H. and Liu, C. S. (1976). Phys. Rev. Lett. 37 693.Cooney, J. L., Gavin, M. T. and Lonngren, K. E. (1991). Phys. Fluids B3 2758.Cooney, J. L., Gavin, M. T., Williams, J. E., Aossey, D. W. and Lonngren, K. E. (1991).

Phys. Fluids B3 3277.Currie, J., Trullinger, S., Bishop, A. and Krumhansl, J. (1977). Phys. Rev. B 15 5567.Davydov, A. S. and Kislukha, N. I. (1976). Sov. Phys.-JETP 44 571.Desem, C. and Chu, P. L. (1987). Opt. Lett. 12 349; IEE. Proc.-J. Optoelectron. 134 145.Desem, C. and Chu, P. L. (1992). Soliton-soliton interactions in optical solitons, ed. Tay-

lor.J. R., Cambridge University Press, Cambridge, pp. 107-351.Dziarmaga, J. and Zakrzewski, W. (1999). Phys. Lett. A 251 193.Faddeev, L. (1958). Dokl. Akad. Nauk SSSR 121 63.Fei, Z., Kivshar, Yu. S. and Vazquez, L. (1992). Phys. Rev. A 46 5214; ibid 45 6019.Fogel, M. B., Trullinger, S. E., Bishop, A. R. and Krumhansl, J. A. (1976). Phys. Rev.

Lett. 36 1411.Fogel, M. B., Trullinger, S. E., Bishop, A. R. and Krumhansl, J. A. (1977). Phys. Rev. B

15 1578.Frauenkron, H. and Grassberger, P. (1995). J. Phys. A 28 4987.Frauenkron, H. and Grassberger, P. (1996). Phys. Rev. E 53 2823.Frauenkron, H., Kivshar, Yu. S. and Malomed, B. A. (1996). Phys. Rev. E 54 R2344.Friedberg, R., Lee, T. D. and Sirlin, A. (1976). Phys. Rev. D 13 2739.Garbalzewski, P. (1985). Classical and quantum field theory of exactly soluble nonlinear

systems, World Scientific, Singapore.Gardiner, C. W. and Savage, C. M. (1984). Opt. Commun. 50 173.Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miure, R. M. (1967). Phys. Rev. Lett.

19 1095.Gordon, J. P. (1983). Opt. Lett. 8 596.Gordon, J. P. (1986). Opt. Lett. 11 662.Gorshkov, K. A. and Ostrovsky, L. A. (1981). Physica D 3 428.Gorss, E. P. (1961). Nuovo Cimento 20 454.Hasegawa, H. and Brinkman, W. (1980). IEEE. J. Quantum Electron 16 694.Hasegawa, H. and Tappert, F. (1973). Appl. Phys. Lett. 23 171.Hermansson, Y. B. (1983). Opt. Commun. 47 101.Ichikava, Y. (1979). Phys. Scr. 20 296.Ikezi, H., Taylor, R. J. and Baker, D. R. (1970). Phys. Rev. Lett. 25 11.Infeld, E. and Rowlands, G. (1990). Nonlinear waves, solitons and chaos, Cambridge Univ.

Press. Cambridge.Kalbermann, G. (1997). Phys. Rev. E 55 R6360.Kalbermann, G. (1999). Phys. Lett. A 252 33.Kaplan, A. E. (1976). JETP Lett. 24 114.Karpman, V. I. and Solovev, V. V. (1981). Physica D3 487.Keener, J. and Mclaughlin, D. (1977). Phys. Rev. A 16 777.Kivshar, Yu. S. and Gredeskul, S. A. (1990). Phys. Rev. Lett. 64 1693.Kivshar, Yu. S. and Malomed, B. A. (1986). Phys. Lett. A 115 377.


Kivshar, Yu. S. and Malomed, B. A. (1989). Rev. Mod. Phys. 61 763.Kivshar, Yu. S., Fei, Z. and Vazquez, L. (1991). Phys. Rev. Lett. 67 1177.Kivshar, Yu. S., Gredeskul, S. A., Sanchez, A. and Vazquez, L. (1990). Phys. Rev. Lett.

64 1693.Klyatzkin, V. I. (1980). Stochastic equations and waves in randomly inhomogeneous media,

Nauka, Moscow (in Russian).Konotop, V. V. (1990). Phys. Lett. A 146 50.Lai, Y. and Haus, H. A. (1989). Phys. Rev. A 40 844 and 854.Lamb, G. L. (1980). Elements of soliton theory, Wiley, New York, pp. 118-125.Lawrence, B., et al., (1994). Appl. Phys. Lett. 64 2773.Lax, P. D. (1968). Comm. Pure and Appl. Math. 21 467.Leedke, E. and Spatschek, K. (1978). Phys. Rev. Lett. 41 1798.Lifshits, I. M., Gredeskul, S. A. and Pastur, L. A. (1982). Introduction to theory of disor-

dered systems, Nauka, Moscow.Lonngren, K. E. (1983). Plasma Phys. 25 943.Lonngren, K. E., Andersen, D. R. and Cooney, J. L. (1991). Phys. Lett. A 156 441.Lonngren, K. E. and Scott, A. C. (1978). Solitons in action, Academic, New York, p. 153.Louesey, S. W. (1980). Phys. Lett. A 78 429.Makhankov, A. and Fedyamn, V. (1979). Phys. Scr. 20 543.Makhankov, V. (1980). Comp. Phys. Com. 21 1.Makhankov, V. G. (1978). Phys. Rep. 35 1.Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1.Marchenko, V. A. (1955). Dokl. Akad. Nauk SSSR 104 695.Miles, J. W. (1984). J. Fluid Mech. 148 451.Miles, J. W. (1988). J. Fluid Mech. 186 718.Miles, J. W. (1988). J. Fluid Mech. 189 287.Nakamura, Y. (1982). IEEE Trans. Plasma Sci. PS-10 180.Nishida, Y. (1982). Butsuri 37 396.Nishida, Y. (1984). Phys. Fluids 27 2176.Pang, Xiao-feng (1985). J. Low Temp. Phys. 58 334.Pang, Xiao-feng (1989). Chin. J. Low Temp. Supercond. 10 612.Pang, Xiao-feng (1990). J. Phys. Condens. Matter 2 9541.Pang, Xiao-feng (1993). Chin. Phys. Lett. 10 381, 417 and 570.Pang, Xiao-feng (1994). Phys. Rev. E 49 4747.Pang, Xiao-feng (1994). The theory for nonlinear quantum mechanics, Chongqing Press,

Chongqing.Pang, Xiao-feng (1999). European Phys. J. B 10 415.Pang, Xiao-feng (2000). Chin. Phys. 9 89.Pang, Xiao-feng (2000). European Phys. J. E 19 297.Pang, Xiao-feng (2000). J. Phys. Condens. Matter 12 885.Pang, Xiao-feng (2000). Phys. Rev. E 62 6898.Pang, Xiao-feng (2003). Phys. Stat. Sol. (b) 236 43.Pang, Xiao-feng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu.Perein, N. (1977). Phys. Fluids 20 1735.Peyrard, M. and Campbell, D. K. (1983). Physica D9 33.Pitaerskii, L. P. (1961). Sov. Phys. JETP 13 451.Rytov, S. M. (1976). Introduction to statistical radiophysics, Part I. Random Processes,

Nauka, Moscow.Sanchez, A., Scharf, R., Bishop, A. R. and Vazquez, L. (1992). Phys. Rev. A 45 6031.Sanders, B. F., Katopodes, N. and Bord, J. P. (1998). United pseudospectral solution to


water wave equations, J. Engin. Hydraulics of ASCE, p. 294.Satsuma, J. and Yajima, N. (1974). Prog. Theor. Phys. (Supp) 55 284.Scharf, R. and Bishop, A. R. (1991). Phys. Rev. A 43 6535.Scharf, R. and Bishop, A. R. (1992). Phys. Rev. A 46 2973.Scharf, R. and Bishop, A. R. (1993). Phys. Rev. A 47 1375.Scharf, R., Kivshar, Yu. S., Sanchez, A. and Bishop, A. R. (1992). Phys. Rev. A 45 5369.Shabat, A. B. (1970). Dinamika sploshnoi sredy (Dynamics of a continuous Medium, No.

5, 130.Shelby, R. M., Leverson, M. D., Perlmutter, S. H., Devoe, P. S. and Walls, D. F. (1986).

Phys. Rev. Lett. 57 691.Skinner, S. R, Allan, G. R., Allan, D. R., Andersen, D. R. and Smirl, A. L. (1991). IEEE

Quantum Electron. QE-27 2211.Skinner, S. R., Allan, G. R., Andersen, D. R. and Smirl, A. L. (1991). in Proc. Conf.

Lasers and Electro-Optics, Baltimore, (Opt. Soc. Am.).Snyder, A. W. and Sheppard, A. P. (1993). Opt. Lett. 18 482.Sulem, C and Sulem, P. L. (1999). The nonlinear Schrodinger equation: self-focusing and

wave collapse, Springer-Verlag, Berlin.Takeno, S. (1985). Dynamical problems in soliton systems, Springer Serise in synergetics,

Vol. 30, Springer Verlag, Berlin.Tanaka, S. (1975). KdV equation: asymptotic behavior of solutions, Publ. RIMS, Kyoto

Univ. 10 pp. 367-379.Tan, B. and Bord, J. P. (1998). Phys. Lett. A 240 282.Tappert, F. D. (1974). Lett. Appl. Math. Am. Math. Soc. 15 101.Taylor, J. R. (1992). Optical soliton-theory and experiment, Cambridge Univ. Press, Cam-

bridge.Thurston, R. N. and Weiner, A. M. (1991). J. Opt. Soc. Am. B8 471.Tominson, W. J., Gordon, J. P., Smith, P. W. and Kaplan, A. E. (1982). Appl. Opt. 21

2041.Tsuauki, T. (1971). J. Low Temp. Phys. 4 441.Wright, E. M. (1991). Phys. Rev. A 43 3836.Wu, L., Kimble, H. J., Hall, J. L. and Wu, H. (1986). Phys. Rev. Lett. 57 252.Yajima, N., Dikawa, M., Satsuma, J. and Namba, C. (1975). Res. Inst. Appl. Phys. Rep.

XXII70 89.Zabusky, N. J. and Kruskal, M. D. (1965). Phys. Rev. Lett. 15 240.Zakharov, B. E. and Shabat, A. B. (1972). Sov. Phys. JETP 34 62.Zakharov, V. E. (1968). Sov. Phys.-JETP 26 994.Zakharov, V. E. (1971). Sov. Phys.-JETP 33 538.Zakharov, V. E. and Manakov, S. V. (1976). Sov. Phys.-JETP 42 842.Zakharov, V. E. and Manakov, S. V. (1976). Sov. Phys.-JETP 44 106.Zakharov, V. E. and Shabat, A. B. (1973). Sov. Phys.-JETP 37 823.Zhou, X. and Cui, H. R. (1993). Science in China A 36 816.

Chapter 5

Nonlinear Interaction and Localization ofParticles

As mentioned in Chapter 4, in the nonlinear quantum mechanics, microscopic par-ticles have wave-corpuscle duality. Obviously, this is due to nonlinear interactionsin the nonlinear quantum systems. However, what is the relationship between thenonlinear interactions and localizations of microscopic particles? What are the func-tions and related mechanisms of the nonlinear interactions? These are all importantissues, which are worth further exploration. We will discuss these problems in thischapter.

5.1 Dispersion Effect and Nonlinear Interaction

We consider first the effects of the dispersion force and nonlinear interaction, fromthe evolution of the solutions of the linear Schrodinger equation (1.7) and the non-linear Schrodinger equation (3.2), respectively.

In the linear quantum mechanics, the dynamic equation is the linear Schrodingerequation (1.7). When the external potential field is zero (V = 0), the solution is aplane wave,

* (f, t) = I*) = AeW-EtVh. (5.1)

It denotes the state of a freely moving microscopic particle with an eigenenergy of

E=<L = ^{P*+PZ+P*h (-™<P*,Py,P*<™)- (5-2)This is a continuous spectrum. It states that the particle has the same probabilityto appear at any point in space. This is a direct consequence of the dispersion effectof the microscopic particle in the linear quantum mechanics. We will see that thisnature of the microscopic particle cannot be changed with variations of time andexternal potential V.

In fact, if a free particle is confined in a rectangular box of dimension a, b andc, then the solutions of the linear Schrodinger equation are standing waves

* (*,y,z,t)=Asin ( ^ ) s i n ( ^ - ) sin ( ^ ) e~iBt'\ (5.3)

233


In such a case, there is still a dispersion effect for the microscopic particle, namely,the microscopic particle still appears with a determinate probability at each pointin the box. However, its eigenenergy is quantized, i.e.,

E = —— 4 + if + 4 • 5.4)2m \a2 b2 c2 J v '

The corresponding momentum is also quantized. This shows that microscopic par-ticles confined in a box possess evident quantum features. Since all matters arecomposed of different microscopic particles, they always have certain bound states.Therefore, microscopic particles always have quantum characteristics.

When a microscopic particle is subject to a conservative time-independent field,V(f, t) = V(r) ^ 0, the microscopic particle satisfies the time-independent linearSchrodinger equation

-^-V2rP + V(f)i; = E^, (5.5)

where

* = ^{r)e-iEt'h. (5.6)

Equation (5.5) is referred to as eigenequation of the energy of the microscopicparticle. In many physical systems, a microscopic particle is subject to a linearpotential field, V = F • f, for example, the motion of a charged particle in anelectrical field, or a particle in the gravitational field, where F is a constant fieldindependent of f. For a one dimensional uniform electric field, V{x) — —eex, thesolution of (5.5) is

i> = AVSH$(le^, (f=f + A) (5.7)

where H^(x) is the Hankel function of the first kind, A is a normalized constant,I is the characteristic length and A is a dimensionless quantity. The wave in (5.7)is still a dispersed wave. When £ —>• oo, it approaches

^(0 = AT1 / 4e-2*3 / 2 /3 , (5.8)

which is a damped wave.If V(x) — Fx2 (such as the case of a harmonic oscillator), the eigen wavevector

and eigenenergy of (5.5) are

r/;(x) = Nne-a***/2Hn(ax), En^(n+^\hw, ( n = 0 , 1 , 2 , • • • ) ( 5 . 9 )

respectively, where Hn(ax) is the Hermite polynomial. The solution obviously hasa decaying feature.

We can keep changing the form of the external potential field V(f), but soonwe will find out that the dispersion and decaying nature of the microscopic particle

Nonlinear Interaction and Localization of Particles 235

persists no matter what form the potential field takes. The external potential fieldV(f) can only change the shape of the microscopic particle, i.e., its amplitude andvelocity, but not its fundamental property such as dispersion. This is due to thefact that the kinetic energy term, -(H2/2m)V2 = p2/2m itself is dispersive.

Because microscopic particles are in motion, the dispersion effect of the kineticenergy term always exists. It cannot always be balanced and suppressed by theexternal potential field V(f, t). Therefore, microscopic particles in linear quantummechanics always exhibit the wave feature, not the corpuscle feature.

However, nonlinear interactions present in nonlinear quantum mechanics, for ex-ample, in the nonlinear Schrodinger equation (3.2), and such nonlinear interactionsdepend directly on the wave function of the state of the microscopic particle. Itbecomes possible to change the state and nature of the microscopic particle. If thenonlinear interaction is so strong that it can balance and suppress the dispersioneffect of the kinetic term in (3.2), then its wave feature will be suppressed, and themicroscopic particle becomes a soliton with the wave-corpuscle duality.

This can be verified using the nonlinear Schrodinger equation (4.40). Both ki-netic energy dispersion and nonlinear interaction exist in the nonlinearly dynamicequation and the equation has soliton solutions (4.8) and (4.56) with constantenergy, shape and momentum. This demonstrates clearly the suppression of thedispersion effect by the nonlinear interaction, and localization of the microscopicparticle. As can be expected, the localization of the microscopic particle cannot bechanged by altering the external potential field V(x) in (3.2) at A(4>) = 0 . As amatter of fact, when V{x) is a constant, as in (2.148) for the superfluid, its solitonsolution is given by (2.150). Here sech(a;) is a stable function. If V(x') = eex, asin (2.122) in a superconductor, its soliton solution is given by (2.141) and (4.26),respectively. If V(x') — a2x'2, its soliton solution is given in (4.28). These solutionsare all of the form sech(a;), which has localized feature and stable.

For a more complicated external potential V(x), for example V(x) = kx2 +A(t)x + B(t), the soliton solution of (3.2) with A(<j>) = 0 can be written as

(t> = <p(x-u(t))eie(x>t\ (5.10)

whereinn

(f (x - u(t)) = y —sech (ax - «(*)),

0(x,t) = \-2Vksm(2Vki +/?) + y ] * ~ [ { [uo(t') - 2kcos(2Vk¥ + 0)^

+B(t') + [ y - 2Vk sin(2v/H7 + /?)] \dt' + Xot + g0.

and

u(t) = 2cos(2\/ki + /3) +uo(t).


When A{t) = B(t) = 0,

u(t) = 2cos(2Vkt) + uo(t),

6(x, t) = - I | [-fc(2 cos(2v^ + u0(*)]2 (-2Vksin(2Vkti^ + —' 1 dt'

-2V^sin (2Vki + —'a:) + g0.

This solution is similar to (4.28) and is still a sech(a;)-type soliton, and is stable,although it oscillates with a frequency which is directly proportional to A;. It willbe discussed in more details in Chapter 8.

The localization feature or wave-corpuscle duality of a microscopic particle can-not be changed by changing the form of the external potential V(x) in nonlinearquantum mechanics. This shows also that the influence of the external potentialV(x) on soliton feature of microscopic particles is secondary, the fundamental natureof microscopic particles is mainly determined by the combined effect of dispersionforces and nonlinear interaction in nonlinear quantum mechanics. It is the nonlin-ear interaction that makes a dispersive microscopic particle a localized soliton. Tofurther clarify these, we consider in the following the effects and functions of thedispersion force and nonlinear interaction.

What exactly is a dispersion effect? When a beam of white light passes througha prism, it is split into beams with different velocities. This phenomenon is called adispersive effect, and the medium in which the light or wave propagates is referredto as a dispersive medium. The relation between the wave length and frequency ofthe light (wave) in this phenomenon is called a dispersion relation, which can beexpressed as OJ — u(k) or G(LJ, k) = 0, where

detaS^0'or

d2k2 T '

in one-dimensional case. It specifies how the velocity or frequency of the wave(light) depends on its wavelength or wavevector. The equation determines wavepropagation in a dispersive median and is called an dispersion equation. The linearSchrodinger equation (1.7) in the linear quantum mechanics is a dispersion equation.If (5.1) is inserted into (1.7), we can get w — Hk2/2m from (5.2), here E = Hu,p = hk. The quantity vp = w/k is called the phase velocity of the microscopicparticle (wave). The wave vector k is a vector designating the direction of the wavepropagation. Thus the phase velocity is given by ve = (ui/k2)k. This is a standarddispersion relation. Therefore the solution of the linear Schrodinger equation (1.7)is a dispersive wave.


But how does the dispersive effect influence the state of a microscopic particle?To answer this question, we consider the dispersive effect of a wave-packet which isoften used to explain the corpuscle feature of microscopic particles in linear quantummechanics. The wave-packet is formed from a linear superposition of several planewaves (5.1) with wavevector k distributed in a range of 2Ak. In the one dimensionalcase, a wave-packet can be expressed as

•i /-fco+Afc

*(*.*) = 5 - / rp(k,t)e^kx-^dk. (5.11)27r Jko-Ak

We now expand the angular frequency <j(k) at kQ by

If we consider only the first two terms in the dispersive relation, i. e.,

fdu\ &

where £ = Afe = k — ko, then,

fAk<a(x,t)=i>(ko)ei{-koX-WDt)

d^[*-{d»Idk)kot]i ( 5 1 3 )J-Ak

= 2^(fco)Sin{[X"(^//^

ot ]Afc}e^°-^),x - (du/dk)kot

where the coefficient of e*(fco»—">ot) m (5.13) j s the amplitude of the wave-packet.

Its maximum is 2ip(ko)Ak which occurs at x = 0. It is zero at x = xn = nn/Ak(n = ±l ,±2 , . - . ) .

Figure 5.1 shows that the amplitude of the wave-packet decreases with increasingdistance of propagation due to the dispersion effect. Hence, the dispersion effectresults directly in damping of the microscopic particle (wave). We can demonstratefrom (1.11) and (1.13) that a wave-packet could eventually collapse with increasingtime. Therefore, a wave-packet is unstable and cannot be used to describe thecorpuscle feature of microscopic particles in the linear quantum mechanics.

When nonlinear interactions exist in a system, such nonlinear interaction canbalance and suppress the dispersion effect of the microscopic particle. Thus themicroscopic particle becomes a localized soliton as mentioned repeatedly, and itscorpuscle feature can be observed. Therefore, microscopic particles have wave-corpuscle duality as described in Chapter 4. As the nonlinear interaction increases,the corpuscle feature of microscopic particles becomes more and more obvious. Thelocalization nature of the microscopic particle can always persist no matter whatform the externally applied potential field takes. Therefore, the localization andwave-corpuscle duality are a fundamental property of the microscopic particles innonlinear quantum mechanics.

(5.12)


Fig. 5.1 Amplitude of a plane wave propagating along the x-direction.

5.2 Effects of Nonlinear Interactions on Behaviors of MicroscopicParticles

What would be the effects of nonlinear interactions on microscopic particles? Toanswer this question, we first consider carefully the motion of water wave in a sea.When a wave approaches the beach, its shape varies gradually from a sinusoidalcross section to triangular, and eventually to a crest which moves faster than therest, as shown in Fig. 5.2.

Fig. 5.2 Changes of the shape of a water wave approaching a beach.

This is a result of nonlinear nature of the wave. As the water wave approachesthe beach, the wave will be broken up due to the fact that the nonlinear interaction isenhanced. Since the speed of wave propagation depends on the height of the wave,this is a nonlinear phenomenon. If the phase velocity of the wave, vp, dependsweakly on the height of the wave, h, then,

vp = -r = vpo + Sih, (5.14)

where

dh h=h0

ho is the average height of the wave surface, Vpo is the linear part of the phasevelocity of the wave, 5i is a coefficient denoting the nonlinear effect. Therefore, thenonlinear interaction results in changes in both the form and the velocity of the


waves. This effect is similar to that of dispersion, but their mechanisms and rulesare different. When the dispersive effect is weak, the velocity of the wave can bedenoted by

VP = | = V'PO + S2k2, (5.15)

where wp0 is the dispersionless phase velocity, and

2 ^ 2 fc=fe0

is the coefficient of the dispersion of the wave. Generally speaking, the term pro-portional to k in the expansion of the phase velocity gives rise to the dissipationeffect, and the lowest order dispersion is proportional to k2.

To further explore the effects of nonlinear interaction on the behaviors of micro-scopic particles, we consider a simple situation

<f>t + <M>x = 0 , (5.16)

where the time evolution is given solely by the dispersion. In (5.16), if><j>x is anonlinear interaction. There is no dispersive term in this equation. It is easy toverify that an arbitrary function of the variable x — (f>t,

<j> = $'(x- <j)t) (5.17)

satisfies (5.16). Fig. 5.3 shows (j> as a function of x for various values of t, obtainedfrom (5.16) and (5.17). In this figure, each point (j> proceeds with the velocity 4>.Thus, as time elapses, the front side of the wave gets steeper and steeper, until itbecomes a triple-valued function of x due to the nonlinear interaction, which doesnot occur for a normal wave equation. Obviously, this wave deformation is theresult of the nonlinear interaction.

Fig. 5.3 Form of the wave given in (5.16).

If we let 4> = $ ' = COSTTX at t — 0, then at x = 0.5 and t — n'1, cf> = 0and 4>x = oo. The time te = n~1 at which the wave becomes very steep is calleddestroy period of the wave. The collapsing phenomenon can be suppressed byadding a dispersion term <j>xxx, as in the KdV equation. Then, the system has a


stable soliton, sech2 (X) in such a case. Therefore, a stable soliton, or a localizationof particle can occur only if the nonlinear interaction and dispersive effect existsimultaneously in the system, so that they can be balanced and their effects canceleach other. Otherwise, the particle cannot be localized, and a stable soliton cannotbe formed.

However, if 4>xxx is replaced by (f>xx, then (5.16) becomes

4>t + 4>4>x=v<f>xx, {v>0). (5.18)

This is the Burgur's equation. In such a case, the term v<j)xx cannot suppress thecollapse of the wave, arising from the nonlinear interaction (jxpxx- Therefore, thewave is damped. In fact, using the Cole-Hopf transformation

^ = ~ 2 7 £ ( i o g i / / ) 'equation (5.18) becomes

dip1 _ d2ip'

This is a linear equation of heat conduction (the diffusion equation), which hasa damping solution. Therefore, the Burgur's equation (5.18) is essentially not aequation for solitary wave, but a one-dimensional Navier-Stokes equation in viscousfluids. It only has a damping solution, not soliton solution.

This example tells us that the deformational effect of nonlinearity on the wavecan only be suppressed by the dispersive effect. Soliton solution of dynamic equa-tions, or localization of particle can then occur. This analysis and conclusion arealso valid for the nonlinear Schrodinger equation and the nonlinear Klein-Gordonequation with dispersive effect and nonlinear interaction. This example also demon-strated that a stable soliton or localization of particle cannot occur in the absenceof nonlinear interaction and dispersive effect.

In order to understand the conditions for localization of microscopic particlesin the nonlinear quantum mechanics, we examine again the property of nonlinearSchrodinger equation (3.2). Prom (3.17) - (3.19), we get

E{p) KE0-bp = E0 + {-bp), (5.19)

where EQ is the eigenenergy of Ho, corresponding to the linear Schrodinger equation.To be more accurate, it is the energy of the microscopic particle depicted by thelinear quantum mechanics. If we consider E(p) as the energy of a microscopicparticle in nonlinear quantum mechanics, then it should depend on the mass andstate wave function of the microscopic particle. From (5.19), we know that theenergy is lower than that of the particle in the linear quantum mechanics by bp.This indicates that in the nonlinear quantum mechanics, the microscopic particle ismore stable than that in linear quantum mechanics. Obviously, this is due to thelocalization of the microscopic particle. Thus an energy gap between the normal


and the soliton states exists in the energy-spectrum of the particles which is seenin superconductors in Chapter 2. The term -bp (< 0) here is the binding energy orlocalization energy of the microscopic particle (soliton). Since the energy decreaseis related to the state wave function of the particle, —b\4>\2, this phenomenon iscalled a self-localization effect of the particle, and this energy is referred to asself-localization energy. If there is no nonlinear interaction, i.e., 6 = 0, then theself-localization energy is zero, and the microscopic particle cannot be localized.Therefore, the nonlinear interaction is a necessary condition for the localization ofthe microscopic particle in nonlinear quantum mechanics.

Known that nonlinear interaction is essential for localization of microscopic par-ticle, the next question would be how could a microscopic particle be localized?To answer this question, we look further at the potential function of the system.Equation (3.17) can be written as

- ~ V 2 < ^ + [F(f) -E]<f>- b\<j>\2<j> = 0. (5.20)

For the purpose of showing the properties of this system, we assume that V[f) and bare independent off. Then in one-dimensional case, equation (5.20) may be writtenas

with

V,«{4>) = \b\4>\i-\{V-E)\<j>\2. (5.22)

When V > E and V > 0, the relationship between Ves{4>) and 4> is shown inFig. 5.4. From this figure we see that there are two minimum values of the potential,corresponding to the two ground states of the microscopic particle in the system,i.e.,

This is a double-well potential, and the energies of the two ground states are — (V —E)2 /4b < 0. It shows that the microscopic particle can be localized because it hasnegative binding energy. The localization is achieved through repeated reflection ofthe microscopic particle in the double-well potential field. The two ground stateslimit the energy diffusion. Obviously, this is a result of the nonlinear interactionbecause the particle is in the normal, expanded state if b — 0. In that case, there isonly one ground state of the particle which is <f>'0 — 0. Therefore, only if b 0, thesystem can have two ground states, and the microscopic particle can be localized.Its localization or binding energy is negative, —bp. This nonlinear interaction is anattractive interaction. The nonlinear attraction can be produced by the followingthree mechanisms by means of interaction among particles and interaction between

(5.21)


Fig. 5.4 The nonlinear effective potential in (5.22).

the medium and the particles. In the first mechanism, the attractive effect is dueto interactions between the microscopic particle and other particles. This is calleda self-interaction. A familiar example is the Bose-Einstein condensation mechanismof microscopic particles, which is due to attraction among the Bose particles, andis also referred to as self-condensation. In the second mechanism, the mediumhas itself anomalous dispersion effect (i. e., k" = d2k/dui2\UJo < 0) and nonlinearfeatures resulting from the anisotropy and nonuniformity. Motion of the microscopicparticles in the system are modulated by these nonlinear effects. This mechanismis called self-focusing. The third mechanism is referred to as self-trapping. Itis produced by interaction between the microscopic particles and the lattice ormedium. In the subsequent sections, these mechanisms will be discussed in moredetails.

Prom (5.22), we know that when V > 0, E > 0 and V < E, or |V| > E,E > 0 and V < 0 for b > 0, the microscopic particle may not be localized by themechanisms mentioned above. On the other hand, we see from (5.20) - (5.22) thatif the nonlinear self-interaction is of a repelling type (i.e., b < 0), then, equation(3.2) with A(</>) = 0 becomes

ih^t+^^-\b\\4>\24> = V(x,t)4>. (5.23)

It is impossible to obtain a soliton solution, with full matter features, of this equa-tion. However, if V(x,t) = V(x) or a constant, solution of kink soliton type exists.

Inserting (3.15) into (5.23), we can get

^ 0 - | % > 3 + [ £ - W ] V = O. (5.24)

If V is independent of x and 0 < V < E, equation (5.24) has the following solution

^ggEii^yEEn^^ (5.25)This is the kink soliton solution when \V\ > E and V < 0. In the case of V(x) = 0,Zakharov and Shabat obtained dark soliton solution (see Section 4.3) which was


experimentally observed in optical fiber and was discussed in the Bose-Einsteincondensation model by Bargeretal.

Equation (5.24) may be written as

E<p = (Ho + \b\\tp\2) ip = {E0 + \b\p)<p. (5.26)

Then, E = EQ + \b\p. This corresponds to a upward shifting of energy levels. It isassociated with nonlinear localization of holes in nonlinear semiconductors.

5.3 Self-Interaction and Intrinsic Nonlinearity

Self-interaction and intrinsic nonlinearity of the microscopic particles often occur inmany body and many particle systems, and are described in quantum field theoryand condensed matter physics. For the purpose of discussing self-interaction andintrinsic nonlinearity of microscopic particles, we briefly introduce nonlinear fieldtheory here.

According to quantum field theory, the Hamiltonian operator corresponding tointeraction between two fermions can be expressed using the creation and annihila-tion operators as

Wint = Gty+7+^Bi&+7'iik + c.c, (5.27)

where the subscripts refer to proton, neutron, electron, and neutrino, respectively,tp£(ipp) and ip+(ipe) are the creation (annihilation) operators of the proton andelectron, respectively, 7^ is the Dirac matrices, and G is the coupling constant forthe interactions of these fields which is the analog of the electric charge. This is alocal interaction which occurs at a point in the space-time.

Shortly after Fermi's theory appeared, Yukawa proposed that the interactionbetween an electron and a proton was mediated by an intermediate Boson field -the meson. The meson was coupled directly and locally to both baryons and leptonswith

Hb = gip+ipn<fi, and « i = ^ > + i M , (5.28)

where <f> is the meson field and g is their coupling constant. In the field theory,each particle is represented by a separate field, but an unified theory has beenestablished. One method for the unification is the intrinsically nonlinear and self-interacting theory. An early example of such a theory is the nonlinear spinor theoryproposed by Heisenberg et o/., in which the basic field describes a spin 1/2 system.Bosons are expected to appear as excitations of Fermion-antifermion pairs. Theinteraction current density in such a case has the form

j = niP+aâutp, (5.29)


where 7] is a coupling constant and av = (1,<T), with a being the Pauli matrices.This is an intrinsically nonlinear theory since only one type of field appears inthe field equation. Recent unification schemes are to unify interactions, such asweak and electromagnetic in combination with local, intrinsic symmetries in orderto reduce the number of distinct fields. These theories also rely on the intrinsicnonlinear ities.

Fig. 5.5 Feynman diagrams of various virtual processes leading to the interaction (see book byBurt).

Self-interactions of mesons are commonly observed in experiments. For example,a pair of 7r-mesons is observed to form a /9-meson, which subsequently decays intoa pion pair. Self-interactions have been studied in connection with the self-energyproblem of an interacting field theory, which exists in any interaction field theory,whether it is quantum or classical. It introduces the persistent interaction, whichis not present in the classical scattering of two point particles, into the field theory.In quantum theory, the concept of an isolated system becomes meaningless. Thefamiliar three-stage interaction of particles, i.e., initial noninteracting, interacting,final noninteracting, or "essentially" free motion, interaction, essentially free mo-tion, no longer holds. For example, electrons described by the Daric theory coupledwith Maxwell's theory in quantized form, are always interacting. Due to the pos-sibility of virtual transition, a single electron continuously experiences interactionswith its own electromagnetic field, showing that the electron is in self-interaction orpersistent interaction. The Feynman diagrams of self-interactions of some virtualprocess are shown in Fig. 5.5. These processes can be described by conventionalperturbation theory. That is, "bare" electrons and "free electromagnetic fields" arecombined in the perturbation theory to "dress" the electron, to provide its physicalmass and charge. The persistent interaction leads to the physical particle.

The field equation including self-interaction of Bosons can generally be expressedas

d^d^ + G(m2,Xi:<l>) = 0, (5.30)

where m2 is the mass and A* (i = 1, • • -n) are coupling constants, cf> is the Boson


field. The common element in these interactions is that they create and annihilatethe constituents in some processes. Burt proposed the following classes of intrinsicnonlinearities generated from the self-interactions.

(1) The self-interaction Hamiltonian A</>4. This interaction can lead to a creationprocess in which a proton-antiproton pair annihilates into a virtual pion pair withsubsequent creation of a real pion pair through the self-interaction mechanism. TheFeynman diagram corresponding to the interaction A< 4 is shown in Fig. 5.6. Thebaryon-pion interaction can be depicted by, for example, (5.28).

Fig. 5.6 A possible mechanism for conversion from a proton-antiproton pair to a pion pair medi-ated by the Air self-interaction, (see book by Burt)

(2) The self-interaction Hamiltonian a<j)h. This interaction can lead to the situ-ation where a 2w system annihilates into a K°-K° system which decays into a 3nsystem. The Feynman diagram of the acp5 interaction is shown in Fig. 5.7.

Fig. 5.7 A 2TT system can be related to a 3TT system through a K°-K° intermediate state whichcan describe the self-interaction on the right side (see book by Burt).

(3) Polynomial currents of the form Jp(<f>) — Yi otn<j>n- This interaction can leadto processes of higher multiplicity. If pions have a Yukawa coupling to nucleons,as low energy nuclear physics seems to indicate, TT-TT interactions occur throughthe creation of a virtual nucleon-antinucleon state. This process has a scattering


amplitude which in the lowest order is given by

/

d^k 17—r^ — - ; — (5.31)(™)2 d(k - p^dik - q^dik - fc - q2)d(k) V ^

/

j4jL

™,\k\*

as shown in Fig. 5.8. In (5.31),

d(k) = k2 - m2 + ie, (5.32)

when the interaction Hamiltonian is

Hnn = igif)+jsip(f> + c.c.

Fig. 5.8 Pions, coupled to necleons, have a self-interaction depicted in lowest order perturbationby annihilation diagrams (see book by Burt).

Since the interaction in (5.31) can extend over all momentum space, the ma-trix element is infinite. However, the Yukawa coupling is accompanied by a self-interaction, H-mr = A >4. Then both interactions lead to a finite result in the per-turbation theory, if the value of A is adjusted to cancel the parts which are infinitein (5.31).

(4) The self-interactions consisting of a transcendental current

m2

SG = —sin(5</>), (5.33)

and a rational current

JT = acj)^+1 + (3<t?+l + 7 4 T p + 1 + Sif>-2p+1 (5.34)

are also interesting. They were the basis for nonperturbative mass generation andnondispersive solitary waves. All of the three currents, polynomial, transcendentaland rational correspond to a generalized solitary wave field. Each class, however,has its unique features, and different applications in physics.


For a Boson field, when the self-interaction is taken into account, the field equa-tion with nonlinear self-interaction (5.31) is

dfid^cf) + m2<t> + J(X, (/>) = 0, (5.35)

where J is one of the self-interaction currents described above. If J = 0, the solu-tions of this equation, containing either creation or annihilation operators, describequanta of the field. Here we consider the persistent self-interaction by first seek-ing solution of (5.35) that also contains either creation or annihilation operators,but not both. Obviously, equation (5.35) is a nonlinear field equation, includingboth the (^-equation and the Sine-Gordon equation. It can describe microscopicparticles. Burt pointed out that the solutions of this equation have the followingproperties, (i) <f> = <f>i '(x) = f(A). 'e±tk'x), (ii) <f>i contains the coupling con-stants for all times, (iii) <fii ' is not perturbative. The exact solutions of the fieldequation are special cases of nonlinear wave solutions. Thus, these new nonlinearfields differ from classical fields by having the integration constants replaced bycreation or annihilation operator. In this respect, they are nonlinear generalizationof the particle solutions of the free field solutions.

Whitham proved that these particular solutions of the nonlinear, dispersive fieldequations are similar to classical solitary wave solutions. They propagate with con-stant phase speed, and are nonperturbative and exact solutions of the field equationsfor all times. They describe new, intrinsically localized nonlinear modes of the sys-tems. A major difference between these solutions and the solitary waves of classicaltheory is that the matrix elements of the quantized fields describe unlocalized so-lution containing oscillating tails.

Letting the current in (5.35) take the form

Jp = a<t>2p+1 + \4>4p+1, (5.36)

Burt obtained the solution of (5.35) as

{ 2 \ -l/2p

where p ^ 0,-1/2,-1, ui±](x) = Aê^îDcjV)-^2, u = Vfc2 + m2, D is

an arbitrary constant, V is the volume of the system, A^ is a coefficient. Thesolution (5.37) has the first two features mentioned above. The exponentials arepurely oscillatory. So the field satisfies the nonlinear field equation at all times. Ifa — A = 0, the field reduces to the solution of the free field equation, UJ- (x). Inorder for the persistently interacting fields to be nonperturbative, it is necessary towrite (fri (x) as a series of positive powers of U~(x). To do this Burt used the

(5.37)


generating function for the Gegenbauer polynomials

(1 - 2wZ + Z2y1/{2p) = Y/C1Ji2p)(w)Zn, (5.38)

n

where C™ is a Genenbauer polynomial. Letting

\[4{p+l)m2\ 4(2p+l)m2J '

Burt then obtained the followingoo

4 ± ) (£) = ^2 Cn/{2P) Hb'nUi±] (x)2pn+1. (5.39)n=0

Therefore, the operator solution of the nonlinear field equation is a positivepower series expansion of creation or annihilation operators. With this expansion,we can show that this solution is nonperturbative. In practice, if we choose thefollowing new solution

when a —> 0 and A —>0, we can find that the result depends on the order in whichthe limit is taken. First, let a = 0, such that

b' I A

Y 4(2p+l)m2'Then

*g°w-f r V n<>™- (5"41)

If a ^ 0 and A = 0, the result is

where

w 1(p + l)m2'

In the first case, the solutions are singular for A = 0 and the corrections areproportional to a2/A for small a. In the second case the solutions are singular for

(5.42)

(5.40)


a = 0 and the corrections are proportional to A/a2 for small A. Hence, no suchsolutions could be found from a perturbation series in a and A. This shows that thesystem has intrinsic nonlinear modes. Obviously, it results from the self-interactionconsisting of the nonlinear current.

These intrinsic nonlinear modes have many classical counterparts. The analogywith classical solitary waves is very useful. In the presence of nonlinear interactionsa quantized system can respond by either a perturbative or a persistent interactingmode. The intrinsic nonlinearity has important meanings in nonlinear fields. Theintrinsic nonlinear field is not a perturbation of the quantum field. It also cannotbe reached from the field by successive infinitesimal motions.

The above solutions, (5.39) - (5.42), show that there are intrinsically nonlinearmodes in the self-interaction fields. For a classical system, its feature can be des-ignated by the wave amplitude. This can mean a small (linear) oscillation, a largeamplitude solitary wave, in which the amplitude is so large that discontinuities de-velop in the systems. Quantum systems also have these features, but the matrixelements of the persistently interacting field can become singular, correspondingto overgrowth in the classical systems. Davidson proved that when the density ofthe system or the interaction coupling constant becomes too large or the mass ofthe particle in the system is too small, the persistently interaction field has singu-lar matrix elements. This development of a singularity is similar to solitary waveinstability and breaking up in the classical systems. If the wave amplitude or non-linear strength is very large, or the dispersion, denoted by m2(j> in (5.35), is so smallthat it cannot balance the nonlinearity, the system cannot support the persistentinteraction. Therefore, this balance depends on the form of the interaction current.

The intrinsic nonlinear fields discussed above possess many features in commonwith classical solitary waves, but there are also considerable differences betweenthem. The most important difference is the relationship between speed of a solitarywave and that of accompanying linear waves. The classical systems have acoustic(linear) modes in which the solitary waves are supersonic. For the quantum systems,the linear and nonlinear fields have identical (superluminal) phase speeds. Thisfeature of the persistent interacting quantized fields enables us to interpret thesefields as coherent excitations of the systems. From (5.39) we see that the field creates2pn+l quantum states from the vacuum, each quantum having the same momentumand mass. This is similar to the coherent many-photon states of the electromagneticfield. However, for the persistent nonlinear fields there is no velocity dependencein the amplitude, which is contrary to the classical solitary waves. Burt provedfurther that the persistent self-interacting fields describe noncanonical excitationsof the systems because the field equation cannot be obtained from Hamiltonian ofthe systems, likely the linear equations. Thus the intrinsically nonlinear fields wouldform a class of motions canonical with respect to the new Hamiltonian.


5.4 Self-localization of Microscopic Particle by Inertialess Self-interaction

In the previous section we discussed the self-interaction among the elementaryparticles and the intrinsic nonlinearity in the classical and quantum field theo-ries. In practice, the self-interactions occur also in many condensed matters. Thedetection of quasi-one-dimensional metal conductivity in metal-organic complexesand some organic salts such as TTF (tetrathiafulvalene), TCNQ (tetralyanoquin-odimethane), TCNE (tetracyanoethylene), TTF-TCNQ, Q (quinolinium) - TCNQ2,NMQ (N-methylquinolinium) - TCNQ and HMTTF (hexamethylenetrathiafulva-lene) - TCNQ, and so on, stimulated significant interest in the study of nonlinearlocalization of the concerned microscopic particles, the electrons. This mechanismwas first proposed by Frohlich who discussed the superconductivity of electrons inone-dimensional systems without using the concept of Cooper pairs.

Frohlich pointed out that charge density wave (CDW) could reduce the energyof electrons in metals, when the electrons are restricted to one-dimensional metalchains in which the Fermi surface consists of two parallel planes. This effect resultsfrom the interaction of the electrons with a sinusoidal potential field due to displace-ments of the ionic charge density in metals ("the jellium model") which lead to anincrease of the energy gap in the electronic energy spectrum and also to a periodicchange in the electronic density in the form of nondamped waves (CDW) under thecyclic boundary conditions. The undamped wave is a soliton, which arises from theself-localization of the electrons due to self-interactions in the one-dimensional con-ductors. Davydov and Pestryakov studied the nonlinear localization phenomenonof a complex scalar field of spinless quasiparticles with inertialess self-interaction.The Hamiltonian density of the scalar field with self-interaction in one-dimensionalinfinite space is given by

where b is a nonlinear parameter which is independent of the velocity, the "—"and "+" signs correspond to attraction and repulsion self-interactions, respectively.Such a self-interaction can be realized by local interaction between the microscopicparticles which correspond to the field or by local interaction of these quasiparticleswith the field of inertialess displacements of the density of other particles whichare not considered explicitly here. This model resembles that of the A$4-type self-interaction which is similar to that discussed in Section 5.3.

If the self-interaction is absent, 6 = 0, the field <p(x, t) describes the states ofnoninteracting quasiparticles with mass m. Their motion is described by a planewave with wave vector k and energy h2k2/2m. lib ^0, the corresponding equation

(5.43)


of motion is

This is a standard nonlinear Schrodinger equation. If we assume that <f>(x, t) satisfiesthe periodic condition (f>(x, t) = <j> (x + r0L, t) and the normalization condition

/ \<j>{x,t)\2dx = l,Jo

equation (5.44) has soliton solution as that given by (4.8). Therefore, electrons inthese conductors are self-localized due to self-interaction. In other word, equation(5.44) has stationary solutions in the form of modulated plane waves, with aninhomogeneous spatial density distribution of the microscopic particle moving alongthe x-axis with a small constant velocity v. In order to elucidate the nature of theparticle density distribution, we rewrite the solution in the following form

<p{x,t) = — <p(£)exp j i \ k x - (UJ + - ^ ) t + ' n \ \ , (5.45)

where £ is a dimensionless coordinate in the reference frame, £ = (x — xo — vt)/r\,moving with the speed v = hk/m, hui is the energy of the particle relative tothe reference frame, r) is an arbitrary phase, ro is length unit in the system beingstudied, which is given by ro = h/V4mb. The periodic condition given above holdsrelative to the moving reference frame, if the wave vector k and the velocity v takesdiscrete values,

k = ^ , (n = 0,±l ,±2,±3-.-) .

Therefore, k is quantized.Substituting (5.45) into (5.44), we can get

^+£ + f2(O]f = 0, (5.46)

where e — fkjj/e0, and e0 = 2b. The corresponding energy E and momentum P ofthe field-excited state are

fL

P(v) = mv V2{i)di = mv-Jo

Multiplying (5.46) on the left by <p(£) and integrating from 0 and L, we get

'•jr[G?H*

(5.44)


Thus the energy of the field is

E(v) = l-mv2 + e0 | e ± ± £ ^(0^] .

The momentum of the field is quantized because v is quantized.In the case of attractive forces, Davydov et al. found that <pa(£, k) = Adn(a£, k)

and

*•<••*>-7^-ii | i j [2-**-!$]• (547)

if e = a2(2 — k2) and A = 2a2, where A and o are determined by the periodicityand normalization conditions mentioned earlier. In (5.47)

Qa(k) = | [2(2 - k2)E(k) - (1 - A;2) K(k)] ,

dn(u, fc), K(k) and E(k) are the Jacobian elliptic function, and the complete ellipticintegrals of the first and second kinds, respectively. When k2 — 1 — k2 -> 0 andk2 -> 0, <pa(€,k) w sech(^/4)/-\/8, Sa » mv2/2-e0/48 in an infinite system. When

Ea(v,k) = -mv2-^(l--k2j.

In the case of a repulsive force, Davydov et al. found that

(pr = —7= < sin(u) + -k2 cos(tt) [sin(u) cos(u) — u] > ,V27T [ 4 J

and

Er(v,k) = \mv2 + ^ ( l + k2),

if k2 -C 1, where u = ^/(2TT), |^| «; L.

5.5 Nonlinear Effect of Media and Self-focusing Mechanism

Many physical systems themselves have nonlinear effects resulting from theiranisotropy and nonuniformity, etc. Such systems could provide nonlinear interac-tions to microscopic particles moving in the media. To understand the mechanismof such nonlinear interaction, we consider a familiar dispersive medium, an opticalfiber, in which microscopic particles (photons) or light waves propagate. In thiscase, Hasegawa obtained a nonlinear equation for propagation of the electric fieldE(x, t) of the light due to the nonlinearity of the medium (fiber) - the Kerr effect.


In this section, we derive the equation of motion of microscopic particle in a non-linear medium following Hasegawa's approach, and give its localization propertiesdue to the self-focusing mechanism arising from this nonlinearity.

The refractive index of the medium, n, is defined by the ratio of the speed of light,c, in the vacuum to the phase velocity of the wave in the medium, n = c/vp = ck/ui.The refractive index of a dielectric medium is related to the frequency of the a wavein the medium. This phenomenon is called the dispersive effect which arises frommolecular structure of the medium as described in Section 5.1. Therefore, in thedispersive medium, the quantities such as dn/du), d2n/du)2, etc. always have finitevalues. We can expand the frequency of the wave u>(= ck/n) around the wavevectorko for the carrier wave shown in Fig. 4.1

"-«-sL»-*>>+a^L(*-*'''+5^L<*-fc''+--<"i«It gives the change of the frequency of the modulated wave due to the deviation ofits wavevector k from the carrier wave vector ko-

Due to the fact that the envelope function of the wave, p(x,t) (or E(x,t), theelectric field of the light wave) in Fig. 4.1 is a slowly varying function in both x andt, we can perform a Fourier transform on this function and then use the Fourierspace variables, Ak(= k - fco) and Aw(= u - w0), which are defined by

ip(AK,Au>) = ~ f f <p(x,t)ei{Akx-Ault)dtdx. (5.49)

The corresponding inverse transform is

¥W) = 7 - / / vKAfc.AaOe-^'-^dtAfcMAw).^"" J — oo J—oo

From (5.48) and (5.49), we see that dtp/dt and d<p/dx can be identified as theFourier transform of iAw<p and —iAkip, respectively. Making use of these, we canrewrite AUJ and Afc as -id/dt and id/dx, respectively. In terms of these operators,equation (5.48) can then be written as

.d . , d LJ" d2 V " 93 / e c .

- ^ = l w ^ - ^ ! ^ - l i r ^ + -"- (5-50)

Operating (5.50) on the envelope function (p(x,t) (or E(x,t)) and keep termsup to the order of the second derivative with respect to the carrier frequency, wethen have

where

, _ dw d2ujU ~ ~dk ' W ~ OP 'OK ko OK k0

(5.51)


and the group velocity vg of the envelop modulated wave is defined by

du , dvgV°=dk=UJ> u =~dk-

Thus w" is given by the wavevector dependence of the group velocity of the wave,and it expresses the dispersion feature of the group velocity of the wave. If UJ" = 0,the solution of (5.51) can be represented by an arbitrary function of x — vgt, i.e.,ip(x—vgt). This shows that the envelope wave of the microscopic particle propagatesat the group speed. Furthermore, we may introduce a new coordinate system whichmoves with the group velocity, r = e2t, £ = e(x - u't), where e = Ako/ko is a smallquantity which characterizes the width of the spectrum. Then (5.51) becomes

i^_^5V_0 (552)

Equation (5.52) is a dispersive equation which is obtained from the dispersive rela-tion (5.48). It shows that the envelope wave of the microscopic particle deforms inproportion to the distance of propagation due to the group velocity dispersion. Ifa microscopic particle with a short wavelength is transmitted through a medium,d2tp/dx2 (or d2E/dx2 in optical fiber) may become large, and inversely propor-tional to the square of the width of the microscopic particle (pulse). Thus the formof the microscopic particle (pulse) distorts during the transmission. In the caseof a medium (fiber) with a certain cross-sectional size which is comparable to thewavelength of the microscopic particle (or light), the group velocity dispersion u"is determined by the property of the medium itself. For a shorter wavelength, ui" isnegative, i.e., the group velocity decreases with increase of the wave number. But ata longer wavelength, cu" is positive and the group velocity increases with increasingwave number. The domain of the wavelength corresponding to negative UJ" is calleda normal dispersion region, while that corresponding to positive to" is called ananomalous dispersion region. If one chooses the wavelength that corresponds to theminimum loss of the medium (or fiber), then the group dispersion w" may becomepositive and the shape of the microscopic particle (or pulse) distorts considerablydue to the group velocity dispersion.

However, for some nonlinear media, for example, optical fiber, the nonlinearinteraction can change the property of propagation of the microscopic particle (orlight). Kerr discovered that the refractive index of a fiber increases in proportionto the square of the electric field E(x,t) (or <p(x, t)). This is called effect and itis caused by a deformation of the electron orbits in the glass molecules due tothe applied electric field. This effect can be expressed by n — no + n2\f\2 (orn = no + ri2|.E|2), where ri2 is called the coefficient in the case of a fiber. Obviously,this is a nonlinear effect because n is a nonlinear function of the state wave functiontp(x,t) (or the electric field E(x,t)). For a grass fiber, n2 is about 1.2 x 10~22

m2/V2. For a typical optical fiber with a 69 jum2 affective cross-sectional area and106 V/m electric field produced by a 100 mW optical pulse, the refractive index


increases by a factor of about 1O~10. Because of this, the change in the frequencyof the microscopic particle (or light wave) in the medium variates by a factor of—n,2\tp\2ck/(nno) = —27rn2|<p|2w/no. When the microscopic particle (or the lightwave) propagates in the medium, it experiences this effect. Therefore, the nonlinearinteraction should be included in the above equation of motion (5.52). With this,equation (5.52) becomes

where b = 2nri2puj/(no£2) and p denotes the reduction factor due to variationof the microscopic particle (or intensity of light wave) in the cross-section of themedium (or fiber), and is approximately taken as constant in most case (or 1/2for the fiber). Evidently, equation (5.53) is a nonlinear Schrodinger equation. Inthis model, the nonlinear interaction results from the interaction of the microscopicparticle (or light wave, photons) with the nonlinear medium. The nonlinear mediummakes the microscopic particles (or photons) localized, or a light self-focused to alight soliton. This mechanism of localization of microscopic particles is called aself-focus. Hasegawa was the first one to obtain the above nonlinear Schrodingerequation for the electric field E(x, t) and its soliton solution in optical fibers.

This self-focusing mechanism is a new type for generating nonlinear interactionin nonlinear quantum mechanics. It is also the mechanism of nonlinear interactionin hydrogen-bonded molecular systems and DNA in living systems. In these systemsthe neighboring heavy ion groups (or the bases in DNA) provides a nonlinear double-well potential (or the Morse potential in DNA) for the hydrogen atom (or proton).Motion of protons in these systems can self-localize to form a soliton which travelsalong the molecular chains. These systems will be discussed in Chapter 10.

To further clarify this mechanism, we derive the equation of motion of a mi-croscopic particle in a nonlinear medium following the approach of Sulem. In thelinear quantum mechanics, a linear wave equation often involves a linear operatorL which consists of dt and V,

L(fit,V)* = 0.

This equation has an approximate monochromatic wave solution,

with a constant amplitude etp. The frequency UJ and the wave vector are realquantities, related by the dispersion relation

L(—ioj,ik) = 0,

which is often written as ui = u>(k).To understand the accumulative effect or canonical resonant character of non-

linear interaction, which arises from the long time and long distance propagation of

(5.53)


microscopic particles (or wave) in the nonlinear quantum systems, it is convenientto rewrite the above linear dispersion relation in the following form

[idt - ui-ids)}^-*-^ = o, (5.54)

where ds is the gradient with respect to x and tj(-ids) is the pseudo-differentialoperator obtained by replacing k with — ids in u>(k).

In a weak nonlinear medium which responds adiabatically (or instantaneously)to a wave of finite amplitude, the nonlinearity is expected to affect the dispersionrelation of the carrying wave (in addition to the generation of harmonics of smalleramplitudes). The frequency of the microscopic particle (or wave) becomes inten-sity dependent. It is then necessary to replace the frequency u>(k) by a functionn(k,e2\tp\2) with fl(k,0) = u>(k). This is called a self-focusing effect. Moreover, acomplex wave amplitude <p of a microscopic particle is no longer a constant, but ismodulated in space and time, and becomes dependent on the slow variables X = exand T = et. In (5.54), the derivatives dt and ds are thus replaced with dt + edrand dg + eV, where V now denotes the gradient with respect to the slow spatialvariable X. As a result, equation (5.54) becomes

[idt + iedT - il{-idx - ieV,e2|^|2)] ipe'^*-"*) = 0.

It is then natural to have the weakly nonlinear dispersion relation

L + iedr - to(k - ieV, e2\ip\2)] <p = 0.

Here the parameter e is small. If we expand various quantities in this equation inpowers of e and keep terms up to the second order, the linear dispersion relationobeyed by the carrying wave becomes

i (dr + vg • V) yj + e [V • Vtp + &MV] = 0, (5.55)

where vg = V^CJ is the group velocity, the coupling coefficient b is related to theexpansion coefficient of intensity of the microscopic particle (or wave) and is givenby dQ/d(\tp\2) evaluated at \ip\2 = 0 and at the carrier wave vector k. If we consider(5.55) as an initial value problem in time, it can be written in a reference framemoving at the group velocity, by denning £ = X — Tvg. Rescaling the time in theform of T = eT, we get the following nonlinear Schrodinger equation

where the spatial derivatives are now taken with respect to £. This equation is thesame as (5.53).

This nonlinear Schrodinger equation was also derived by Zakharov, L'yov, andFalkovich (1992) in their work on plasma physics (see Sulem and Sulem 1999).Their approach was based on the Fourier-mode coupling formalism in which themodulation corresponds to broadenings of frequency and wave vector spectra of the


carrier wave, u and k. This mechanism also results in the localization of microscopicparticles or waves, to form solitary wave, and is referred to as self-focusing as well.

Considering now a wave vector K close to k and denning K = K — k such that|«| <C k — \k\, the corresponding frequency of the wave can be approximated by

U)(K) S3 Uj(k) +Vg • K+ -UJjiKj • Kl

in such a nonlinear medium.In the linear theory, the related Fourier-mode <p(K, t) is proportional to e~lu(K)t

and satisfies (5.54). At present, it is of the form

dt<p{K, t) + iu{K)<p{K, t) = 0. (5.56)

The nonlinearity contributes mainly through four-wave interactions in this mech-anism where approximately equal wave vectors are assumed. This leads to thereplacement of (5.56) in the nonlinear regime by

dtif{K,t) +iw(K)<p(K,t)

= J f A A A / A ) ^ ) ^ ) ^ + KX-K2- K3)dK123,

where dKi23 = dKid^dK^. For a narrow wave packet centered at the wave vectorfc, the kernel g is approximately given by

b9K,RUK^R3 *> 9kXk,k = J^Y •

Thus we have

dt(p{K) +i\cjk) + vg-K+ -u'jiZj • «i <p{K)

~ ( ^ s / ¥>*(£iMtf2Mtf3)<5 (& + Kx-K2- K3) dK123 = 0.

For a fixed k, we can define

4>'(K,t) = eiw^t<p(K + k,t).

This is a slowly varying function of t and obeys

dt<t>'{K) + i{vg • K ) 0 ' ( K ) + ûi'^Kj • KI4>'{K) (5.57)

ib f- T ^ T J / 4>'*(KI)(/)'(K2)<J>'(K3)6(K + it! - K2 - ^3)dK123 = 0.


The complex field can then be obtained

= J^6'^^ I*&*)***•**&,which appears as the envelope of the wave packet (2TT)~3/2 / ip(K, t)eiR'3dK. From(5.57), we know that <j>'{x,t) satisfies the nonlinear Schrodinger equation

idt<p' + ivg • V 0 ' - ^"jdidrf - ib\<j>'\2<j>' = 0.

Since the amplitude of the wave is small, the function (f>' varies slowly in spaceand time. The simple rescaling (f>'(x,t) = e(f>(X,T) with X — ex and T = et thenreproduces (5.53) or (5.55).

The above discussion illustrates that the motion of a microscopic particle (orwave) is always described by the nonlinear Schrodinger equation under this self-focusing mechanism which results in the nonlinear interactions in the systems.

5.6 Localization of Exciton and Self-trapping Mechanism

We are interested in the collective excitations in one-dimensional molecular systemsin which excitons generated by intramolecular excitation or extra electrons can beself-trapped as a localized exciton-soliton, through its interaction with the molecularlattice. These problems were first studied by Davydov and co-workers. Examples ofone-dimensional systems include a-helical protein molecules in living systems andorganic molecular crystal-acetanilide (ACN). In these systems, the peptide groups(or amino acid molecules) which consists of four atoms (N, H, 0 and C) boundby chemical forces, are arranged along the molecular chains in the form of • • • H-N-C=O • • • H-N-C=O • • • H-N-C=O • • •. The neighboring peptide groups in thechain are linked by hydrogen bonds. In acetanilide, two close chains of hydrogenbonded amide-I groups run through the crystal. In a-helix protein molecules thereare three amino acid molecular chains, of which • • • H-X is a hydrogen bond, X is N-C=O in which an amide group, C=O is a carbohyl group. The collective excitationin these systems are intramolecular excitations which has a very specific natureand is different from ordinary elementary excitations in three-dimensional crystals.In protein molecules and acetanilide the intramolecular excitations include amide-Ivibrational quantum with an energy of 0.205 eV and an electric dipole momentof 0.3 Debye directed along the axis of the molecular chains. This intramolecularexcitation or quantum of amide-I vibration was referred to as an exciton by Davydovwho studied its property.


Consider an infinite chain of weakly bound molecules (or groups of atoms) ofmass M, which are separated at a distance TQ. We assume that the excitons inthe chains have energy £0 and electric dipole moment d directed along the chains.When the molecule is excited the forces of the excitons interacting with neighboringmolecules are changed. This results in a change in the equilibrium distance betweenthe molecules. If the molecules are identical, there will be some additional resonanceinteractions which cause transfer of the excitations from one molecule to another.The states of the excitons with energy e0 can be described by the Hamiltonian

Hex = J > o - D)B+Bn - J {B+Bn+1 + B++1Bn), (5.58)n

where B£ (Bn) is the exciton creation (annihilation) operator at site n with anenergy e0 (0.205 eV). They satisfy the commutation relation,

[Bn,B+] = Snm, [Bn,Bm] = 0.

Also in (5.58), J = iS1 jr% is the resonance (or dipole-dipole) interaction thatdetermines the transition of excitation from one molecule to another, D is thedeformation excitation energy, and is approximately a constant.

The collective excitations in the systems are represented by the wave function

l*> = Z>n(t)B;t|0)ex, (5-59)n

where |0)ex is the ground state of the exciton, the coefficient ipn(t) satisfies thenormalization condition

£lV>n(*)|2=0.n

We assume that the wave functional |\£) satisfies the following Schrodinger equation,

t f t | |*> = ff|*>. (5.60)

Then we can get

i h ^ = (so - D)4>n - J (WH-I - Vn-i) • (5.61)

In the continuum approximation,

Vn(t)=lKz,i),

^n±i(t)=iP(x,t)±ro—T \fl^-j,

we can get the equation of motion of the excitons

j B ^ i ) = ( e o _ D _ 2 J M l , ( ) _ | l^ fe i ) , (5.62)


where m = h2/2Jrl is the effective mass of the exciton. Equation (5.62) showsthat the exciton, an elementary excitation, satisfies the linear Schrodinger equationin the linear quantum mechanics. Therefore, the exciton is a standard microscopicparticle with wave features. It is described by linear quantum mechanics and hasthe same properties of the electron.

We now determine the energy of the exciton. Making the transformation

B" = ^Y,A»eiknr°' A" = 7^£ B * e ~ i f c n r o > (5-63)the periodic boundary condition requires that

k-—m, (m = 0,±l ,±2---) .

Inserting (5.63) into(5.58), we get

Hex = Y,E(k)A+Ak, (5.64)

where E(k) = e — D — 2JCOSKXO, which is the energy of the exciton. The wavefunction of the exciton is given by

^k{x,t) = -j= ^ e ^ - O V l x - nr0), (5.65)

where hw(k) = E{k), and ip'(x - nr0) is the amplitude of the exciton at site n.Equation (5.65) shows that once the exciton is formed, all components in the chainsare excited, i.e., the exciton state with energy E(k) spreads over the entire chain,rather than localizes at one molecule. Hence, it is an expanded and collectiveexcitation states.

Equation (5.65) may be rewritten as

rl>k (x, t) = A(k)Skx-"ikW, (5.66)

where

A M = ^= zyfc<np°-*y (z - wo).

This shows that the exciton moves as a plane wave with a wavelength A = 2TT/A;,

but its amplitude is modulated. When its wavelength is larger than the spacingbetween the molecules, i.e., kro C 1, the energy of the exciton can be written as

7J 2 PE(k)=E0 + — , (5.67)

where Eo = e - D — 2J. Therefore, the energy spectrum is continuum. In sucha state, the excitation energy is uniformly distributed over the whole chain, and


thus cannot propagate along the chain. Excitation encompassing a small part ofthe chain, to, is not stationary.

Quasi-stationary excitation localized over a segment £Q is described by the wavepacket

¥ e x ( z , t ) = / A{ky[kx-"(k)t]dk, (5.68)Jko-Ak

where Afc = n/2l0. The group speed of the wave packet is

_ dw(k) _ MoVg ~ dk k=ko ~ m •

Here hko is the average momentum of the wave packet. When the resonant inter-action \J\ increases, the effective mass m* decreases, and the velocity vg increasesfor a given fco. Therefore, the exciton with energy E(ko) localized in a segment £oat time t becomes the wave packet with a length

at time t + r . This indicates that the wave packet is spread out and dispersedwith increasing time. When r > mlo/k, the wave packet will collapse gradually asdescribed in Chapter 1, and (5.11) - (5.13). The exciton is essentially unstable.

However, since the exciton is produced by vibrations of the amide-I (or the C=Ostretching) in the peptide groups, it is a intramolecular excitation. When it movesand decays, the states and positions of the peptide groups will be changed. If weconsider this effect and the low-frequency vibration of the peptide group, and assumethat the displacement of the peptide groups is un, then the low-frequency vibrationalHamiltonian of the peptide groups and the interaction Hamiltonian between theexciton and the vibration of the peptide group can be written as

flint = E Ûn+1 ~ ^n-i)B+Bn, (5.70)n

respectively, where M is the mass of the peptide group, w is the force constantof the molecular chain, Pn is the conjugate momentum of un, x = dJ/dun is thecoupling constant. In such a case, the Hamiltonian of the system can be written as

H = Hex + Hph + HiDt. (5.71)

Davydov used the following wave function to describe the collective excitation

(5.69)


state

Mt)}=J2^(t)B+\0)exexpl-l-Y/Wn(t)Pn-Mt>n}\\0)^, (5.72)

where |0)ph is the ground state of the phonon. Using the variational approach andthe functional ($(t)\H\$(t)), we can get

ih~gf = [eo-D + W + x (/Wi ~ /?n-i)] K - J («W + <j>n-i), (5.73)

Q~& + ¥ { 2 P n - P"-1 - ?n+l) = M (l^"+l12 - ' * n - l a ) ' (5'74)

where

Pn(t) = <*(t)|«n|*(*)),

TTn = <$(t)|Pn|*W> = M ^ ,

n L '

In the continuum approximation, equations (5.73) and (5.74) become

^-<«£r]«*.fl-^5W-OI" = o. (5-76)

where A = £o — 27 — D + W, v0 = roûi/M is the sound speed of the molecularchain. Now let f = x - x0 - vt, G = 4x2/[w(l - s2)], s = w/u0, /i = G/4J, we canget from (5.75) and (5.76) that

dgx2t)__ 2roX\<l>(x,t)\2

~dx~ ~ ~ w(l-s*) • ( 5 - 7 8 )

Equation (5.77) is a nonlinear Schrodinger equation. It has a soliton solution asgiven in (2.150). we now write the solution as

4>(x,t) = ^ | sech ]^- (x-x0- vt)] exp | i [ ^ a (s - a?o) - y ] } ' (5-79)

P{x,t) = Y° ax tanh|^-(g-ao-t;t)1 . (5.80)

Equations (5.77) - (5.80) show clearly that the exciton now is localized and becomesa soliton due to the nonlinear interaction G|</>|2. The energy of the exciton-soliton

(5.75)

(5.77)


is

E = E0 + -msoiv2, (5.81)

where v is the soliton velocity, Eo is the rest energy of the soliton, given by

Eo = eo - D - 2 J - -^j <Eex = e0-D- 2J, (5.82)

and

4x4(l + 3s 2 /2-s 4 /2)mso\ = m + - ^ £—*- > m.3u;2 J^2 (1 - S2)3

Equation (5.82) shows that the soliton energy is lower than that of the excitongiven in (5.64) by EB — —"X^l^w^J, which is the binding energy of the soliton.Obviously, EB increases with increasing coupling interaction between the excitonand vibration of the peptide group for a fixed chain. This result reveals not only thatthe soliton is more stable, but also the mechanism of the soliton formation. Thismechanism of exciton localization is called a self-trapping of the exciton due to thenonlinear interaction, G\4>\2, generated by the interaction between the exciton andthe vibration of the peptide groups. The self-trapping mechanism of excitons worksas follows. The exciton or vibrational energy of the amide-I (or C=O bond stretchingoscillation) acts through a phonon (vibration of the peptide group) coupling effectto distort the structure of the molecular chains. The chain deformation reacts,again through phonon coupling, to trap the exciton and to prevent its dispersion.Thus the exciton is localized on the chain and becomes a soliton. This is the self-trapping mechanism of the exciton. This concept or mechanism of self-trappingwas early proposed by Landau on the motion of an electron in a crystal lattice.He suggested that an effect of a localized electron would be to polarize the crystalwhich, in turn, would lower the energy of the electron. Landau's suggestion wasdiscussed in detail by Pekar, Frohlich and by Holstein. Other examples of the self-trapping include electromagnetic energy in a plasma and hydrodynamic energy ina water tank which will be discussed in Section 5.8. Besides Davydov, transport ofvibrational energy in a-helix proteins and acetanilide have been studied extensivelyby many scientists. An improved theory was proposed by Pang for energy transportin protein molecules which will be discussed in Chapter 9.

5.7 Initial Condition for Localization of Microscopic Particle

As it is known from earlier discussion, the necessary and sufficient conditions forlocalization of microscopic particle in nonlinear quantum mechanics are that thedispersion effect and nonlinear interaction occur simultaneously and balance eachother. However, an appropriate initial perturbation or excitation is also an impor-tant factor for the localization of microscopic particle. Proper initial perturbation


will be able to enhance the nonlinear interaction on microscopic particle. Therefore,it is necessary to study the influence of time evolution of the initial excitation onthe localization. Brizhik translated the mechanism of soliton generation in a molec-ular chains into a problem of time evolution of an initial excitation, in the formof an exponential damping in space in accordance with the nonlinear Schrodingerequation (4.40) in which b is now replaced by 2b, with the following initial condition

where a is a constant. The nonlinear Schrodinger equation was integrated by theinverse scattering method by Zakharov and Shabat. The same method can alsobe used to investigate the time evolution of the initial impulses in the form ofrectangular steps. According to this method, the nonlinear Schrodinger equationwith an initial state corresponding to an arbitrary function (p{x', 0) which decreasesrapidly at infinity is consistent with the linear scattering problem for the two-component eigenvector I/J, satisfying (4.42) with eigenvalues £ = £ + ir] and potentialq(x) determined via the initial state (5.83),

q(x')=iVb4>(x',0).

In the region of x' < 0, equation (4.42) has the solution

^ = ( 0 ) *'**' ' ( 5 ' 8 4 )

while in the region of x' > 0, the solution is expressed in terms of the Besselfunctions,

V>x = \AJ-m{y) +BJm{y)}e^-a^'/\

V>2 = -i [AJ-m(y) - BJm-i(tf)] e-(a+ikW2,

where

• -ax' • /26 (a-ik)/2-iC , _ „ .y = jeax, J = y — . m= ^ • (5-85)

The constants of integration are determined by matching the solutions (5.84) -(5.85) at x' = 0.

A Jm-i(j) J-m+i(j)

where

D = Jm(j)J-m+l(j) + J-m(j)Jm-l(j)-

(5.83)


The asymptotic behavior of this solution at x' -> oo is of the form

^ = a(0(J)e-<''+6'(0(J)e^1,

where a(£) and 6(£) are the transmission and reflection coefficients, respectively,given by

where F(m) is the gamma function.According to the inverse scattering method, the bound states or localization of

the microscopic particle corresponds to solitons and are determined by the zeros ofthe transmission coefficient. The velocities of the solitons are proportional to thereal part of the corresponding eigenvalues, vi = —4&, & = 3ftCti and the amplitudes(and widths) are proportional to the imaginary parts, a = 2r}i, Jjj = S&. Prom(5.86) we know that the zeros of the transmission coefficient are determined fromthe equation

Jm-i ( j )=0. (5.87)

According to (5.85), the soliton velocity is proportional to the wave number of theinitial excitation, £ = —fc/2. Thus

m - l = /x= 2 r ? ~ Q . (5.88)a

According to the theorems on the absence of multiple zeros (except for j = 0),and on the alternation of the roots of linearly independent real Bessel functions,(5.87) yields the amplitude of the microscopic particle (soliton), r), as a function ofthe width of the initial impulse a. The lower and upper bounds of the amplitudeof the microscopic particle are given by the Shafheitlin's theorem,

^ - V2ba < T) < ^ + y/2ba.

Because 7? is related to the index of the Bessel function, /z, which in turn dependson the root j according to (5.87), one can get the a dependence of TJ as follows

*L = 1 + bFâlda a 2r) — a

where

From (5.89), Brizhik obtained that at very small values of a the amplitude ofthe microscopic particle increases with a, till the value of ao which is determined

(5.89)

(5.86)


from the transcendental equation

a0F(r},a0)b= '-.

Beyond ao, the derivative in (5.89) changes sign. Therefore, we can conclude from(5.87) and (5.88) that soliton solutions exist in the region j > 1.5, i.e.,

a<aCT = 0.8896. (5.90)

In the interval a[sT < a < aCT, where a1^ = 0.086, a single soliton is generated,

and its amplitude increases from zero to 77 = 0.1766, and eventually reaches themaximum value 77max = 0.1806 at a0 = 0.126 (see Fig. 5.9).

Fig. 5.9 Dependence of soliton amplitude 77 on the inverse width a of the initial condition (6 isthe nonlinearity parameter of the system).

When a has values in the interval a%* < a < a'csT, where 0%* — 3.6 x 10~26,

equation (5.87) admits a two-soliton solution with parameters A*f in < Pi < Mi max >(i = 1,2). In this case, the amplitude of the soliton with larger initial amplitudedecreases with decreasing a, while the amplitude of the second soliton increaseswith decreasing a, but is always smaller than that of the first soliton, 772 < 771. Ifa -C 1, the initial component decays with time into N solitons with small amplitudesr]i/b <t; 1, which approach to zero at a —> 0.

The propagation of microscopic particle (soliton) is accompanied by an oscillat-ing tail which decreases with time, and is determined by the reflection coefficient(5.86).

The above discussions show clearly that for a given initial condition the expo-nentially decreasing initial impulse admits soliton solution, or localization of themicroscopic particle, if the nonlinearity of the system exceeds the critical value,which, according to (5.77), is inversely proportional to the width a of the initialimpulse, 6 > 6cr = 1.125a.

The existence of the critical value of the nonlinearity parameter was shown bynumerical calculations for Davydov's solitons in molecular systems by Hyman et al.


and Scott, and by analytical investigation of the rectangular step evolution (SeeChristiansen and Scott 1990).

5.8 Experimental Verification of Localization of Microscopic Par-ticle

The theoretical prediction on localization of microscopic particles discussed aboveis subject to experimental verification. However, it is almost impossible to ex-perimentally observe the localization of microscopic particles in a material usingavailable instruments. How then is it possible to verify these predictions? It isfortunate that the nonlinear Schrodinger equation describing localization of mi-croscopic particle in nonlinear quantum mechanics can also be used to depict thenonlinear behaviors of condensed matter consisting of a large number of molecules,such as water, and transmission of light in optical fibers. The nonlinear Schrodingerequation describing the self-focus mechanism of microscopic particle in a nonlinearmedium such as an optical fiber has been given in Section 5.5. The fluid dynami-cal form of the nonlinear Schrodinger equation was also given in (3.110) - (3.111).As a matter of fact, a nonpropagating solitary wave in water can be described bya time-independent nonlinear Schrodinger equation. Therefore, water-soliton andlight-soliton (or photon-soliton) can be used to verify the nonlinear behaviors orlocalization of microscopic particles.

Nonpropagating solitary wave was first found in water by Wu et al, in 1984.Subsequently, it was confirmed experimentally by Cui et al.. Larraza et al., Miles,and Pang et al. demonstrated theoretically that this phenomenon can be very welldescribed by a nonlinear Schrodinger equation. Larraza et al. derived that thesurface wave in water troughs satisfies the following nonlinear Schrodinger equation

C 2 ^ + K-W2)<A-vl | ( /O = 0, (5.91)

based on the velocity potential of the fluid which satisfies the Laplace's equationand the corresponding boundary conditions.

For A > 0 and k<f> > 1.022, the soliton solution is of the form

• .yESES^psi^], (,92)

where

A = h* (oT4 - 5T2 + 16 - 9f-2) ,

and d is the depth of the water, k = 7r/b, where b is the width of the water troughs,T — tanh(fcd), to\ = gkT, where g is the acceleration due to gravity, and w is the


frequency of the applied field.A slightly different form was obtained by Miles

B<f>xt + (/? + A\4>\2) 4> + vp = 0, (5.93)

which has the following soliton solution

<t> = eieSechU—XJ, (5.94)

at v = 0, where

B = f + kd ( l - f2) , A' = - (6T4 - 5T2 + 16 - 9T ~2) ,

2 2 2

£ = " 0 ~"1> 7 = — , wi = V/5fctanh(A;d), X = 2\^t kx,

ao and ^ are constants, e is a small, positive scaling parameter.Pang et a/, obtained the following nonlinear Schrodinger equation

^-K2<j>n{x)+i\<t>n\2cl>n=Q, (5.95)

which has the following soliton solution

with

K{x) = BJ™ sech [Vk^(x - xo)] C0Sc^s"(

(^)d)] ™s(kny) (5.96)

where

K2 = k2n + k'l kn^jn, k'n^^n, ( n = l , 2 , 3 - - - ) ,

7 P2(w2-WgrHere WQ = P/p, /3 is surface tension constant, p is the density of the water, a is thenonlinear coefficient of the surface water, Bn is a constant. Prom (5.91) - (5.96),we see that the water-soliton satisfies the time-independent nonlinear Schrodingerequation, and has a bell-shaped form.

Likewise, the discussions in Section 5.5 showed also that the light transmis-sion in optical fibers satisfies the nonlinear Schrodinger equation and light can beself-focused to form a light soliton which was obtained by Hasegawa et al. Thesephenomena can be clearly observed experimentally. In the following, results of thesetwo experiments are briefly presented.


5.8.1 Observation of nonpropagating surface water soliton in wa-ter troughs

The experimental apparatus used by Cui et al. was quite simple. It included aorganic glass trough of 38 cm long and 2.0 - 5.0 cm wide, which is filled with variousliquids to a depth of d = 2.0 - 5.0 cm; a loudspeaker, whose cone is driven by a lowfrequency (7 —15 Hz) signal in the vertical direction and another signal (12 — 25 Hz)in the horizontal direction, respectively; a power amplifier and some measurementinstruments. The water trough was placed on the vibrational platform which wasdriven by the power amplifier. During the initial period when the frequency of thedriving signal was below a certain threshold, only some small ripples were observedand no significant wave generation took place in the surface of the water trough.In this case, the density of water molecules were uniform. However, above thethreshold driving frequency, if a nonlinear initial disturbance was applied to thesurface of water trough, a parametrically excited wave was observed at half thedriven frequency. A large number of water molecules assembled and got togetherto form a bell-shaped nonpropagating solitary wave on the surface of the water inthe trough, due to nonlinear interaction among the water molecules arising fromthe surface tension, as shown in Fig. 5.10. The soliton was the same type as thatof the nonlinear Schrodinger equation in nonlinear quantum mechanics.

Fig. 5.10 The bell-type nonpropagating surface water soliton occurred in the water trough.

An obvious peculiarity in this phenomenon is that the density of water moleculeswhere the soliton occurs is much larger than other places in the water trough, asshown in Fig. 5.11. Therefore, the mechanism forming the water soliton throughgathering of the water molecules can be called self-localization of water moleculesdue to the nonlinear interaction arising from surface tension.

With different liquids and different initial disturbances, one, two, or even moresolitary waves can be established on the water surface. However, the shapes of thesesolitary waves are different in different liquids. If the driving amplitude is furtherincreased, more solitary waves can be formed.

From these experiments, the following conditions for forming solitary wave werederived.


Fig. 5.11 The change of density of water and illustration of direction of motion of water molecules.

(1) The liquid must have a certain depth.(2) The coefficient of surface tension of the liquid must be smaller than a certain

value. This upper bound depends on the characteristics of the liquid.(3) The vertical driving frequency is about twice the intrinsic frequency, and the

horizontal frequency is about the same as the intrinsic frequency. The drivingamplitude must be in a certain range.

(4) There must be a nonlinear initial excitation.(5) The channel should not be too wide.

Another interesting behavior is that the soliton has a fixed position in the xdirection and does not propagate, but it oscillates in the y direction, and the fre-quency of oscillation is about the same as the intrinsic frequency. In the x direction,the profile of the soliton is a bell-shaped curve. In glycerine-water, the soliton curvecan be fitted to the expression: (f> = 1.7sech(a;/1.25) cm. The measured profiles ofsoliton for four other liquids are shown in Fig. 5.12. They are given in the order ofincreasing surface tension. Curve I is that of an ideal fluid whose surface tensionis zero. Curve V is the wave of glycerine-water. It is clear that the amplitude ofthe soliton increases, while its width decreases as the surface tension increases. Ifthe coefficient of surface tension is greater than the upper bound, no soliton can beformed.

Fig. 5.12 Measured profiles of soliton-waves.

Nonlinear Interaction and Localization oj Particles 271

The surface water solitons can move under the effect of external forces, forexample, when they are pushed by a little oar, or when they are blown by wind.If saltwater is added into the channel so as to increase the surface tension, thesoliton moves in the direction of greater surface tension. If the channel is tiltedto a slope of 0.05, the soliton moves toward the shallow side and the height of thesolitary wave is reduced. As soon as the soliton reaches the point where the depthof water is the minimum for forming solitons, the motion of the soliton stops. Itoften occurs that the motion of a soliton can be prevented by another soliton whichis out of phase and has a small amplitude. These dynamic properties of watersoliton are the same as those of classical particles as well as microscopic particles innonlinear quantum mechanics, as described in Sections 4.1 and 4.2. Therefore, theseresults were regarded as an experimental verification of the dynamic properties ofthe microscopic particle in nonlinear quantum mechanics discussed in Chapter 4.

The following two types of interaction or collision between two solitons shouldbe distinguished. (1) Out-of-phase solitons repel each other and do not merge, asshown in Fig. 5.13. (2) Interactions between two solitons that are in-phase show acyclic process of attraction —» merging —> separation —*• re-attraction, as the drivingamplitude is increased, as shown in Fig. 5.14.

These properties of collision between solitons are basically the same as thoseobtained in Sections 4.3 - 4.7 for microscopic particles. Thus the properties ofcollision of microscopic particles in nonlinear quantum mechanics, and the factthat the forms of the microscopic particles can be retained after a collision, areexperimentally confirmed.

Fig. 5.13 Collision between two solitons which are out-of-phase.

In the case of horizontal driving force, the driving frequency required for solitonformation increases when the depth of the liquid increases. If the channel is rotatedhorizontally by upto 40 degrees while the driving signal remains the same, thesoliton still remains at its original place. Similar phenomena can also be observedin non-rectangular channels, for example, in round, ring or trapezoid channels, andV- or X-shaped channels.

Solitons can be easily generated and are very stable if the liquid is magnetized.The collision process between two solitary waves can also be easily observed inmagnetized liquid.


Fig. 5.14 Collision between two solitons which are in-phase.

5.8.2 Experiment on optical solitons in fibers

From earlier discussion, we know that optical solitons generated by nonlinear in-teraction arising from the Kerr effect in optical fibers can be well described by thenonlinear Schrodinger equation. The form and properties of the solitons can beverified experimentally. Thus, we can understand the mechanism of localizationof microscopic particles in nonlinear quantum mechanics by examining the opticalsolitons in details.

In order to experimentally verify the propagation of a soliton in an optical fiber,it is necessary to generate a short optical pulse with sufficiently high power anduse a fiber which has a loss rate less than 1 dB/km, and it is required that thespectral width of the laser is narrower than the inverse of the pulse width in time.In other words, it requires the generation of a pulse with a narrow spectrum. In1980, Mollenauer et al. at the Bell Laboratories succeeded for the first time inexperimentally verifying soliton transmission in an optical fiber. This was achievedby using an F2+ color center laser which is pumped by a Nd:YAG laser. Using afiber with a relatively large cross-section (10~6 cm2) and a length of 700 meters,they transmitted an optical pulse with a 7 ps width and measured the shape of theoutput pulse by means of autocorrelation. Their experimental results are shown inFig. 5.15. The pulse shape is measured by autocorrelation at the output side ofthe fiber, for different input power levels. It can be seen clearly from Fig. 5.15 thatwhile the output pulse width increases for a power below the threshold of 1.2 W,it continuously decreases for an input power above 1.2 W. The appearance of twopeaks in the case of an input power of 11.4 W is a result of phase interference of threesolitons, which are produced simultaneously in such a case. This is consistent withresults of numerical calculation obtained by Hasegawa et al.. The periodic behaviorof the higher order solitons was confirmed by Stolen et al. in a later experiment.Most experimental observations of light solitons were achieved by using the color


center laser because of the need to produce the Fourier transform limited pulse.However, light solitons have also been successfully observed using other types oflasers, such as the dye laser or laser diodes. But it is necessary to control frequencychirping and to have a narrow spectrum for a well-defined soliton.

Fig. 5.15 Experimental observation of soliton formation by Mollenauer et al.

The interaction between optical solitons has been experimentally observed byMitschke and Mollenauer in 1987. Both the repulsion and the attraction regimeshave been observed, depending on the soliton phase difference. In this experiment,solitons were generated using a soliton laser. The pulse first passed through aMichelson interferometer, giving a pair of pulses. The length of one arm of theinterferometer permitted a change in the distance between the pulses from zero upto several picoseconds. The soliton phase difference was also measured. In thisexperiment, they used a polarization preserving 340 m low-loss (> 0.3 dB/km)fiber with D = 14.5 ps/(nm-km) at an operating wavelength of 1.52 /irn. The pulseduration was sa 1 ps. Therefore the fiber length is 10 soliton periods. Such along fiber enables the researchers to clearly observe interaction between two well-separated soliton pulses. The results of the measurements are shown in Fig. 5.16 forboth the repulsive and attraction regimes (See Abdullaev et al. 1994). As it can beseen, the observed result is different from that predicted by theory of optical solitonin the case of attractive interaction and in the region where there is a small initialdistance between the pulses. According to the theoretical calculation discussed inChapter 4, two interacting microscopic particles (solitons) pass through each other.This is denoted in Fig. 5.16 by the oscillating structure of the theoretical curve. Theexperimental data deviate somewhat from such a feature. In the unstable regionthe attractive force becomes repulsive, as a result of the influence of the Ramanself-frequency shift.

Mollenauer and Smith (1989) discovered that between optical solitons separatedat long distances (more than 1000 km), there exists a long-range phase-independentinteraction. Dianov et al. (1990) showed that the electrostrictional mechanism maybe responsible for the observed anomalous interaction. Reynaud and Barthelemy(1990) observed interaction of spatial solitons between two soliton beams. Theseresults demonstrated the existence of a mutual interaction between two close propa-gating soliton beams which depends on the relative phase and the distance between


Fig. 5.16 Pulse separation at output (<rOut) vs. the pulse separation at input (cin) i n a 340 mfiber. Solid curve: theory; dashed line: experiment. Without interaction, all points would fall onthe 45° line (Courtesy of Mitschke and Molenauer, 1987)

the beams. (See Abdullabev et al. 1994)In addition, Vysloukh and Cherednik (1986) developed an effective method

based on a combination of the inverse scattering theory and numerical methods todescribe soliton interaction, and to follow the evolution of an arbitrary pulse. Themethod is useful for studying dynamics of multi-soliton states, and for studyingthe interaction of optical solitons in nearly integrable systems. These experimen-tal results of soliton collisions are basically consistent with that of the microscopicparticles described by nonlinear Schrodinger equation in Chapter 4.

Prom Fig. 5.15, we see that a bell-type soliton, which is the same as that inFig. 4.1, can exist in optical fibers. This exhibits clearly that light wave with dis-persive feature can be self-localized or self-focused due to the nonlinear interactionto form a stable soliton which is described by the nonlinear Schrodinger equation.This makes us believe that microscopic particles in nonlinear quantum mechanicscan be self-localized or self-trapped as solitons with wave-corpuscle duality undernonlinear interactions and the theories discussed in Chapters 4 and 5 give a appro-priate description of the microscopic particles in nonlinear quantum mechanics.

Bibliography

Abdullaev, F., Darmanyan, S. and Khabibullaev, P. (1994). Optical solitons, Springer,Berlin.

Barut, A. D. (1978). Nonlinear equations in physics and mathematics, D. Reidel Publish-ing, Dordrecht.

Beehgaard, K. and Jerome, D. (1982). Sci. Am. 247 50.Beehgaard, K., Jerome, D. (1981). J. Am. Chem. Soc. 103 2440.Borken, J. D. and Drell, S. D. (1964). Relativistic quantum mechanics, McGraw-Hill, New


York.Borken, J. D. and Drell, S. D. (1965). Relativistic quantum fields, McGraw-Hill, New York.Brizhik, L. S. (1993). Phys. Rev. B 48 3142.Brizhik, L. S. and Davydov, A. S. (1983). Phys. Stat. Sol. (b) 115 615.Burger, S.,et al. (1999). Phys. Rev. Lett. 83 5198.Burt, P. B. (1974). Phys. Rev. Lett. 32 1080.Burt, P. B. (1977). Lett. Nuovo. Cimento 18 547.Burt, P. B. (1978). Proc. R. Soc. London A359 479.Burt, P. B. (1979). Phys. Lett. A 71 19; Phys. Lett. B 82 423.Burt, P. B. (1981). Quantum mechanics and nonlinear waves, Harwood Academic Pub-

lishers, New York.Christiansen, P. L. and Scott, A. C. (1990). Self-trapping of vibrational energy, Plenum

Press, New York.Cui, Hong-nong,et al. (1988). J. Hydrodynamics 3 43.Cui, Hong-nong, Yang Xue-qun, Pang Xiao-feng and Xiang Longwan, (1991). J. Hydro-

dynamics 6 18.Davidson, R. C. (1972). Methods in nonlinear plasma theory, Academic Press, New York.Davydov, A. S. (1979). Phys. Scr. 20 387.Davydov, A. S. (1980). Phys. Stat. Sol. (b) 102 275.Davydov, A. S. (1985). Solitons in molecular systems, D. Reidel Publishing, Dordrecht.Davydov, A. S. and Kislukha, N. I. (1973). Phys. Stat. Sol. (b) 59 465.Davydov, A. S. and Kislukha, N. I. (1976). Phys. Stat. Sol. (b) 75 735.Davydov, A. S. and Pestryakov, G. M. (1981). Excitation states of the field with inertial

self-action, preprint TTP-112R, Inst. Theor. Phys. Kiev (in Russian).Dianov, E. M., Mamyshev, P. V., Prokhorov, A. M., and Chernikov, S. V. (1990). Opt.

Lett. 14 18.Dianov, E. M., Luchnikov, A. V., Pilipetskii, A. N., and Starodumov, A. N. (1990). Opt.

Lett. 15 314.Erdely, A. (1953). Bateman manuscript project, McGraw-Hill, New York.Fermi, E. (1951). Elementary particles, Yale, New Haven.Feynman, R. P. and Hibbs, A. R. (1965). Quantum mechanics and path integrals, McGraw-

Hill, New York.Frohlich, H. (1954). Adv. Phys. 3 325.Frohlich, H. (1954). Proc. Roy. Soc. London A 223 296.Guo, Bai-lin and Pang Xiao-feng, (1987). Solitons, Chin. Science Press, Beijing.Hasegawa, A. (1983). Opt. Lett. 8 342.Hasegawa, A. (1984). Opt. Lett. 9 288; Appl. Opt. 23 3302.Hasegawa, A. (1989). Optical solitons in fiber, Berlin, Springer.Heisenberg, W. (1966). Introduction to the unified field theory of elementary particles,

Interscience, London.Holstein, T. (1959). Ann. Phys. 8 325 and 343.Hyman, J. M., Mclaughlin, D. W. and Scott, A. C. (1981). Physica D3 23.Keller, H. J. (1974). Low dimensional conductive phenomena, Plenum Press, New York.Landau, L. D. (1933). Phys. Z. Sowjetunion 3 664.Landau, L. D. and Lifshitz, E. M. (1987). Quantum mechanics, Pergamon Press, Oxford.Larraza A. and Putterman, S. (1984). J. Fluid. Mech. 148 443; Phys. Lett. A 103 15.Miles, J. W. (1984). J. Fluid Mech. 148 451.Mitschke, F. M. and Mollenauer, L. F. (1986). Opt. Lett. 11 659.Mitschke, F. M. and Mollenauer, L. F. (1987). Opt. Lett. 12 355.Mittal, R. and Howard, I. A. (1999). Physica D 125 79.


Mollenauer, L. F. and K. Smith (1988). Opt. Lett. 13 675.Mollenauer, L. F. and K. Smith (1989). Opt. Lett. 14 1284.Mollenauer, L. F., Gordon, J. P. and Islam, M. N. (1986). IEEE. J. Quantum Electron.

22 157.Mollenauer, L. F., Stolen, R. H. and Islam, M. N. (1986). Opt. Lett. 10 229.Mollenuaer, L. F.,et al. (1979). Opt. Lett. 4 247.Mollenuaer, L. F., Stolen, R. H. and Gordon, J. P. (1980). Phys. Rev. Lett. 45 1095.Pang, Xiao-feng (1994). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing, 1994.Pang, Xiao-feng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu.Pang, Xiao-feng,et al. (1984). Proc. ICNP, Shanghai, p. 168.Pekar, S. (1946). J. Phys. USSR. 10 341 and 347.Reynaud, F. and Barthelemy, A. (1990). Europhysics Lett. 12 401.Schuster, H. (1975). One-dimensional conductors, Springer, Berlin.Scott, A. C. (1984). Phys. Scr. 29 284.Stolen, R. H. and Lin, C. (1978). Phys. Rev. A 17 1448.Sulem, C and Sulem, P. L. (1999). The nonlinear Schrodinger equation: self-focusing and

wave collapse, Springer-Verlag, Berlin.Toda, M. (1989). Nonlinear waves and solitons, KTK Scientific, Tokyo; Kluwer, Dordrecht.Vysloukh, V. A. and Cherednik, I. V. (1986). Dokl. Akad. Nauk SSSR 289 336, (in Rus-

sian).Vysloukh, V. A., Ivanov, A. V. and Cherednik, I. V. (1987). Izv. Vysch. Uchebn. Zaved.

Radiofiz 30 980.Whitham, G. B. (1975). Linear and nonlinear waves, Wiley, New York.Whitham, G. B. (1979). J. Phys. A 12 LI.Wu, R. J., Keolian, R. and Rudnik, I. (1984). Phys. Rev. Lett. 52 1421.

Chapter 6

Nonlinear versus Linear QuantumMechanics

In this chapter we discuss the relations between the nonlinear quantum mechanicsand the linear quantum mechanics in relation to the self-consistency of the nonlinearquantum mechanics, the differences of the uncertainty relation and causality inthe two quantum mechanical theories, and the method of calculating eigenenergyspectra of the Hamiltonian in the nonlinear quantum mechanics, which are alsodifferent from that in the linear quantum mechanics. The calculated eigenenergyspectra can be compared with experimental data. Incorporating results obtainedin Chapters 2, 4 and 5, we can conclude that the nonlinear quantum mechanicaltheory presented in Chapter 3 is a correct, complete and self-consistent theory.

The validity of the theory means that it has solid theoretical and experimentalfoundation, as mentioned in Chapter 2, that the laws of conservation and invari-ance in the nonlinear quantum mechanics are consistent with natural laws, thatthe motion of particles in the nonlinear quantum mechanics can be simulated andverified by experiments, and the energy spectra obtained in the nonlinear quantummechanics can be compared with experimental data. The completeness means thatit can completely describe the properties of the microscopic particle including itswave-corpuscle duality in nonlinear systems. Finally, self-consistency means thatit is compatible and correspondent with the linear quantum mechanics and it isitself self-compatible. These are necessary conditions for the nonlinear quantummechanics to be a correct theory.

6.1 Nonlinear Quantum Mechanics: An Inevitable Result of De-velopment of Quantum Mechanics

As far as the validity of the nonlinear quantum mechanics is concerned, it can bedemonstrated from the following four aspects.

(1) The nonlinear quantum mechanics proposed here is a complete and entiretheory because it can give the properties and states of microscopic particles in dif-ferent systems and under various conditions, including their wave-corpuscle duality,which was confirmed by the experiments discussed in earlier chapters, and the linear

277


quantum mechanics failed to explain.(2) The nonlinear quantum mechanics can give the universal laws of motion of

matters in nature, for example, conservation laws of the mass, energy and momen-tum, as mentioned in Section 4.1. This shows that the nonlinear quantum mechanicscan correctly describe motion of microscopic matters.

(3) The nonlinear quantum mechanics is an inevitable result of the developmentof the linear quantum mechanics in nonlinear systems. It is consistent with thelinear quantum mechanics.

(4) The nonlinear quantum mechanics is self-consistent or self-correspondent.In this and the following sections we will discuss the last two aspects.To show that the nonlinear quantum mechanics is a necessary result of develop-

ment of the linear quantum mechanics in nonlinear systems, we focus our discussionto the evolutions of dynamic equations, the Schrodinger equation, from linear tononlinear cases in quantum mechanics. We know now that the nonlinear Schrodingerequation and the linear Schrodinger equation, are fundamental equations in the non-linear quantum mechanics and linear quantum mechanics, respectively. The linearSchrodinger equation describes the motion of microscopic particles in linear sys-tems. In these systems, there are no nonlinear interactions or very small nonlinearinteractions. In such a case, microscopic particles move in the form of waves in thewhole space-time due to the dispersion effect of the media. According to the theoryof nonlinear interactions described in Chapter 4, these linear quantum mechanicalsystems can only be simple atoms and molecules consisting of a few elementary par-ticles such as the hydrogen atom, or hydrogen molecule, or the helium atom, andso on. In these simple systems there are no mechanisms that generate nonlinearinteractions. In other words, they have only very small self-interaction. In fact,we confirmed that the nonlinear two-body and three-body self-interaction energyin these systems is small, compared to its dispersive energy, through calculation ofthe corresponding Feynmann diagrams using the method given in Section 5.3. Sucha small nonlinear interaction cannot balance and cancel the dispersion effect of themicroscopic particle. Thus the microscopic particle cannot be localized, and it canonly has wave or dispersive feature. The linear quantum mechanics is then sufficientto describe the states of motion of the particles. As a matter of fact, linear quantummechanics indeed gives very accurate description for these systems, and the resultscompare well with the experimental data, as mentioned in Chapter 1. Therefore,the linear quantum mechanics is appropriate for describing these systems, and itwould be unnecessary to use the nonlinear quantum mechanics in such cases.

However, for systems consisting of many bodies or many particles and othercomplex systems in which there are significant nonlinear interactions generated byself-interaction, self-focusing, self-trapping, self-condensation and self-localization,etc., these nonlinear interactions cannot be neglected. These nonlinear interactioncan be so large that it can balance and suppress the dispersion effect of these media.In such a case, the nature of the microscopic particles changes. It becomes a soliton

Nonlinear versus Linear Quantum Mechanics 279

with the wave-corpuscle duality. Then, the microscopic particle must be describedby the nonlinear quantum mechanics. This is a very natural transition for themotion of the particle.

Vice versa. For nonlinear quantum systems depicted by the nonlinear quantummechanics, if the nonlinear interactions are very small relative to the dispersioneffect, we may use the linear quantum mechanics to describe them. But strictlyspeaking, all physical systems have nonlinear interactions to a certain extent. There-fore, the nonlinear quantum mechanics is an more general theory, and the linearquantum mechanics can be considered as a special case or an approximation of thenonlinear quantum mechanics in the case of weak nonlinear interactions. We cantherefore say that the relation between the nonlinear quantum mechanics and thelinear quantum mechanics resembles that between Einstein theory of relativity andNewton's classical mechanics. Furthermore, the dynamic equations, including thenonlinear Schrodinger equation and the nonlinear Klein-Gordon equation, can beobtained by adding nonlinear interactions to the corresponding dynamic equations,the linear Schrodinger equation or the linear Klein-Gordon equation, in the linearquantum mechanics. This correspondence relation advances also the consistencybetween the nonlinear quantum mechanics and the linear quantum mechanics. It istherefore clear that the nonlinear quantum mechanics is consistent with the linearquantum mechanics.

If we chose other approaches or other dynamic equations to establish the nonlin-ear quantum mechanics, then we could not obtain the consistency and compatibilitybetween the nonlinear and the linear quantum mechanics. Therefore, the presentnonlinear quantum mechanics is a correct theory and it is a necessary result of thedevelopment of the linear quantum mechanics.

However, we must remember that solutions of dynamic equations in nonlinearquantum mechanics in the limit of weak nonlinear interaction are not exactly thesolutions of the dynamic equations in the linear quantum mechanics. To see thisclearly, we first examine the velocity of the periphery of the soliton given in (4.8),which is rewritten as

<p(x',t') = 2V2fc sech [>/2Jb (a/ - « e 0] eiv'^'-v^l2, (6.1)

for 6 = 1 , V(f,t') = A(<f>) - 0 and x1 = y/2m/h2x, t' = t/H in the nonlinearSchrodinger equation (3.2). In (6.1), ve and vc are the group and the phase veloc-ities, respectively. As it is known, the nonlinear term in (3.2) sharpens the peak,while its dispersion term has the tendency to broaden it. Thus for weak nonlinearinteraction and small periphery <j>(x' ,t'), it may be approximated by (for x > vet).

4> = 2s/2ke-^k{x'-v*t">eiv'{x'-v<t')/2. (6.2)

The small term \<t>\24> in (3.2) can be approximated by

i<pf + 4>x'x" « 0. (6.3)


Substituting (6.2) into (6.3) we get

ve sa 2V2k, (6.4)

which is the group speed of the soliton. (Near the top of the peak, we must takeboth the nonlinear and dispersion terms into account because their contributionsare of the same order. The result is the group speed.) Here we have only consideredthe region where (j>(x,t) is small, that is, when a soliton is approximated by (6.2),it satisfies the approximate wave equation (6.3) with ve « 1\flk.

However, if (6.3) is treated as an linear equation, its solution is of the form

•<f>'(x,t)=Aeilkx-utK (6.5)

We now have u> = k2, which gives the phase velocity u>/k as vc = k and thegroup speed du/dk — vgr = k. Apparently this is different from ve = 2^/2k.This is because the solution (6.2) is essentially different from (6.5). Therefore, thesolution (6.5) is not the solution of the nonlinear Schrodinger equation (3.2) withV(x) = 0 and A{cf>) = 0 in the case of weak nonlinear interactions. Solution (6.2) isa "divergent solution" {(f>{x,t) —> oo at x —> — oo), which is not an "ordinary planewave". The concept of group speed does not apply to a divergent wave. Thus we cansay that the soliton is made from a divergent solution, which is abandoned in linearwaves. The divergence is development by the nonlinear term to yield waves of finiteamplitude. When the nonlinear term is very weak, the soliton will be diverge, andsuppression of divergency will result in no soliton. These circumstances are clearlyseen from the following soliton solution in the case of the nonlinear coefficient b ^ 1

<f>(x, t) = 2J^k sech \V2k(x' - vet1)] e<».(*'-^')/2. (6.6)

If the nonlinear term approaches zero (b —> 0), the solitary wave diverges (<f)(x, t) —>oo). If we want to suppress the divergence, then we have to set k = 0. In such a case,we get (6.5) from (6.6). This illustrates that the nonlinear Schrodinger equation ornonlinear quantum mechanics can reduce to the linear Schrodinger equation or linearquantum mechanics if and only if the nonlinear interaction and the group speed ofthe soliton are zero. Therefore we can conclude that the solutions of the nonlinearquantum mechanical dynamic equations in the weak nonlinear interaction limit isnot the same as that in the linear quantum mechanics, and only if the nonlinearinteraction is zero, the nonlinear quantum mechanics reduces to the linear quantummechanics.

However, real physical systems or materials are made up of a great numberof microscopic particles, and nonlinear interactions always exist extensively in thesystems or among the particles. The nonlinear interactions arise from interactionsamong the microscopic particles or between the microscopic particles and the envi-ronment. Therefore, the nonlinear quantum mechanics should be the correct andmore appropriate theory for real systems. It should be used often and extensively,


even in weak nonlinear interaction cases. The linear quantum mechanics, on theother hand, is an approximation to the more general nonlinear quantum theory andcan be used to study motions of microscopic particles in systems in which thereexists only very weak and negligible nonlinear interaction.

6.2 Relativistic Theory and Self-consistency of Nonlinear Quan-tum Mechanics

As we know, there are nonrelativistic and relativistic theories in the linear quantummechanics. In the relativity theory the dynamic equations are the linear Klein-Gordon equation and Daric equation, which can reduce to the linear Schrodingerequation in nonrelativistic case. Therefore, the linear quantum mechanics is itselfself-consistent. In the nonlinear quantum mechanics there are also relativistic andnonrelativistic theories. The nonlinear Klein-Gordon equation in (3.3) and (3.4) isthe dynamic equation of microscopic particles in the relativity case in the nonlinearquantum mechanics, which have been well studied by many scientists and have beenextensively used to test various rules concerning motions of elementary particles innonlinear fields and nonlinear systems, in particle physics and condensed matterphysics. It is also a fundamental equation in the nonlinear field theory. Thisequation can reduce to the nonlinear Schrodinger equation in the nonrelativisticcase, i.e., there is also a consistency between the nonlinear Schrodinger equationand nonlinear Klein-Gordon equation, which will be the topic of discussion in thissection.

The nonlinear Klein-Gordon equation in the relativistic case in the nonlinearquantum mechanics can be, in general, represented as

<kt-d>zx = ~ = -U'(<t>), (6.7)

and the corresponding Hamiltonian is given by

H = Jdz^4t-l4>l+U(4>)\, (6.8)

where F(<f>) is a field force, U{cj>) is the potential of an applied field which is anonlinear function of the wave function or the field <j>. The nonlinear interaction isgenerated by self-interactions between the microscopic particles which are describedin Section 5.3. The natural unit system is used here. The solution of (6.7), as afunction of £ = x — vt, can be obtained by integration,


where

_ 1

Under the "zero derivative" boundary condition

i.e., C = 0, the solution becomes

The exact solution can be obtained for a given potential and it depends only on£ = x — vt. Its dependence on v follows from the covariant form of the originalequation (6.7). The energy associated with a single soliton is easily obtained,

Ek(v) = y/El(o)+P* = >yEk(o),

where P = JMI-V and

Ek(o) = Mk = 2 [if [MO]dx = V2 f * ^\U\dc/>J J$x

is the soliton energy and mass in its rest frame, <pi and <f>2 are minima of thelocalized potential. The + and the — signs in the above equation correspond to thekink (d<j>/d£ > 0) and antikink (d<f>/d£ < 0) soliton solutions, respectively, and allowone to define the topological charge N = N+ — N-, i.e., the difference between thekink and antikink numbers.

There are many forms of the nonlinear Klein-Gordon equation. For example,if U'{cj>) = sin</>, it becomes the Sine-Gordon equation. U'(<j>) — sin<j> + A sin 2^,gives the dual Sine-Gordon equation. If U'(<f>) = <j> — <j>3, it becomes the ^-fieldequation. If U'(4>) = —(j) + <j>z, it is the <j>t. -field equation. In practice, the <j>4 -fieldis a special case of the Sine-Gordon equation because sin< > « <j> — (/>3/3! -1- • • •. Inthe relativistic nonlinear quantum mechanics we concentrate mainly on the Sine-Gordon equation and the (/>4-field equation. These are fundamental equations innonlinear field theory in particle physics and condensed physics and they have wideapplications. However, when U'{4>) = <j>, (6.7) reduces to fat - (j>xx = —, which isan equation of motion of the filed for a free Boson with a unit mass, i.e., it is thelinear Klein-Gordon equation in the linear quantum mechanics, and its solution isa plane wave. Therefore, the nonlinear Klein-Gordon equation (6.7) is a naturalgeneralization of the linear Klein-Gordon equation in the nonlinear case. Thus, thisrelativistic theory in the nonlinear quantum mechanics is a necessary result of thedevelopment of that in the linear quantum mechanics.


6.2.1 Bound state and Lorentz relations

We can obtain the solutions of the nonlinear Klein-Gordon equation. For example,for the </>i -field equation,

4>tt ~ 4>xx - + <f>3 = 0 , (6.9)

its kink soliton solution is given by

4>{x, t) = ± tanh (x~^f) , (6.10)\ 2v l -v2j

where the -I- (-) sign is for the positive (negative) kink.For the $5_-field equation,

4>tt -4>xx + <t>-<l>3=o, (6.H)

its solution is of the form

<j>(x,t)=<posech(X-J?Lllty (6.12)

For the Sine-Gordon equation

<kt - <l>xx + m2 s i n </> = 0, (6.13)

its solution is given in (2.106), and is written down here for convenience,

J>(x,t) = ±4tan-1 | ± [exp ( i m 1 ^ ) ] } , (6.14)

where the + (—) sign is for the positive (negative) kink, x(t) = xo+vt is the positionof the kink at time t. The width of the running kink is \ / l - v2/m, i.e., y/1 - v2

times the static kink width. In (6.14), m is the mass of the microscopic particle.We can show the dependence of 4>{x,t) on m, (6.14), using a graph, see clearlythat the region of variation of (j>{x, i) is centered at the point x(t) = q = XQ + vt.Furthermore, the "length" of the variational region decreases as m increases. Thus,if the function <p(x, t) is used to denote a localized disturbance, then the quantityqa — Xo + vt and m~1 can be considered as its "position" and "width", respectively.The function (^{x, t) in (6.14) is the static solution of the Sine-Gordon equationwith mass m at t = 0. Because there is a Lorentz factor j = Vl — v2 in (6.14) andthe position of the microscopic particle q(t) = xo+vt is moving with a linear velocityv, we have a "Lorentz" contraction for the length of the microscopic particle whenit moves with velocity v. This is exactly analogous to the motion of a relativisticparticle.

The analogy between the kink (+<f)(x,t)) and a positively charged particle, orbetween the antikink (-(f>(x,t)) and a negatively charged particle is also readilyexhibited for the Sine-Gordon equation. In Fig. 6.1, we show results obtained byDodd et al. for the time evolution of an initial state consisting of two static kinks


Fig. 6.1 Interaction between kinks: Two iV = 1 kinks are initially at rest at x = — q and x = qrespectively. The dotted line is <£(z,t) for t > 0 but small, showing that the kinks repel each other.

localized at x = +q and — q, respectively. The boundary conditions, <f>(t, -co) =4>{t, +oo) = 0, or 2TT, were used. We see that the kinks move away from each other.Fig. 6.2 shows the time evolution of an initial state consisting of a kink at x = +qand an antikink at x = —q. In this case, the two disturbances move toward eachother. This is an attractive effect for the kink-antikink pair. This diagram makes itintuitively clear why this must happen. It arises as a direct consequence of boundaryconditions. For two like-kinks the solutions of the Sine-Gordon equation whichevolves out of the initial configuration has to change by 2n over a distance. Thisresults in high field gradients and consequently large contributions to the energy.The system moves so as to reduce its energy and as a result the kinks separate. Forthe solution which develops from the initial condition consisting of two kinks withopposite helicities, no such change is required and the energy is lower if the kinksare nearer. Hence, two dissimilar kinks approach each other. These properties ofmicroscopic particles described by the Sine-Gordon equation show that microscopicparticles have corpuscle feature.

Fig. 6.2 Same as in Fig. 6.1, except that now we have an antikink at x = q (N2 = —1). Here<t>(x, t) shows that a kink and an antikink attract each other.

The energy of the running kink is given by

E=r<j>Vx+^+4m2 sin2 ( 0 1 d x = v B ^ • (6-i5)The rest mass of the kink is a constant, given by

M= f \<l>\2dx = 8m.J-00

These also show the corpuscle feature of microscopic particles.


Clearly, there is an important factor 7 = y/l - v2 in (6.9) - (6.15). This is theLorentz factor, or the Lorentz contraction coefficient. It's appearance in the aboveequations represents the relativistic invariance of the nonlinear quantum mechanicaltheory which exhibits the right relativistic dependence on the speed v of the micro-scopic particle (kink). Therefore, the model described above has Lorentz invariance.We can write down its Lagrangian, Hamiltonian and momentum as follows,

L = JdxC = J d x ^ ( # - <&) - m2(l - cos</>)] , (6.16)

H= IdxU = fdx\^cl)l + m2{l-cos(f>)\ , (6.17)

P = [ dxfafa. (6.18)

Applying the Lorentz transformation,

we can get

(7)-^(i7)tf)-This shows the invariance of the momentum and Hamiltonian of the microscopicparticle under the Lorentz transformation. The >4-fieId equation also has Lorentzinvariance. These are just the Lorentz or relativistic invariance for the nonlinearfield theory. Therefore, the above theory or equations are actually appropriate todescribe the motion of the microscopic particle in relativistic case in the nonlinearquantum mechanics.

The Sine-Gordon equation satisfies the following Heisenberg equation,

, ,. „, [fdfadH dcj>tdH\

and the corresponding conservation laws are

jtCn(4>) + yxf{4>) = 0, (6.22)

where

C1=H-P = j4%dx,

Cz = I {<P\ - Ml*) dx,

C5 = I (4>l - 20<t>l<j>2xx + B<f>2

xxx) dx.

These problems were studied by Fogel, Olsen, Rubisten and Finkelstein, and others.

(6.19)

(6.20)

(6.21)


6.2.2 Interaction between microscopic particles in relativistic the-ory

Interaction between microscopic particles in relativistic theory is a very interestingproblem. As a matter of fact, collision of microscopic particles described by theSine-Gordon equation or the $+-field equation also has similar properties as those ofmicroscopic particles described by the nonlinear Schrodinger equation as mentionedin Chapter 4. Interaction between two microscopic particles in relativistic theorywill be discussed here, following the approach of Rubinstein.

For short periods of time the deformation of several interacting microscopicparticles (kinks) will be small. Rubinstein looked for solutions of the form

n

4>{x, t) = <H{x, t)+Y, Mx, *), (6-23)i= l

with

^ , t ) = 4tan-êxp[^(a-i^-^0))]|> ft = ±1, (6.24)

and (f>'(x,t) <C 1. In fact, with

Vi =%<!>'(x,0) = 4>'(x,0)=0, (6.25)

such a solution can determine the direction and strength of the force acting oneach kink at rest at t = 0. For simplicity we restrict ourselves t o n = 2, and let—x\ = x2 = q. Substituting (6.23) into the Sine-Gordon equation (6.13), Rubinsteinobtained, at t = 0, that

•\%4>'{x,0) = V s i n & - sin ( $ > i )

_ J>\ sinh[m(a: + q)] + 82 sinh[w(a: - q)]cosh2[m(x + q)] cosh2[m(a; — q)]

In Figs. 6.1 and 6.2, we plotted (j)(x,0) = <f>i(x,0) + 4>2{x,0) using solid line and<j>(x,t) w (j>{x,Q) + d?4>'(x, 0)£2/2 (t is small) using dotted line. The figures showthat the kinks repel if 5\ = 82 and attract if 8\ = —82 (the cases 81 = 82 = — 1 and—Si = 82 = +1 are represented in Figs. 6.1 and 6.2 by 4> -y —(/>). If q » 1/m andx ~ q, Rubinstein obtained, from (6.26), the change in fa as

1 ^ ( 1 , 0 ) - - ^ i s i n h M s + g)] _, -8fte-2m« ( g 2 ? )

m? t ' cosh2[m(x + q)] cosh2 [m(x — q)] cosh2[m(x — q)\

From (6.24), the change in ^2(^,0) generated by a small change in q, i.e., a rigiddisplacement of the kink is

d<j>2{x,0) f -282mo<P2 = ^ "1 = Tr—7 vf"9, (6.28)

dq cosh[m(a; - g)J

(6.26)


and, if Sq changes in time,

dHSfa) = ~26r ,M (6.29)cosh[m(a; - q)}

Comparing (6.29) and (6.27) at x ~ q, we see that due to the interaction themicroscopic particles tend to contract and that the force acting on <fo is

(Sm)(dq) ~32m2(51J2e-m(29). (6.30)

This formula approximates the classical law of motion of the particle. It showsthat microscopic particles described by the Sine-Gordon equation obviously havecorpuscle feature.

Two exact solutions of (6.13) were obtained by Perring and Skyrme. Lettingm = 1 and 4> + 2v = <p", the solutions for interaction between the two kinks are

tan_x r cos j^ iL v sinh(7x) J

ro "1—> 4 t a n " 1 - e 7 1 co sh (7 ^ ) (6.31)

x->-oo \_v J

—¥ 4 tan"1 -e^x-vtA

t->-oo v J

—> 4 tan"1 rie-r(-+-*)l ;t-»oo [v J

C = 4tan-1[=^l+27r[ cosh(7^t) J

—• 4 tan"1 [ V(f* 1 + 2TT (6.32)s->oo [2cosh(7Di)J

—> 4 tan"1 [ve7(a!+l'*)l +2n

—> 4 tan"1 [ t /e7^-^] + 2TT;

0 » = 4 t a n - 1 f ^ ^ l[ sinh(7i;t) J

—> 4 tan"1 [2vcosh(7a;)e7''t] (6.33)

—> 4 tan"1 \ve-^x~vtAX—> — OO L J

—> 4tan"1 \ve<(x+vtA ;I-+O0 L J


4>'j = 4 tan"1 - y - Z + 2TT[ cosh^z) J

F elvt 1—• 4 tan"1 — — + 2TT (6.34)t->oo L2ticosh(7a;)J

—• 4 tan"1 [ i e ^+^) l + 27r

—> 4 tan"1 fle-Tt*-"*)] + 2TT.z->oo [i>

Except for the unimportant factors l/v and v (which can be absorbed in the phase),(6.31) and (6.32) show that (/>" represents two 6=1 kinks approaching the originwith speeds ±v for t —> —oo and moving away from it for t —t +oo. If v is smallenough (< 0.05), </>" departs little from a linear combination of two kinks, with thedistance of the closest approach larger than twice the kink size, so that </>" can berecognized as the bouncing of two kinks. For larger v's, <f>" is strongly distortedand such an interpretation is impossible.

Similarly, </>'b' represents, at t = —oo, a S = — 1 kink approaching the origin withspeed v from x = —oo and a 6 = 1 kink coming from x = oo, with speed —v. Theypass through each other, and, at t = oo, <f>-i is at x = oo and </>+i at x = — oo,4>'l(x, +oo) has the form of <j>{x,Q) in Fig. 6.2.

The above discussions show that the interaction or collision of microscopic par-ticles described by the Sine-Gordon equation is the same as that described by thenonlinear Schrodinger equation, i.e., the form and energy of the microscopic parti-cles can be retained after the collision. Considering also results of the Sine-Gordonequation discussed in Chapter 4, we can conclude that microscopic particles in therelativistic theory also have the wave-corpuscle duality.

6.2.3 Relativistic dynamic equations in the nonrelativistic limit

In the following, we show that the dynamic equations in the relativistic theory canreduce to the nonlinear Schrodinger equation in the nonrelativistic limit. We startfrom the following 04-field equation

with the boundary conditions, (y) = Asin(Ky) + Bcos(Ky), (6.37)

(6.35)


where K2 = ui2 — k2 - m2. Applying the boundary conditions, we get

5 = 0, K2=(™)\ (n = l,2,3---).

Thus there are many modes of motion given by the dispersion relation ui(k, n) =0 in such a case. For given k and n, OJ can be determined. We now consider thenonrelativistic case of slow changes or small velocity in (6.35). Thus we can assumethat £ = E{X — vt), T — e3t in such a case, where e is a small factor. Let

9 9 d 8 d 3 d d /ono.

Y^d-x+£dl' di-"di + £^-£vdi- (6-38)

Then the solution of (6.35) becomes

oo

0 =£>*</><">, (6.39)n=l

where

f=-oo

From the dispersion relation we get

Substituting (6.39) into (6.35), the equation in e is of the form

<t>^=ip(^r)sm(Ky), ^ = 0 , (\i\ ± 1). (6.41)

The equation in e2 is

o ,(1) 32^(2)

2it{k -vu)^ + d-j±- + {£2u2 - fk2 - m2) 0 f = 0. (6.42)

Because v = k/u, the first term in (6.42) is zero. Thus we get <frp = 0 for \l\ 1.The equation in e3 now becomes

^ j f - + K2<f>f + [2 iu;^ + (1 - v2) 0 ] 8in(Ky) - 3k\<p\2tp sm(Ky) = 0. (6.43)

Multiplying (6.43) from left by sin(Ky) and integrating again over y from 0 toa, then the first and the second terms become zero in (6.43) according to matchconditions of functions. Finally we get

U2 - k2 &<p U . ,2 n . A ^

^-~^^w~^l<plip=o- (6-44)This is a nonlinear Schrodinger equation. Therefore, the relativistic equation ofmotion reduces to the nonlinear Schrodinger equation in the nonrelativistic case.

(6.40)


This again shows that the nonlinear quantum mechanics established in Chapter 3is self-consistent.

For the Sine-Gordon equation, 4>xx — cf>tt = sin 0, we can also derive the nonlinearSchrodinger equation in nonrelativistic and small amplitude cases following a similarprocedure, and making the same assumptions, (6.38) - (6.40). For the equation ine3, we obtain the following nonlinear Schrodinger equation,

where a/ = ±Vl + fc2, v' = ±k/Vl + k2.In addition, the above Sine-Gordon equation has the following breather solution

, /</>\ _ v sin \ / l - v2£'

\V " Vd-»2) cosh^) 'with £ = dx + t/d, £' = dx - t/d, where d > 0. If the function is expanded in termof tanh"1 $ to the first order, then we get

(j> » . 2 W sech^Qe^1-"2/2)^' + c.c. = f ^ ^ ' ) ^ ' + ex.,V1 — v2

where

^,C') = îfe-^'/2SechK).V1 — v2

Inserting ip(£, £') into the Sine-Gordon equation given above, in the nonrelativis-tic case, keeping only the O(i>3) term, we get the following nonlinear Schrodingerequation

*V«'(£,O-2V«-gMV = 0.

Therefore, both the 04-field equation and the Sine-Gordon equation can reduce tothe nonlinear Schrodinger equation in the nonrelativistic case.

This can also be demonstrated in other ways. In fact, for the 0+ equation

<Ptt - <t>xx + m23 = 0 ,

we can assume that its solution is of the form

<j>{x,t)=lp{x,t)e-imt. (6.45)

Inserting (6.45) into (6.44), and taking into account the relation, idip/dt <IC rmp, inthe nonrelativistic case, we then get

.dip _ 1 63

This is also the nonlinear Schrodinger equation. Therefore, we can always reducethese relativistic equations of motion to the nonlinear Schrodinger equation in the


nonrelativistic limit, and thus conform that the nonlinear quantum mechanics es-tablished in Chapter 3 is self-consistent.

6.2.4 Nonlinear Dirac equation

In linear quantum mechanics, Fermi particles are described by the following Diracequation in the relativistic theory

ih—<j> = (ca • p)4> + mc2f3<t>. (6.46)Ob

It is natural to extend the above equation by including self-interaction discussed inSection 5.4, to arrive at the following nonlinear Dirac equation

ih~(j> = (ca-p)<j> + mc2P(j) + b\<j)2\<f>. (6.47)

The corresponding matrix equation is

fiWi\ ( me? 0 CPZ C(Pz-iPy)\ /fc\ih4>2 I _ 0 me2 c(px + ipy) cpz I <f>2

I iHz I cpx c(px - ipy) -me2 0 I <fo\ih4i) \c(pz+ipy) cpx 0 -me2 ) \<$>A)

+ \ UWMl) X , J • (6-48)\<A4/J \</>4/

This is a vector nonlinear equations. Its solutions have not been obtained previously.Here we are interested in whether this equation can have a soliton solution, and if so,what properties it has. These problems cannot be answered yet. But nevertheless,they are interesting and worth further investigation.

For the nonlinear Dirac equation in terms with a cubic nonlinear term in theDirac matrix 7^ (/z = 1,2,3,4)

7 ^ ^ ^l^lv^Wlv-lv^)) = 0,

Heisenberg obtained its single soliton solution which is given by

, 757A [A 77A \" (% A \

^ = — V 2^sech {-R1^)exp U e " 2hlvXv)'where


6.3 The Uncertainty Relation in Linear and Nonlinear QuantumMechanics

6.3.1 The uncertainty relation in linear quantum mechanics

The uncertainty relation in quantum mechanics is an important formula and alsoa problem that troubled many scientists. Whether this is an intrinsic property ofmicroscopic particles or an artifact of the linear quantum mechanics or measuringinstruments has been a long-lasting controversy. Obviously, this is related to thewave-corpuscle duality of microscopic particles. Since we have established nonlinearquantum mechanics which differs from the linear quantum mechanics, we couldexpect that the uncertainty relation in the nonlinear quantum mechanics is differentfrom that in the linear quantum mechanics. The significance of the uncertaintyrelation can be revealed by comparing the linear and nonlinear quantum theories.

It is well known that the uncertainty relation in the linear quantum mechanicscan be obtained from

HO = j | (f A i + iAfl) i/>(f, t) |2 dr > 0, (6.49)

or

P(0 = I d?p(?,t)F [A(r, t), B(r, tj\ $(?,t). (6.50)

In the coordinate representation, A and B are the operators of two physical quan-tities, for example, position and momentum, or energy and time, and satisfy thecommutation relation [4,B] = iC, tl>{x,t) and ip*(x,t) are wave functions of themicroscopic particle satisfying the linear Schrodinger equation and its conjugateequation, respectively, F = (AAf + AB)2, (A A = A-A, AB = B - B, A and Bare the average values of the physical quantities in the state denoted by x[>(x,t)), isan operator of physical quantity related to A and B, £ is a real parameter.

After some simplifications, we can get

I = F = Ai2f2 + 2AiABf + AB2 > 0,

or

Ai2^2 + d£ + A~¥ > 0. (6.51)

Using mathematical identities, this can be written as

A i 2 A B 2 > ^ p (6.52)

This is the uncertainty relation in the linear quantum mechanics. Prom theabove derivation we see that the uncertainty relation was obtained based on thefundamental hypothesizes of the linear quantum mechanics, including properties ofoperators of the mechanical quantities, the state of the particle represented by the


wave function, which satisfies the linear Schrodinger equation, the concept of aver-age values of mechanical quantities and the commutation relations and eigenequa-tion of operators. Therefore, we can conclude that the uncertainty relation (6.52)is a necessary result of the linear quantum mechanics. Since the linear quantummechanics only describes the wave nature of microscopic particles, the uncertaintyrelation is a result of wave feature of microscopic particles, and it inherits the wavenature of microscopic particles. This is why its coordinate and momentum cannotbe determined simultaneously. This is an essential interpretation for the uncertaintyrelation (6.52) in the linear quantum mechanics. It is not related to measurement,but closely related to the linear quantum mechanics. In other word, if linear quan-tum mechanics could correctly describe the states of microscopic particles, then theuncertainty relation should also reflect the peculiarities of microscopic particles.

Equation (6.51) can be written in the following form

#=z*(<+S2Y+zi; - <aS£ > o,V Ai2 J Ai2

or

A3*U+^£=) +A^2-4£L>0. (6.53)V 4Ai2 / 4 A A2

This shows that A A2 ^ 0, if (AAAB)2 or C2/4 is not zero. Or else, we cannot

obtain (6.52) and A J 2 A B 2 > (ABAB)2 because when AA2 = 0, (6.53) does not

hold. Therefore, (AA2) ^ 0 is a necessary condition for the uncertainty relation

(6.52). A A2 can approach zero, but cannot be exactly zero. Therefore, in the linear

quantum mechanics, the right uncertainty relation should take the form

AA*AB2 > &-. (6.54)

6.3.2 The uncertainty relation in nonlinear quantum mechanics

We now return to the uncertainty relation in the nonlinear quantum mechanics.Since microscopic particles are solitons and they have wave-corpuscle dualityin thenonlinear quantum mechanics, we can also expect that uncertainty relation in non-linear quantum mechanics is different from (6.52).

We now derive this relation for position and momentum of a microscopic particledescribed by the nonlinear Schrodinger equation (3.2) with V = 0, and A = 0, givenin (4.40), with a soliton solution, <ps, as given in (4.56). The function cj>8{x',t') is asquare integrable function localized at x'o = 0 in the position space. If the micro-scopic particle (soliton) is localized at x'Q ^ 0, it satisfies the nonlinear Schrodinger


equation,

i<t>f + 2</>x-x> + \(l>\24> = 0

for 6 = 1 , where t' = t/h, x' = y/mx/h. The Fourier transform of this function is

0,(p,O = 4 = f ° <Ps(x'ê-^'. (6.55)V2TT J-OO

It shows that <j>s(p,t') is localized at p in the momentum space. For (4.56), theFourier transform is explicitly given by

MP,t') = 7|sech U-(p- 2V2O| e^^+^-P^2^)*'-^-2^)^. (6.56)

The results in (4.56) and (6.56) show that the microscopic particle is localized notonly in position space and moves in the form of soliton, but also in the momentumspace, as a soliton. For convenience, we introduce the normalization coefficient AQin (4.56) and (6.56), then A% = l/(4y/2rj). According to the definition in Section4.2, the position of the center of mass of the microscopic particle, (x1), and itssquare, (x12), at t' = 0 are given by

(x1) = / dx'\4>s{x')\\ (x12) = / dx'xl2\4>3{x')\2.J—00 J — oo

We can thus find that

(x1) = 4V2VA2x'o, (x12) = ^ - + AÂlnx12, (6.57)

respectively. Similarly, the momentum of the center of mass of the microscopicparticle, (p), and its square, (p2), are given by

(P) = r P\Mp)\2dp, (p2) = r P2\j>s(p)\2dP,

j—00 j—00

which yield

 = 164)!tf, (P2) = ÂW + 32V2A2r,e. (6.58)

The standard deviations of position Ax' = \/{x'2) — {x')2 and momentum Ap =y/ip2) - (p)2 are given by

(Ax')2 = A2 [ ^ + 4 ^ ( 1 - 4Â 2 ) ] = ^ ,

(Ap)2 = 32V2A2 [ir,3 + tfa(l - 4 ^ ^ ) ] = |i,2, (6.59)


respectively. Thus we obtain the uncertainty relation between position and momen-tum for the microscopic particle, (4.56), in nonlinear quantum mechanics,

Ax'Ap = ~. (6.60)0

This result is not only valid for microscopic particle (soliton) described by thenonlinear Schrodinger equation, but holds in general. This is because we did notuse any specific property of the nonlinear Schrodinger equation in deriving (6.60)n in (6.60) comes from the integral constant l/^/2n. For a quantized microscopicparticle, n in (6.60) should be replaced by nh, because (6.56) is replaced by

4>s{p,t') = -±= f°° dx'U^,t')e-ipx>/h-V2nh J-oo

The corresponding uncertainty relation of the quantum microscopic particle (soli-ton) in the nonlinear quantum mechanics is given by

AxAp=f = ±. (6.61)

The relations (6.60) or (6.61) are different from that in linear quantum mechanics(6.54) i.e., AxAp > h/2. However, the minimum value AxAp = h/2 has not beenobserved in practical systems in linear quantum mechanics up to now except for thecoherent and squeezed states of microscopic particles. The relation (6.61) cannotbe obtained from the solutions of the linear Schrodinger equation. Practically, wecan only get AxAp > h/2 from (6.54), but not AxAp = h/2, in linear quantummechanics.

As a matter of fact, for an one-quantum coherent state,

oo n

\a) = exp(ab+ - a*b)\0) = e~^2 Y ° 6+n|0),n t j V i - 1

which is a coherent superposition of a large mumbler of microscopic particles(quanta). Thus

, .„, . I h~ . „. . , . . hmcj. „ .(a|a; |a) = y 7 ^ ( « + a), (a\p\a) = i\j - ^ — (a* - a),

and

(a\x2\a) = ^—{a*2 + a2 + 2aa* + 1), (a\p2\a) = ^(a*2 + a2 - 2aa* - 1),

where

/ h ft i+\ , . I hum / - , t \

and b+ (b) is the creation (annihilation) operator of the microscopic particle (quan-tum), a and a* are some unknown functions, u is the frequency of the particle, m


is its mass. Thus we can get

{AX) = W ( A ^ = — ' (Ax)2(Ap)^T >

and

Ax _ 1

Ap urn'

or

Ap = {um) Ax. (6.62)

For the squeezing state,

|/3)=exp[/J(6+2-b2)]|0),

which is a two-microscopic particle (quanta) coherent state, we can obtain

</?|A*2|/?> = ^ / ^ </?|Ap2|/J> = fcê-*

using a similar approach as the above. Here /? is the squeezing coefficient and|/3| < 1. Thus

A A h Ax 1 oflAxAp=-, —— = e8^1,^ 2 ' Ap mw '

or

Ap = Ax(ojm)e-80. (6.63)

This shows that the momentum of the microscopic particle (quantum) is squeezedin the two-quanta coherent state compared to that in the one-quantum coherentstate.

From the above results, we see that both the one-quantum and the two-quantacoherent states satisfy the minimal uncertainty principle. This is the same as thatof the nonlinear quantum states in nonlinear quantum mechanics. We can concludethat a coherent state is a nonlinear quantum state, and the coherence of quanta isa nonlinear phenomenon, instead of a linear effect.

As it is known, the coherent state satisfies classical equation of motion, in whichthe fluctuation in the number of particles approaches zero, i.e., it is a classical steadywave. According to quantum theory, the coherent state of a harmonic oscillator attime t can be represented by

\a,t) = e-ifltlh\a) = e-M*+M/2)t|a) = c-*,t/2-Ma/a f^ ""C^V)n=0 ^

= e-^2\ae-iut), (|n> = (6+)»|0».


This shows that the shape of a coherent state can be retained during its motion,which is the same as that of a microscopic particle (soliton) in the nonlinear quantummechanics. The mean position of the particle in the time-dependent coherent stateis

(a,t\x\a,t) = {a\eiHtlhxe-iHtlh\a)

= (a x-^[x,H}+{-^[[x,H},H} + --â)

= (a x + ^ - ^t2u2x + ••• a) (6.64)

= (a x cos u)t H sin ujt a)mui

= \ \a\ cos(u>t + 9),V mu>

where

6 = tan"1 ( - ) , x + iy = a,

and

[x,H\ = —, \p,H} = -ihmu2x.

Comparing (6.64) with the solution of a classical harmonic oscillator,

x=\ r-cosM + 0), E= £- + -moj2x2,V mu2 2m 2

we find that they are similar, with

E = hua2 = (a\H\a) - (0\H\0), H = hw fb+b + M .

Thus we can say that the center of the wave packet of the coherent state indeedobeys the classical law of motion, which happens to be the same as the law of motionof microscopic particles in nonlinear quantum mechanics discussed in Section 4.2.

We can similarly obtain

(a,t\p\a,t) = —V2mEuJ\a\sm(ujt + 9),

(a, t\x2 \a, t) = — fH2 cos2 (wt + 6) + ]] ,urn \_ 4J

(a,t\p2\a,t) = 2rnHu \\a\2sin2(ut + 0) + - ,

[Ax{t)]2 = 2 ^ ' [Ap{t)]2 = \ m u K


and

Ax(t)Ap(t) = \ .

This is the same as (6.62). It shows that the minimal uncertainty principle for thecoherent state can be retained at all time, i.e., the uncertainty relation does notchange with time t.

The mean number of quanta in the coherent state is given by

n = (a\N\a) = (a\b+b\a) = a2, (a\N2\a) = |a|4 + \a\2.

Therefore, the fluctuation of the quantum in the coherent state is

An = \J(a\N2\a)-((a\N\a))2 = \a\,

which leads toAn 1n \a\

It is thus obvious that the fluctuation of quantum in the coherent state is verysmall. The coherent state is quite close to the feature of soliton or solitary wave.

These properties of coherent states of microscopic particles described by thenonlinear Schrodinger equation, the (j>4-equation, or the Sine-Gordon equation innonlinear quantum mechanics are similar. In practice, the state of a microscopicparticle in the nonlinear quantum mechanics can always be represented by a co-herent state, for example, the Davydov wave functions, both |£>i) and \D2) (see(5.72)), the wave function of exciton-solitons in protein molecules and acetanilide(see Chapter 9), and the BCS wave function (2.28) in superconductor, etc. Hence,the coherence of particles is a kind of nonlinear phenomenon which occurs only innonlinear quantum mechanics. It does not belong to systems described by linearquantum mechanics, because the coherent state cannot be obtained by superposi-tion of linear waves, such as plane wave, de Broglie wave, or Bloch wave, which aresolutions of the linear Schrodinger equation in linear quantum mechanics. There-fore, the minimal uncertainty relation (6.63) as well as (6.60) and (6.61) are onlyapplicable to microscopic particles in nonlinear quantum mechanics. In other words,only microscopic particles in nonlinear quantum mechanics satisfy the minimal un-certainty principle. It reflects the wave-corpuscle duality of microscopic particlesbecause it holds only if the duality exists. This uncertainty principle also suggeststhat the position and momentum of the microscopic particle can be simultaneouslydetermined in certain degree and range. A rough estimate for the size of the uncer-tainty is given in the following.

We choose f = 140, 77 = x/300/0.253/2%/2 and x0 = 0 in (4.56) or (6.56), sothat <ps(x,t) or <f>(p,t) satisfies the admissibility condition, i.e., </>s(0) « 0. (In fact,in such a case, <A«(0) « 10~6, thus the admissibility condition can be consideredsatisfied). We then get Ax « 0.02624 and Ap « 19.893, according to (6.59) and


(6.60). These results show that the position and momentum of a microscopic particlein nonlinear quantum mechanics can be simultaneously determined within a certainapproximation.

Finally, we determine the uncertainty relation of the microscopic particle de-scribed by the nonlinear Schrodinger equation, arising from the quantum fluctuationeffect in the nonlinear quantum field. The quantum theory presented in Section 4.6was discussed by Lai and Haus based on the nonlinear Schrodinger equation. As dis-cussed in Chapters 3 and 4, a superposition of a subclass of the bound states |n, P)which are characterized by the number of Bosons, such as photons or phonons, n,and the momentum of the center of mass, P, can reproduce the expectation val-ues of the field of the microscopic particle (soliton) in the limit of a large averagenumber of Bosons (phonons). Such a state formed by the superposition of \n,P) isreferred to as a fundamental soliton state. In quantum theory, the field equation isgiven by (3.62), with the commutation relation (3.63). The corresponding quantumHamiltonian is given by (3.65). In the Schrodinger picture, the time evolution ofthe system is described by (3.66). The many-particle state \tp') can be built up fromthe n-particle states given by (3.69).

The quantum theory based on (3.69) describes an ensemble of Bosons interactingvia a ^-potential. Note that H preserves both the particle number.

N = f ci>+(x)4>(x)dx (6.65)

and the total momentum

P = i\J [^WiW ~ t+Wj^fa)] dx- (6-66)Lai et al. proved that the Boson number and momentum operators commute,

so that common eigenstates of H, P and N can exist in such a case. In the caseof a negative b value, the interaction between the Bosons is attractive and theHamiltonian (3.63) has bound states. A subset of these bound states is characterizedsolely by the eigenvalues of N and P. The wave functions of these states are

( oo , oo \

ip^xj + - Y^ ki-Zjl , (6.67)j=l l<i,j<n I

where

Thus

/„(*!, •"*»,*) = /^n(p)/ n ,p(zi , - - -zn ,*)e~ i B ( n ' p ) t , (6.68)


where

fc-rfW.-"~. and 9(P)^HP-^/[2(Ari']},

Using / n i P given in (6.67), we found that \n, P) decays exponentially with sep-aration between any pair of Bosons. It describes an n-particle soliton moving withmomentum P — hnp and energy E(n,p) = np2 - \b\2(n2 - l)n/12. By construction,the quantum number p in this wave function is related to the momentum of thecenter of mass of the n interacting Bosons, which is now defined as

X = lim (xcj>+(x)(t>{x)dx{e + N)'1, (6.69)

with

[x,P] = ih.

The limit of e -» 0 is introduced to regularize the position operator for the vacuumstate.

We are interested in the fluctuations of (6.65), (6.66) and (6.69) for a state\ip'(t)) with a large average Boson number and a well-defined mean field. Kartnerand Boivin decomposed the field operator into a mean value and a remainder whichis responsible for the quantum fluctuations.

4>(x) = W(0)\j>+(x)W(0)) + 4>i(x),

[Mx),$t(x')]=6(x-x'), [jn(x)MJ)]=0- (6-70)

Since the field operator <j> is time independent in the Schrodinger representation,we can then choose t = 0 for definiteness. Inserting (6.70) into (6.65), (6.66) and(6.69) and neglecting terms of second and higher order in the noise operator, Kartneret al. obtained that

N = no + An, P = hnopo + hno^p, X — XQ (1 I + Ax,

with

n0 = [dx(4>+{x))(j>(x)), An= (dx{4>+{x))4>x{x) +c.c,

po = — [ dx(fc{x))($(x)), Ap=— [dx(4>i(x))Mx)+c.c,n0 J n0 JIf - 1 f

xo = — dxx(4>+{x))(cf>(x)), Ax = — / dxx{<t>+(x))Mx) + c-c->n0 J no J

where Ax is the deviation from the mean value of the position operator, An, Ap,and Ax are linear in the noise operator. Because the third- and fourth-order cor-relators of 4>i and <ft are very small, they can be neglected in the limit of large n0.


Note that Ap and Ax are conjugate variables. To complete this set we introduce avariable conjugate to An,

Ad = ±Jdx{i [(i+(x)} + x(4>+(x))] -pox(4>+(x))}Mx) + c.c

As it is known, if the propagation distance is not too large, then to the first order,the mean value of the field is given by the classical soliton solution

with

4>o,nQ(x, t) - - ^ — exp [iClni - ip\t + ipo(x - x0) + i0o]

XSeCh \ ^ ( x -xo- 2pd*)l . (6-71)

where the nonlinear phase shift flnj = no|6|2t/4. If po = XQ = 60 = 0, we obtain thefollowing for the fluctuation operators in the Heisenberg picture,

Afi(t) = Jdx[f_n(xyF^ + c.c],

A0(t) = Jdx{f_g(x)*F^+c.c),

Ap(t)= Jdx[f-p(x)*F^+c.c},

Ax(t)= Jdx[f_x(xyF^+c.c],

with

Kt=e«inl4>1(x,t),

and the set of adjoint functions

/-n(x) = —^—sech(a;no),

f-e(x) = — — \sech{xno) + xno- sech(xno) ,& [ o,xno j

f-p(x) = f 1 sech(a;no),* axna

f-x(x) = 7=a;nosech(a;71o),

noVl lwhere

1 ...Zn0 = 2no \b\ x-


For a coherent state defined by

4>(x)\$0,n0) =</)0,n0(a;)l*0,no). o r 0l(aOl0O,no) = 0 ,

where

|$o>no) = exp y dx [^no(x)4>+(x) - <^,no(*)<£+(x)] } |o>,

and (/>o,no has been given (6.71), Kartner and Boivin further obtained that

(A^)=nO l <A6>2> = — , <APg> = 3 ^ , (A^) = - ^ J

where TQ = 2/no|6| is the width of the microscopic particle (soliton). The un-certainty products of the Boson number and phase, momentum and position are,respectively,

(Ah20){A92) = 0.6075 > 0.25, and n2

Q(Ap2)(Axl) = 0.27 > 0.25.

Here the quantum fluctuation of the coherent state is white, i.e.,

{4>i(x)My)) = (it(x)My)) = o.

However, the quantum fluctuation of the soliton cannot be white because interactionbetween the particles introduces correlations between them. Thus, Kartner andBoivin assumed a fundamental soliton state with a Poissonian distribution for theBoson number

and a Gaussian distribution for the momentum (6.68) with a width of (Apg) =no|6|2/4/i, where fj, is a parameter of the order of unity compared to n0. Theyfinally obtained the minimum uncertainty values

..22. 0.25 0.25 I" _ / 1 \ ]

{n20Ap2} n0 [ \noj\

up to the order of l/n0, for the corresponding initial fluctuations in soliton phaseand timing. Thus at t = 0 the fundamental soliton with the given Boson numberand momentum distributions is a minimum uncertainty state in the four collec-tive variables, the Boson number, phase, momentum, and position, up to terms of


O(l/no),

(Ang)(A 02) = 0.25[l + o ( ^ ] !

ng(Apg><A > = 0.25 [l + O ( £ ) ] . (6.72)

These are the uncertainty relations arising from the quantum fluctuations in non-linear quantum field of microscopic particles described by the nonlinear Schrodingerequation. They are the same as (6.61) - (6.63). Therefore, we conclude that theuncertainty relation in the nonlinear quantum mechanics takes the minimum valueregardless a state is coherent or squeezed, a system is classical or quantum.

Pang et al. also calculated the uncertainty relation of quantum fluctuationsand studied their properties in nonlinearly coupled electron-phonon systems basedon the Holstein model but using a new ansatz which includes correlations amongone-phonon coherent and two-phonon squeezing states and polaron state. Manyinteresting results were obtained. The minimum uncertainty relation takes differentforms in different systems which are related to the properties of the microscopicparticles. Nevertheless, the minimum uncertainty relation (6.63) holds for boththe one-quantum coherent state and two-quanta squeezed state (see Jajernikov andPang 1997, Pang 1999, 2000, 2001, 2002). These work enhanced our understandingof the significance and nature of the minimum uncertainty relation.

In light of the above discussion, we can distinguish the motions of particles inthe linear quantum mechanics, nonlinear quantum mechanics, and classical me-chanics based on the uncertainty relation. When the motion of the particles satisfyAxAp > h/2, the particles obey laws of linear quantum mechanics, and they haveonly wave feature. When the motion of the particles satisfy AxAp = h/12 or TT/6,the particles obey laws of motion in the nonlinear quantum mechanics, and the par-ticles are solitons, exhibiting wave-corpuscle duality. If the motion of the particlessatisfy AxAp = 0, then the particles can be treated as classical particles, with onlycorpuscle feature. The nonlinear quantum mechanics introduced here thus givesa more complete description of physical systems. Therefore, we can say that thenonlinear quantum mechanics bridges the gap between the classical mechanics andthe linear quantum mechanics.

6.4 Energy Spectrum of Hamiltonian and Vector Form of the Non-linear Schrodinger Equation

Like in the linear quantum mechanics, it is useful to obtain the energy spectrum ofthe Hamiltonian operator for a given system in the nonlinear quantum mechanics.The energy of a microscopic particle satisfying the nonlinear Schrodinger equation


can be obtained from/•OO

E= / Hdx,J — OO

where H. is the classical Hamiltonian density which depends on the wave function<t>{x,t). However, this only gives the energy of the microscopic particle, not theeigenenergy spectrum of the Hamiltonian operator. As discussed in Chapter 3,we can obtain the eigenvalues of the L operator corresponding to the nonlinearSchrodinger equation (3.2) following Lax's approach. But these eigenvalues are notthe eigenenergy spectrum of the Hamiltonian operator of the systems. We must,therefore, use alternative method to calculate its eigenenergy spectrum. In linearquantum mechanics, the eigenenergy spectrum was obtained from the eigenequa-tion of the Hamiltonian operator. Because the latter is independent of the statewavefunction of the particle, there is little difficulty in calculating the eigenenergyin linear quantum mechanics. However, this is not the case in nonlinear quantummechanics in which the Hamiltonian operator depends on the state wave vector ofthe microscopic particle, as mentioned in Chapter 3. How do we then obtain theeigenenergy spectra in nonlinear quantum mechanics?

6.4.1 General approach

As discussed in Chapter 3, the wave function of a microscopic particle can be quan-tized by the creation and annihilation operators of the particle in nonlinear quantummechanics. The Hamiltonian of a system described by the wave function <p(x, t) canbe quantized by introducing creation and annihilation operators in the particle num-ber representation or the second quantization representation. We can then calculatethe eigenenergy spectrum using the eigenequation of the quantum Hamiltonian andthe corresponding wavevector in the particle number representation. This is ba-sically how the eigenenergy spectrum in the nonlinear quantum mechanics can beobtained. For convenience, we express the nonlinear Schrodinger equation (3.2)with A(<j)) = 0 in the following discrete form

iKlH = ~2^{lt>i+1 ~ 24>i + ^-l) ~ b^2(t>i + V ^ ^ {j = L 2 . 3 . ' • • . J)>(6.73)

where r0 is a spacing between two neighboring lattice points, j labels the discretelattice points, J is the total number of lattice points in the system.

The vector form of the above equation is

\ih%- - - ^ - V(j, t)} 4> = -eM4> - &diag.( |0i |2 , \<j>2\2 • • • \4>a\

2)4>, (6.74)I Ob TflTn J

where <j> is a column vector, ^ = Col.(<j>i,</>2 • • • <Pa) whose components are complex.Equation (6.74) is a vector nonlinear Schrodinger equation with a modes of motion.In (6.74), b is a nonlinear parameter and a is the number of modes that exist in the


systems, M = [Mnt] is an a x a real symmetric dispersion matrix, e = H2/(2mrl).Here n and t are integers denoting the modes of motion.

The Hamiltonian and the particle number corresponding to (6.74) are

H = J2 {ô\K\2 - U<j>n\4) - e J2 Mne<j>nct>t, (6.75)

n=l V l / r#l

N=^2\<f>n\2, (6.76)

n=l

where

We have assumed that V(j, t) are independent of j and t. In the canonical secondquantization theory, the complex amplitudes (4>*n and <pn ) become Boson creationand annihilation operators (B+ and Bn) in the number representation. If \mn) is aneigenfunction of a particular mode, then B£\mn) = y/mn + l\mn + 1), Bn\mn) =y/^\mn - 1) and Bn|0> = 0.

Since the no particular ordering is specified in (6.76) and (6.76), we use theaverages,

\<t>n\2^\(BtBn + BnB+),

and

| ^ | 4 -»• i (B+B+BnBn + B+BnB+Bn + B+BnBnB+

+BnB+BnB+ + BnBnB+B+ + BnB+B+Bn) ,

with the Boson commutation rule BnB+ - B+Bn — 1. Equations (6.76) and (6.76)then become

H = J2[(*»o- \b) (B+Bn + - \bB+BnB+B^ -ej^MraB+Bt, (6.77)

N = JT (B+Bn + 1) . (6.78)71=1 ^ '

Prom now on, we will use the notation \m\m2 • • • ma] to denote the product ofthe number states |mi)|m2) • • • \ma). The stationary states of the vector nonlinearSchrodinger equation (6.74) must be eigenfunctions of both N and H. Consider anm-quantum state (i.e., the mth excited level, m — mi + rri2 H— • rrij), with m < a.


A n e i g e n f u n c t i o n o f N c a n b e e s t a b l i s h e d a s

| * m > = C i [ m , 0 , 0 , - - - , 0 ] + --- + C l2 [ 0 , m , 0 , 0 , - - - , 0 ] + --- +

C i [ 0 , 0 , 0 , - - - ,m] + ••• + Ci+1[(m - I),1,0,- •• ,()] + ••• + ( 6 . 7 9 )

C p [ 0 , 0 , - - - , 0 , 1 , 1 , - - , ! ] •

(m times)

The number of terms in (6.79) is equal to the number of ways that m quanta canbe placed on the a sites, which is given by

_ (m + a — 1)P~ m ! ( a - l ) ! '

The wave function |$m) in (6.79) is an eigenfunction of N for any values of the C'as.Thus we are free to choose these coefficients such that

H\*m) = E\*m). (6.80)

Equation (6.80) requires that the column vector C = Col.(Ci,C2, • • -Cp) satisfiesthe matrix equation

\H - IE\C = 0, (6.81)

where H is a p x p symmetric matrix with real elements. / is a p x p identitymatrix, E is the eigenenergy. Equation (6.80) is an eigenvalue equation of thequantum Hamiltonian operator (6.77) of the system. We can obtain the eigenenergyspectrum Em of the system from (6.81) for given parameters, e, w0, and b. Scott,Bernstein, Eilbeck, Carr and Pang et al. used the above method to calculate theenergy-spectra of vibrational excitations (quanta) in many nonlinear systems, forexample, small molecules or organic molecular crystals and biomolecules. Theseresults can be compared with experimental data, and will be discussed in Chapter9.

6.4.2 System with two degrees of freedom

We now discuss a simple instance and consider a system with two degrees of freedom(or two modes of motion). This was studied by Scott et al. and Pang. It takes theform

H^m<r«)(:;)- <->The corresponding N and H, in such a case, are given by

JV = £+!? !+5^52 + 1, (6.83)

H = (tkj0 - ^ N - | (B+^B+Bi + B+B2B+B2) - e (B+B2 + BXB%) ,


respectively.We seek eigenfunction |$) of both the operators. For the ground state, |$o) =

|0)|0), which is the product of the ground state wavefunctions of the two degrees offreedom, we have

iV|$0> - |#o>, H\$o) = Eo\*o),

where

JEJo = fkoo - -.

For the first excited state, we have 1$^ = Cx jl)|O)+C2|O)|l) with |Ci|2 + |C2|2 =1. Thus iV|$i) = 2|$i). From # |$ i ) = Ei|$i), we can get

{2huo - Ei) --b e (CA_

e {2Huo-E1)-h \C2J

The first excited state splits into a symmetric and an antisymmetric states,

I*I.> = ^ ( | 1 > | 0 > + |0>|1»,

l*la> = ^ ( | l ) | 0 ) - | 0> | l ) ) ,

with

Eis = - o + wo — b — e,

Ela = E0 + tlLJo -b + E.

respectively.For the second excited state, we have |$2) = Ci|2)|0) + C2|1)|1) + C3|0)|2), with

i=l

Ar|$2)=3|$2), H|*2> = ^|*2>,

and

(3hwo-E2 + lb y/2e 0 \ .„ N

V2e 3 wo - E2 + ^b y/2e \ C2 = 0.

K 0 Vê 3hujo-E2 + -bJ V 3 /

Diagonalizing this matrix equation, we get the eigenenergies with E2a = 2ftu>o +Ei + 76/2 corresponding to the antisymmetric state Ca = (1,0, - 1 ) / A / 2 , and#2± = 2/kj0 + î + 36 ± 3\/b2 + 16^2 corresponding to the symmetric state


C± = (e, Vb2 + 16£2/2, y/2e), respectively. In the limit of b » e, C+ = (0,1,0),C- = (l / \ /2,0,l / \ /2), and the state Ca is a localized mode which has all the energyand the particle has equal probability to be in each of the two sites. In this case,C± have the same structure, but a different relative phase. Linear combinations ofthese states will first localize all the energy on a particular site.

For the mth excited state, we have

\*m) = Ci|m>|0> + C2\m - 1)|1> + • • • + Cm\l)\m - 1) + Cm+1|0)|m) (6.84)

with

m

#|*m) = (m+l)|*m>, H\$m)=Em\*m),

and

[(m + l)fiuo -Em-nm]C = 0,

here C = Col.(Ci,C2,• • • ,Cn+i), ^m is an (m + 1) x (m + 1) tridiagonal matrix.For an odd number n, we have

(D{1) Q(l) 0 ••• 0 ••• 0 • • • 0 0 0 \g ( l ) I>(2) 0(2) ••• 0 ••• 0 • • • 0 0 0

0 Q(2) ••• 0 ••• 0 • • • ( ) 0 0

0 0 0 • • • D[(m + l)/2] • • • Q[(m + l)/2] • • • 0 0 0n ~ 0 0 0 • • • Q[(m + l)/2] • • • D[(m + l)/2] • • • 0 0 0 '

0 0 0 ••• 0 ••• 0 • • • 0 Q(2) 00 0 0 • • • 0 ••• 0 ••• 0 ( 2 ) D(2) 0 ( 1 )

\ 0 0 0 ••• 0 ••• 0 • • • 0 0(1) £>(!)/

where

D{i) = h [m + 1 + (m + 1 - if + (t - I)2] , Q{i) = ey/i(m + l-i).


For an even number m, we have

/D(l ) Q(l) 0 ••• 0 0 0 • • • 0 0 0 \0 ( 1 ) D(2) Q(2) • • • 0 0 0 ••• 0 0 0

0 0 ( 2 ) 0 ••• 0 0 0 ••• 0 0 0

0 0 0 ••• 0 Q ( m / 2 ) 0 ••• 0 0 0

Qn- 0 0 0 • • • Q(m/2) D(m/2 + 1) Q ( m / 2 ) • • • 0 0 0

0 0 0 ••• 0 Q(m/2) 0 • • • 0 0 0

0 0 0 ••• 0 0 0 ••• 0 Q(2) 0

0 0 0 ••• 0 0 0 - - - 0 ( 2 ) D(2) 0 ( 1 )

\ 0 0 0 ••• 0 0 0 ••• 0 Q(1)D(1)J

where

„ (m \ b ( m2\ _ /m\ frnrrn ~\D(-2+i) = 2{m+i+-r)> « ( T W T ( T + 1 ) -

We can see that if (m + l)ftu>o — E\ is one of the (ra + 1) real eigenvalues offin, and C1 is the corresponding eigenvector, the wave function for the mth excitedstate can be established by (6.84). For example, for a localized mode with itsenergy concentrated on a single degree of freedom, the classical system (6.82) hasharmonic solutions of the form 4>i = 4>'ie~%ut • For iV > 2e, there is a local modebranch with hw = hu0 - bN. Along this branch, E = Hu)0N - bN2/2 - e2 jb.In the quantum case here (let e < 7), there are two states corresponding to thelocalized modes. One is symmetric with C\ = Cm+i = 0(1), C2 = Cm = O(e/b),C3 = Cm-i = O(e2/b2), etc. The other is antisymmetric with C\ = — Cm+i = 0(1),C2 = -Cm = 0(e/6), C3 = -C m _i = O(e2/62), etc. The gap between them isabout Ems — Ema = O(em/bm~1). To the order of e2, the energy of these modes is

Ema -Eo= Ems - Eo = [Y/ô - \ b \ m - ±bm2 - " ^ J . (6.85)

When m > 4, there could be overtone spectra at integer values of m. Asm-> 00,it approaches the classical limit given above except that the frequency is reducedby a factor of 6/2. Therefore, the energy of the local mode remains constant on aparticular state in the classical case, but oscillates between the two states with aperiod r = 0 (b"1"1 /em) in the quantum case. When m —> 00, r —> 00, it returnsto the classical result.

6.4.3 Perturbative method

From (6.73) - (6.76), we know that foujo contains the applied potential V(x), whichin general depends on x. Strictly speaking, the method discussed above is only


applicable to the case of V(x) = 0 or a constant. If V(x) depends on x, we cannot usethe above method to obtain the eigenvalue spectrum, but we can use perturbationmethod to find an approximate eigenenergy spectrum, when \V(x)\ <boi h2/2mr2

).Scott et al. and Pang studied the eigenenergy spectrum of a Hamiltonian operatorwith V{x) using a perturbation approach. Their work are briefly described in thefollowing.

In accordance with the general perturbation theory, the Hamiltonian operatorof the system in (6.74) is written as

H = H0 + eV. (6.86)

At the same time, we assume that the eigenvectors and eigenvalues of H can bewritten as

C = C o + e C i + £ 2 C 2 + ••• ,

E = EQ + i E i + e 2 E 2 + ••• . (6.87)

Inserting (6.87) into (6.81) we can get

LC0 = 0,

LCy = (Ei - V)C0,

LC2 = (Ex - V)Ci + E2C0, (6.88)

LC3 = (Ei - V)C2 + E2Cr + E2C0,

with L = Ho — EQ, where the matrix Ho is diagonal and its first a elements areequal. Since we are mainly interested in the dependence of the lowest a eigenvalueson e, Co can be considered the eigenvector corresponding to the eigenenergy Eo.Then L has a null space of dimension a which we must take into consideration aswe seek solutions of (6.88).

Solution of this problem was given by Scott et al.. The first excited state (m = 1)has a wave function of the form

| * 0 = Ci[l,0,0, • • • ,0] + • • • + Ca[0, • • • ,0,1].

Measuring energy with respect to the ground state

Ho = (hw0 - 6)diag.(l, 1,1,1, • • • , 1). (6.89)

Thus Eo = (huj0 — b),L = [0] and the first equation in (6.87) places no constraintson the selection of Co-

Next, Scott et al. defined an inner product

p

(v,Q) - ^Vi'î'-


Since the CV 's may be complex, and L is self-adjoint, from the Fredholm alternativetheorem, we know that (6.88) has solutions if and only if the right hand side of thesecond equation is orthogonal to (a dimensional) null space of L, as pointed outby Strang. Assuming this null space is spanned by the orthogonal set [(CQ )],i' — 1,2, • • • , a and that Co = CQ , the second equation in (6.87) has a solution ifand only if

Ei (ciil},€P) - (CP,MC^) = 0, (6.90)

and

* (CP,CP) - (Cp,MCP) = 0, for k> / i>. (6.91)

Condition (6.90) is satisfied if we choose

(cfUc<<'>)

For this value of E\, condition (6.91) is satisfied if CQ is an orthogonal eigenvectorof M. Then C\ = O leads to Ci — 0, etc. and the exact result of the perturbationtheory is C\ = (% and E = Eo + eE\. This is essentially a direct determination ofthe eigenvectors and eigenvalues of E — (huj0 — b)I + eM.

Next, we consider the second excited level (m = 2). With the wave function|n) constructed as in (6.79),

Ho = diag.{EQ,E0,--- , £ Q , h a + i , h a + 2 , - - - ,hp) (6.93)

a times (p-a) times

and

\ O G] }a-[CP R\}(p-a). ( 6 9 4 )

Similarly

\O O]}a[O A j } ( p - a ) . ( 6 9 5 )

a p—a

where

A = diag.(di,d2--- ,dp_a).

(6.92)


All elements of A are nonzero. To be consistent with the partitioning in (6.94) and(6.95), Scott et al. defined

Ci=\WA\™ y (6.96)[Zi\ }(p-m)

Prom the first equation in (6.88) and the structure of L, we have Zo = O. Forsolvability of the second equation in (6.88) and the structure of V, the Fredholmalternative theorem requires that E\ = 0. Thus (6.88) can be expressed as thefollowing two sets

Zi = -A^GjfWb,

Z2 = -A~1(GTW1+RZ1),

Z3 = -A-1(GTW3 + RZ2 - E2Z1), (6.97)

ZA = -A-1(GTW3 + RZ3 - E2Z2 - EkZx)

and

TW0 = 0,

TWi = -GA~1RZ1 - E3W0,

TW2 = -GA-l(E2Z1 - RZ2) - E3Wi - E4W0, (6.98)

TW3 = GA-ÊsZ! + E2Z2 - RZ3) - E3W2 - E±WX - E5W0,

with

T = GA~1GT + E2.

Scott et al. gave the following procedure for solving (6.97) - (6.98).

(1) Choose Wo as an eigenvector of GA~1GT and —E2 the corresponding eigen-value, to satisfy the first equation in (6.98), assuming that E2 is not a multipleeigenvalue.

(2) Obtain Z\ from the first equation in (6.97).(3) Since Eo is not a multiple eigenvalue, the null space of T is simple WQ . From the

Fredholm alternative theorem, a necessary and sufficient condition for the secondequation in (6.98) to have a solution is

(Wo, GA-^RZ,)3 [Wo,W0) •

(4) Solve the second equation in (6.98) for W\ with the requirement of (W\, WQ) = 0 .(5) Obtain Z2 from the second equation in (6.97).


(6) The solvability of the third equation in (6.98) requires

[Wo,GA-1(E2Z1-RZ2)}4 (Wo,Wo)

Following this procedure, we can find the solutions and the corresponding energyspectrum of the system under the perturbation eV in (6.86).

6.4.4 Vector nonlinear Schrodinger equation

Since we are dealing with colum vectors in (6.74), it would be useful to have a clearunderstanding on properties of the vector nonlinear Schrodinger equation. Thisequation can often be written as

t/#t = - | ^ x x - & ( # ) 0 , (6-99)

where

is a column vector of n-components, describing the "isospace" state. The corre-sponding Lagrangian and Hamiltonian densities of the system are given by

C = | (#t - M) ~ (&*») + \ {Wf , (6-100)

n = ^{h^)-\{H>f. (6.101)

Here 4> = 4>+lo a n d 70 is a diagonal matrix, and the internal product (</></>) = (f>+jo<t>is conserved. Such transformations also conserve C and %. That is, they aresymmetry transformations of the systems which do not change (6.99). When 70 isalso an n-component unit matrix, the transformations form a compact group U'(n).On the other hand, if

70 = diag.(l, • • • , 1 , -1 , ••• , - 1 ) ,

i j

which plays the role of isotropic space metric and the dagger sign (*) denotes Her-mitian conjugate, the iso-transformation belongs to the non-compact group U(i,j\.Zakharov and Shabat showed that (6.99) is completely integrable, in the case ofC/(l) group for both positive (b > 0) (the U(l,0) model) and negative (b < 0) (theC/(0,1) model) coupling constants. Furthermore, Manakov integrated (6.99) for theU(2) group (the C/(2,0) model, b > 0), and the integrability for the case of C/(l, 1)group was shown by Makhankov. These can be considered as special cases of thegeneral system.


In the U(1,O) model, Makhankov et al. found that the integrals of the particlenumber, momentum and energy can be written in the field and the action-anglevariables as follows

N = J \4>\2dx = J n(p)dp + ^2 Ns,

P = - 3 [(<Fx)dx = fpn(p)dp+Y^ ^ s N s , (6.102)J J s=l

E = I ( l ^ l 2 - > 4 ) dx = / A(P)*+ E f (** - \b2N2) .The continuous action n{p) and angular variable 0(p) are related to the scattering

matrix element. Prom (6.102), we see that the angular variables 9 and vs arecyclic and the action functionals corresponding to them are integrals of motion. Inquantum language the continuous variables (wave background) correspond to linear"microparticle" or elementary excitations (phonons, magnons, etc.), with dispersionE = P2 and mass 1/2 (in natural unit system). The nonlinear microscopic particle(solitons) (i.e., the discrete variable) can be interpreted as a special form of boundstates with Ns constituent "particles" of mass 1/2 in the nonlinear quantum field.The energy, momentum and mass of a microscopic particle (soliton) are

Es = ±-P?-±b2Nl Ps = \vsNs, Ms=l-Ns, (ATS»1), (6.103)

The first term of Es is the kinetic energy of the microscopic particle (soliton) withmass Ns/2 and the second term is its binding energy.

In the 1/(0,1) model, the dynamic equation is of the form

iHt + ^ r a - b (M2 -p)4> = 0. (6.104)

A term, bpcj>, is introduced into the equation and a corresponding term, —bp\<p\2 ——/i|0|2 is introduced into the Hamiltonian. Here the quantity /x plays the role ofchemical potential. The integrals of particle number, momentum and energy arerenormalized as follows

/•OO

N = / (\<t>\2 - p)dx,J — oo

P = j Q(<f>*(f>x)dx + p0, (6.105)J—oo

where 8 is the phase shift in the solution when the coordinate varies from — oo to+oo. The discrete part of the spectrum corresponding to the hole excitation mode


in quantum case was studied by Lieb. In the classical limit, this mode is denotedby a complex kink solution

Vie

<j)k = ak tanh[ajt(a; - vkt - x0)] + i — ,

where

Thus we get

The negative sign in the expression of N corresponds to a particle "deficiency" inthe condensate, i.e., the presence of holes in the system. The hole number is givenby — Nk = Nh > 0. Therefore, this model can be used to describe the nonlinearexcitation of holes or hole-solitons in the nonlinear quantum mechanical systems.

6.5 Eigenvalue Problem of the Nonlinear Schrodinger Equation

As mentioned in Chapter 3, the eigenvalue problem of the nonlinear Schrodingerequation (3.2) with the Galilei invariance is determined by the eigenequation of thelinear operator L in the Lax system in (3.20) - (3.22). The eigenequation corre-sponding to the nonlinear Schrodinger equation is found by the linear Zakharov-Shabat equation (4.42) from Lax equation (3.20), or

iil>x> + $V = A03V1- (6.106)

This is an eigenequation for an eigenfunction ip with a corresponding eigenvalue Aand a potential $, where,

•=(£)• - ( ; - . ) • - t i t y <"•»>Here <j> satisfies (4.40). It evolves with time according to (3.21). However, whatare the properties of the eigenvalue problems determined by these relations? Thisdeserves further consideration.

As it is known, the eigenequation is invariant under the Galilei transformation.As a matter of fact, if we substitute the following Galilei transformation

{ x = x' - vt',i=t', (6.108)

(j>'(x,i) = eivx'~iv2t'/2</>(x',t')


into (6.107), $ is transformed into

*'(*) = (% e i / 2 J * ( * ' ) ( e0 ^ J , (6-109)

where

0 = vx' - -v2t' + 0O,

and #o is an arbitrary constant. If the eigenfunction ip{x') is transformed as

*&={ 0 e-i/aj^)' (6-U°)then (6.106) becomes

ifa + $V = (A - £) orf. (6.111)

It is clear that in the reference frame that is moving with velocity v, the eigenvalueis reduced to v/2 compared to that in the rest frame. It shows that the velocityof the microscopic particle (soliton) is given by 23?(A«). When & is constant, i.e.,0 = 60, the eigenvalue is unchanged because v = 0. This implies that the nonlinearSchrodinger equation is invariant under the gauge transformation, 0' = e%e°<j>{x').

Satsuma and Yajima studied the eigenfunction of (6.106) that satisfies theboundary condition tp = 0 at |a;| -» oo. The eigenvalues and the correspondingeigenfunctions were denoted by Ai,A2,--- ,XN and tpi,ip2,--- ,I}>N- For a giveneigenfunction, tpn(x'), equation (6.106) reads

i ^ + $ ( i ' ) i ( i ' ) = U i ( i ' ) 1 n = 1,2, •••,7V. (6.112)ax

$(x') was expressed in terms of the Pauli's spin matrices a\ and a2,

$(x') = RMaOfci - 9fe(a;')]"2. (6.113)

Multiplying (6.112) by a2 from left and taking the transpose of the resulting equa-

tion, we get

where the superscript T denotes transpose. Multiplying the above equation by%i>n from right and (6.111) by tym

ci fr°m 1 ^ a n 0 as |x'| -> oo, were used in obtaining theabove equation. The following orthonormal condition was then derived.

/•OO

/ Vm°^ndx' = Snm. (6.114)J — oo

Satsuma and Yajima further demonstrated that (6.112) has the following sym-metry properties.

(I) If <j>{x') satisfies <j>{-x') = 4>*(x'), then replacing x' by -x' in (6.112) andmultiplying it by 02 from left, we can get

{lb ^n(~x'K + ^ [^"(-z')] = Ana3 [<x2i/>n(-z')] •

Since cr2ipn(—x') is also an eigenfunction associated with An, its behavior resemblesthat of ij)n{x') in the asymptotic region, i.e., a^tpni—x') -» 0 as |a;'| -> oo, Thus ipn

has the following symmetry

<r2rl>n(-x') = 6rl>n(x'), or ^n{-x') = S<72^n{x'), (8 = ±1).

Therefore, if <j)(—x') = —<j>*(x'), then tpn(x') satisfies the symmetry propertyipn(-x') — (7iV>n(z') with S = ± 1 . This can be easily verified by replacing <TIwith CT2 in the above derivations.

(II) If 4>(x') is a symmetric (or antisymmetric) function of x', i.e., <j){-x') =±^>(a:'), then ?/>^s'(a:') = a\^{—x') is the eigenfunction belonging to the eigenvalue—\*n, and ij>n {x') = CT2^(-X') is the eigenfunction belonging to the eigenvalueAn. The suffix s {ox a) to the eigenfunction ip'n indicates that (j> is symmetric (orantisymmetric). Since fi(—x') = 4>{x'), replacing x' with —x' in (6.112) and takingcomplex conjugate, we get

i£; [*!#;(-*')] + *(^) ki<(-^)] = -A>3 [a^^-x')] •

Compared with (6.112), the above equation implies that — A£ is also an eigenvalueand the associated eigenfunction ip'n (xf) is oi$*(—i'), with an arbitrary constant.For <j)(—x') = —<f>(x'), the same conclusion is obtained by replacing &i with <?2 inthe above derivations.

These symmetry properties are useful in providing a general view of the solutionof (4.40). As it is known, the real part of the eigenvalue, £n, corresponds to thevelocity of a soliton and the imaginary part, rjn, the amplitude. Then, if <j>(x', t'),whose initial value has the symmetry <j>(x',t' = 0) = ±<^(—x',t' = 0), breaks intoa series of solutions, the decay is bisymmetric, corresponding to the eigenvalues An

and —Xn-If <j>(x') is real, the above symmetry property yields

^H-J) = ffi [-6a2rn(-x')] = 5o^{x%

i>ia\-x') = a2 [ScrM-x')} = -Sa^Hx'),


i.e., ipn(x') has the same parity as ipn(x'), while ip'Jta\x') has the opposite one.When (/>(-x') = ~(p(x'), and An is pure imaginary (An = -A*), the eigenvaluescorresponding to the positive and negative parity eigenfunctions degenerate.

(III) If 4>{x') is real but not antisymmetric, then the eigenvalue An is pure imag-inary, i.e., 3?(An) = 0. Prom (6.112) and its Hermitian conjugate, Satsuma andYajima found that

9J(An)(n|a2|n> = (nm<t>(x')}a3\n), (6.115)

with

(m|<72|n> = / iPln<T2ipndx', (6.116)

where [$,<TI] = 2iQ($)<73 was used. From (6.115), we see that Jt(An) vanishes if 4>is real and (n|<72|n) ^ 0. When </> is a real and an antisymmetric function of x',symmetry property (I) gives

/•OO

(n\a2\n) = d2 / ip+(-x')tTia2crixl)n(-x')dx' = -(n\a2\n).J—oo

Thus (n\a2\n) = 0.(IV) If the initial value takes the form of <fi = elvx R(x'), where R(x') is a real

but not an antisymmetric function of x', all the eigenvalues have the common realpart, —v/2. This can be easily shown by the Galilei transformation. In fact, when<j>(x',t' = 0) = elvx R(x'), the solution does not decay into a series of solitonsmoving with different velocities, but forms a bound state. In this case, the realparts are common to all the eigenvalues, i.e., the relative velocities of the solitonsvanish.

(V) If 0 is a real non-antisymmetric function of x', it can be shown that

rn(x') = idcT3ipn(x'), (6.117)

where 6 = ±1. Because 5ft(An) = 0, from the complex conjugate of (6.112), onecan get ipn(x) a ozipn(x). Substituting (6.117) into the normalization condition(6.114), one then has 8 = ±1. If the eigenvalue of (6.106) is real, i.e., A = £ is real,then

{lt> + = ^ (6-118)

and the adjoint function of ip, ip = ia2ip*", is also a solution of (6.118) i.e.,

di>i— + $ip = fr3i>-

From this and (6.118), Satsuma and Yajima obtained the following

A(^+V) = ±$++) = ^«>+V0 = ±m) = 0. (6.119)


Using the above boundary conditions, they found that the solutions of (6.106),xpi(x',£), i>2{x',0) a n d ifa(x',Q satisfy the following relations

Prom ipi = a(^)%p2 + HOV^, we get a = t/>^Vi &nd & = V'Ji/'i, where

^l(x''e) = (o)e"^''as ar' = —oo and

V'2(x',O=(°)e+^',

as x' = oo.As it is pointed out earlier, if real (not antisymmetric) initial value is consid-

ered, the microscopic particle does not decay into moving solitons, but forms abound state of solitons pulsating with the proper frequency. Satsuma and Yajimadeveloped a perturbation approach to investigate the conditions for the solutionsto evolve and decay into moving solitons.

If the wave function cf> in (6.106) undergoes a small change, i.e., <f> -> </>' = 4>+A<f>,the corresponding change in $ is given by

An and ipn change as An + AAn and tpn + Aipn, respectively. To the first order inthe variation, equation (6.112) becomes

r d x 7 + ^ ~ Xn(T^\ A ^ " + ( A * ~ AAn<73)</>n = 0.

Multiplying the above equation by Vn°2 from left and integrating with respect tox' over (—oo, oo), we get

nOO

AAn = -i VnÂ^Vnda:'J — OO

/•OO /-OO

= - / ^(Aâs^ndx'+i i>lZ{&4>)^ndx'.J~ OO J~ OO

If 4> is a real and non-antisymmetric function of x', equation (6.117) holds and

AAn = 6(n\^{A<j>)a3\n) + W(n|R(A0)|n). (6.120)

Equation (6.120) indicates that if (n|3(A</>)(T3|n) ^ 0, the perturbation A<f> makesthe real part of the eigenvalue finite. That is, for the initial value, 4>{x') + A<f>(x'),


the solution of the nonlinear Schrodinger equation (4.40) breaks up into movingsolitons with velocity 23?(AAn). If tf> is a real and is either a symmetric or anantisymmetric function of x', the above symmetry properties of eigenvalues of thenonlinear Schrodinger equation lead to

(n|3(A#c')V3|n) = -(n\$s(A<j>{-x))a3\n).

Therefore, if %(A<f>) is a symmetric function, (n|3(A0)<73|n) vanishes, i.e.,R(AXn) — 0, and the soliton bound state does not resolve into moving solitonseven in the presence of the perturbation A<f>.

Satsuma and Yajima also obtained the shifts of the eigenvalues of (6.106) underthe double-humped initial values, <f>(x',t' = 0) = <j>o{x' - x'o) + et6°<j>o(x' + x'o),where <p0 is a real and symmetric function of a;', x'o and <j)0 are real. The shifts ofthe eigenvalues were finally written as

AA± = 8 fanfloMrsMz' + 2a£)|n) Tsin (j) <n|<T3ô(i')e2x«(<l/'lB')|f»>] +

iS \cos60(n\<j>o(x' + 2x'0)\n) ± cos (j) {n\<j>0{x')e2x'^dldx^\n)\ . (6.121)

where

-Jcos {^\ (n\Mx')e2x'o{d/dx)\n)-i6Sin (^) {n\az<f,0(x')e2x^dld^\n)

= ^2 CT2$2î d,X '= / Vl 0-2*1^2 dx \J— oo J—oo

-5cos(00)(n\<t>o(x' + 2x'0)\n) + i6 sin eo(n\a34>o(x' + 2x'0)\n)rOO pOO

= I tf{n)Ta2*2tf(n)dx' = I ^n)Ta2^2'{n)dx'.

J—oo J—oo

Here

$(O=*l(z') + $2(*'),

and

$i(a;') = ai<t>0(x' - x'o),

$2(x1) = [cos(0o)"'i - sm(0o)a2]<l>o(x' + x'o).

The corresponding eigenvalue equation is given by


The eigenfunction <f>'n{x') satisfies the following symmetry and orthogonality re-quirements

CtC-z') = ±6 [cos (jfj a2 + sin ( | ) CTl] ^±(x'), S = ±1,

J—oo

when 6Q = 0, <j>(x') is real and symmetric, A An ' is pure imaginary. When 0O = n,<f>(x') is real and antisymmetric, AXh, ' is real,

» [AA±(0o = vr)] = TS(n\a3Mx')e2x'o{d/dxl)\n),

3 [AA±(0o = TT)] = -S(n \<f>Q{x' + 24 ) | n). (6.122)

Thus the solution of the nonlinear Schrodinger equation (4.40) decays into pairedsolitons and each pair consists of solitons with equal amplitude and moving in theopposite directions with the same speed. For arbitrary 6'0, we can see from (6.122)that the solution of (4.40) breaks up into an even number of moving solitons withdifferent speeds and amplitudes.

6.6 Microscopic Causality in Linear and Nonlinear Quantum Me-chanics

The microscopic causality is an interesting problem in quantum mechanics, andunderstanding the difference in microscopic causalities in the linear quantum me-chanics and the nonlinear quantum mechanics is important for a better understand-ing of the essential features of the nonlinear quantum mechanics. The microscopiccausality will be discussed in this section.

In classical physics, the causal relations, which crystallizes the intuitive notionthat a cause must precedes its effects, are often expressed in the Green's functionof a theory or by the Kramers-Kronig type of dispersion relations. These forms canbe easily generalized to the linear quantum mechanics. In fact, the commutatorof two operators or its matrix form contains causality information, and for thisreason, it is referred to as an interference function, and the relations are calledmicrocausal. Using solutions of the dynamic equation represented by the creationand annihilation operators, Burt expressed the microscopic causality in terms ofprobability amplitudes.

The microcauslity concerns two processes or events that must be classified intotwo independent data sets if the events are separated by a space-like interval. Asshown in Fig. 6.3, two such events may appear to have quite different temporal orien-tations in various coordinate systems. Thus, in the absence of superlumimal signals,the description of the events must be independent. In the canonical quantum the-ory, the information concerning the interference of the two events is described by


Fig. 6.3 Minkowski diagram illustrating relativity of temporal order of two events separated bya space-like interval in three coordinate systems related by Lorentz transformations (see book byBurt).

the commutator of the operators denoting the two events. This commutator givesthe interference when the sequence of events is reversed and the difference takenbetween the two orderings. The representation in terms of probability amplitudesresults in an interference function which can be built by the creation and annihila-tion operators. The sequence of events to be considered consists of the creation andannihilation of microscopic particles from the vacuum at two points in space-time.In the reversed sequence the points in space-time are exchanged. The difference be-tween the amplitudes for the two sequences defines the interference function. Usingthe knowledge of quantum field theory, the amplitude for the creation of a particlewith positive energy from the vacuum at x and its subsequent annihilation into thevacuum is represented by

k,p,€

= ^(o i4 + ) 4 + ) t i ° ) ( £ ) 2 w P w ^ 2 )" 1 / 2 e * i "^' s ' (6-123)

where V is the volume of the system. Similarly, the sequence for a particle with annegative energy is

Attp(y,i) = ^(0|4-)4-)t|0)(D2o;pa;fcy2)-1/2/eii •*-**. (6.124)

Here At and A^ are the creation and annihilation operators of the particle withwave vector k, respectively, (+) and (-) indicate the positive and negative energystates, respectively, the field operators 0 of the particles is represented by

where uik is the frequency of the particle, k — (tJk,k), w\ — k2 + m2, x — (x, -ict),\Ar, At 1 = 6tt,, D is a normalization constant, Ay'elk'x is a creation operator.Thus, the probability amplitude that a positive energy particle with momentum k,


created from the vacuum at position x, is found in the intermediate state \e) is

Similarly, the probability amplitude that a negative energy particle annihilated fromthe vacuum at position x with momentum k is found in the state |e) is

Then, the amplitude for annihilation of a negative energy particle from the vacuumat y, followed by the creation (to fill the negative energy state) of a negative energyparticle at x (the time sequency is reversed) is p\~\x)pC'{y). Thus we get therepresentations (6.123) and (6.124). Since the two amplitudes have equal weight,total amplitude is given by the arithmetic average,

Aamp(y, x) = \ [4J)p(j/,x) + A[-)p(y, x)] . (6.125)

However, from the definition of the vacuum state, it is required that

AJ,~)t|O)=O. (6.126)

Therefore,

AAmp(y,x) = ±A{+}p(y,x). (6.127)

The amplitude of the reversing sequence can be calculated following the samesteps. The total interference amplitude is then given by

A>lamp(y, x) = Aamp(y, x) - Aamp(x, y) (6.128)

= \ ] T ( 0 | 4 + ) 4 + ) t | 0 ) (e**-**"* - e**-"-*) {D2cjpu;kV2)-1/2 •

For the neutral, spin zero system of particle, using the commutator [/l^^lt] =finkrip and converting the sum into an integral using

the total interference amplitude can be obtained

AA (v x\ - 1 f — [P«*-(*-S) _ p»*-(S-s)l Cfi loq-iAAamp(2/,x; - 2 ( 2 7 r ) 3 y w £ ) [e e j . (b.129)

If the constant D is set to unity, this becomes the interference function dennedin the canonical formalism by Bjorken et al. and Bogoliubov et al. (see book by


Burt)

AA&mp(y, i) = A(y - x) = ^-^ J ^ [>»-*> - e^'*"*)] . (6.130)

This Lorentz invariant function clearly vanishes for space-like intervals. If (y — x)2 <0, a Lorentz transformation can always be found which takes us to a coordinatesystem in which XQ = y0. Replacing k by — k in the second term does not changethe integrand. Hence

A(y - x) = 0, (y - x)2 < 0. (6.131)

Properties of this function were studied by Bjorken et al. and Bogoliubov et al. Thestatement of microscopic causality in terms of probability amplitudes is equivalentto the usual canonical relation for free fields in the linear quantum mechanicaltheory. This idea was generalized to the self-interacting fields in nonlinear quantummechanics by Burt.

As in the establishment of propagator including persistently interacting fields inthe nonlinear quantum mechanics, the interference function describing microcausal-ity with interactions is obtained by employing the superposition principle of quan-tum theory. The algorithm developed in the linear quantum mechanics is utilizedfor the persistently interacting field operators since the latter can be interpretedas creation or annihilation operators. For the representation of the interferencefunction in (6.128), Burt generalized it to the following form

AAamp(y,*rrS = Amp(y,x)perS - A&mp(x, y)*™

= \ E w i p t t o ^ W - 4+)(*)^+)(j')ti0>' (6-132)k,q

where <f>k (x) are persistently interacting fields and

A&mp(y,xr™ = \A{$p(y,xr™ = \ £<0|4+)(y)^+W|0>. (6.133)

Using the general expressions for persistently interacting fields given in section5.3, the current / is denoted by a series containing only positive powers of thecreation or annihilation operators,

<t>f(x) = f>n(Ai;m2;p)C/f >(i)d»+«')>

71=0

where <j>n(\i;m2;p) contains all the coupling constant dependence and depends on

the parameters m2, p and po, the p and po are indeces of interaction which arerelated to the form of the self-interaction current. The interference function can be


written as

AAamp(y,x)Pers = i J2 ^n(Ai;m2;p)0s(Ai;m2;p)x (6.134)

k,q,n,s

{(01 [u?Hy)]pn+P° [u?>t(i)p° _ [uf\i)]pn+P0 [u?»(y)]PS+P° |o>} •Using

UJ^tf) = AWeîDuV)-1'2,

where D is a constant, V is the volume of the system, u = Vk2 + m2, and

with

equation (6.134) becomes

AAamp(£,*)Pers = i 5 ] «^n(Ai;m2;p)</)s(Ai;m

2;p)(£»a;V)-P("+s)/2-''ox

{/Q|aP«+PoatP«+Po IQ\ re-t(pn+po)fc-y+i(ps+po)g-ik 9 I

_e-i{pn+po)k-i+i(ps+po)q-y] \

= \YJ{<l>n{\urn2-,p)2{Du1V)-vn-P<>{im + p0)\xk,n

\e-i(pn+p0)~k-(y-i) _ e-i(pn+p0)*(i-y)l 1 (6.135)

If we consider events for which the interval (y — x)2 is space-like, a Lorentztransformation can be performed to the system in which the events are simultaneous.Prom (6.135), Burt found that

AAAmp(y • x)Pers = \'£<l>n(Xi;m2-,p)2(DwV)-*n-'o(pn + p0)\ xfc,n

fe-i{pn+po)k(y-x)_e-i(pn+po)'k(x-y)\ (6.136)

Replacing k by —k in the second term, the total expression vanishes. Since theinterference function is Lorentz invariant, the result is independent of the coordinatesystems. Thus, Burt obtained that

AAamp(y, x)pers = 0, (x- y)2 < 0. (6.137)


Therefore, the interference between the creation and annihilation operations of thepersistently interacting fields vanishes when the processes are separated by space-like intervals. The principle of microscopic causality, stated in terms of probabilityamplitudes, is satisfied by the persistently interacting fields.

As is the case for a propagator, the causal function AAamp(y,x)pers typicallyreduces to a sum of terms of the form

AAamp(j/, x)*ers = A(y -£) + APers(£ - x; A;), (6.138)

where A(y - x) is the usual free particle interference function obtained by Bjorkenand Bogoliubov, and APers(y - x; A,) contains the interaction effects. Thus, for the\(f>z current or (/>4-field equation, Burt found that

APers(z;Xi) = \J2 Vn + iy.iDajV)-2"-1 (-^-) " {c-<(2n+i)t-« _ ei(2n+i)*.« jk,n=l \ m /

Due to the presence of the factor (DuV)~2n~1, this function is less singular thanA(z). The effect of the interaction is to smear the singularity into a smooth func-tion with a tail characterized by the Compton wavelength of the higher mass con-tributions and by strength proportional to the coupling constant. Therefore, thereare only somewhat differences between the representations of the causalities in thenonlinear quantum mechanics and linear quantum mechanics, but their results arebasically the same.

Bibliography

Asano, N., Taniuti, T. and Yajima, N. (1969). J. Math. Phys. 10 2020.Bjoken, J. D. and Drell, S. D. (1964). Relativistic quantum mechanics, McGraw-Hill, New

York.Bjoken, J. D. and Drell, S. D. (1980). Relativistic quantum fields, McGraw-Hill, New York.Bogoliubov, N. N. and Shirkor, H. (1959). Introduction to the theory of quantized fields,

Interscience, New York.Bogoliubov, N. N. (1962). Problems of a dynamical theory in statistical physics, North-

Holland, Amsterdam.Burt, P. B. (1981). Quantum mechanics and nonlinear waves, Harwood Academic Pub-

lishers, New York.Carr, J. and Eilbeck, J. C. (1985). Phys. Lett. A 109 201.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1996). Chin. Phys. Lett. 13 660.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1997). Chin. J. Atom. Mol. Phys. 14 393; Chin.



Phys. 11 240; Chin. J. Comput. Phys. 16 346; Commun. Theor. Phys. 31 169.Dodd, R. K., Eiloeck, J. C, Gibbon, J. D. and Morris, H. C. (1984). Solitons and nonlinear

wave equations, Academic Press, London.


Drazin, P. G. and Johnson, R. S. (1989). Solitons, an introduction, Cambridge Univ. Press,Cambridge.

Eilbeck, J. C, Lomdahl, P. S. and Scott, A. C. (1984). Phys. Rev. B 30 4703.Eilbeck, J. C, Lomdahl, P. S. and Scott, A. C. (1985). Physica D16 318.Finkelstein, D. (1966). J. Math. Phys. 7 280.Goldstone, J. and Jackiw, R. (1975). Phys. Rev. D 11 1486.Guo, Bai-lin and Pang Xiao-feng, (1987). Solitons, Chin. Science Press, Beijing.Heisenberg, W. (1974). Across the frontiers, Harper and Row.Kartner, F. X. and Boivin, L. (1996). Phys. Rev. A 53 454.Kleinert, H., Schulte-Frohlinde, V. (2001). Critical properties of ^-theories, World Scien-

tific, Singapore.Kolk, W. R. (1992). Nonlinear system dynamics, Reinhold.Korepin, V. E., Kulish, P. P. and Faddeev, L. D. (1975). JETP Lett. 21 138.Lai, Y. and Haus, H. A. (1989). Phys. Rev. A 40 844 and 854.Lieb, E. (1963). Phys. Rev. 130 1616.Lieb, E. and Liniger, W. (1963). Phys. Rev. 130 2605.London, R. (1986). The quantum theory of light, 2nd ed., Oxford University Press, Oxford.Majernikov, E. and Pang Xiao-feng (1997). Phys. Lett. A 230 89.Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1.Makhankov, V. G., Makhaidiani, N. V. and Pashaev, O. K. (1981). Phys. Lett. A 81 161.Manakov, S. (1974). Sov. Phys. JETP 38 248.Pang, X. F. (1994). Acta Phys. Sin. 43 1987.Pang, X. F. (1995). Chin. J. Phys. Chem. 12 1062.Pang, X. F. and Chen, X. R. (2000). Chin. Phys. 9 108.Pang, X. F. and Chen, X. R. (2001). Commun. Theor. Phys. 35 323; J. Phys. Chem. Solids

62 793.Pang, X. F. and Chen, X. R. (2002). Chin. Phys. Lett. 19 1096; Commun. Theor. Phys.

37 715; Phys. Stat. Sol. (b) 229 1397.Pang, Xiao-feng (1994). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing.Pang, Xiao-feng (1999). Acta Physica Sin. 8 598; Chin. Phys. Lett. 16 129; Phys. Lett. A

259 466.Pang, Xiao-feng (2000). J. Phys. Chem. Solid 61 701; Phys. Stat. Sol. (b) 217 887.Pang, Xiao-feng (2001). J. Phys. Chem. Solid 62 491.Pang, Xiao-feng (2002). Inter. J. Mod. Phys. B16 4783.Perring, J. K. and Skyrme, T. H. (1962). Nucl. Phys. 31 550.Potter, J. (1970). Quantum mechamics, North-Holland, Amsterdam.Rubinstein, J. (1970). J. Math. Phys. 11 258.Saji, R. and Konno, H. (1998). J. Phys. Soc. Japan 67 361.Satsuma, J. and Yajima, N. (1974). Prog. Theor. Phys. (Supp) 55 284.Scott, A. C. (1982). Phys. Rep. 217 1.Scott, A. C. (1985). Phil. Trans. Roy. Soc. Lond. A315 423.Scott, A. C. and Eilbeck, J. C. (1986). Chem. Phys. Lett. 132 23; Phys. Lett. A 119 60.Scott, A. C, Bernstein, L. and Eilbeck, J. C. (1989). J. Biol. Phys. 17 1.Scott, A. C, Lomdahl, D. S. and Eilbeck, J. C. (1985). Chem. Phys. Lett. 113 29.Strang, G. (1975). Linear algebra and its applications, Academic Press, New York.Tomboulis, E. and Woo, G. (1976). Nucl. Phys. B 107 221.Zakharov, V. E. and Shabat, A. B. (1972). Sov. Phys.-JETP 34 62.Zakharov, V. E. and Shabat, A. B. (1973). Sov. Phys.-JETP 37 823.

Chapter 7

Problem Solving in Nonlinear QuantumMechanics

In Chapters 3 - 6, we have established the fundamental principles and theory ofthe nonlinear quantum mechanics. The next important step is to discuss methodsfor solving the dynamic equations. In this Chapter, some common approaches forsolving these equations will be discussed.

7.1 Overview of Methods for Solving Nonlinear Quantum Mechan-ics Problems

From earlier discussions, we know clearly that problems in nonlinear quantum phe-nomena are very complicated. This complexity is due to the different mechanismsof nonlinear interactions. Therefore, understanding the mechanism and property ofnonlinear interaction is essential for establishing correct dynamic equations and forsolving these equations. The dynamic equations, (3.2) - (3.5), cannot be solved un-less the exact form of the nonlinear interaction, b, is known. Therefore, the first stepone should take in solving a nonlinear quantum mechanics problem is to carefullystudy the nonlinear quantum phenomenon and its properties, look into the mecha-nism of the nonlinear interaction, and come up with an appropriate model for thenonlinear interaction. Based on these understandings, the Lagrangian, Hamiltonianfunction, and operators of the system can be established. Finally, the dynamicalequations of the microscopic particles can be obtained from the corresponding clas-sical quantities, or Lagrangian, using the Euler-Lagrange equation or the Hamiltonequation, or from the quantum Hamiltonian using the Schrodinger equation or theHeisenberg equation in the second quantization representation in which the Hamil-tonian is given in terms of creation and annihilation operators of the microscopicparticle. Once the dynamical equation is known, we can solve the equation to obtainits the solution and study the nonlinear-phenomena. Related systems can also bestudied based on the fundamental dynamic equation. For example, external fieldsor forces can be considered and included in the dynamical equations. This is ageneral approach for solving nonlinear quantum mechanics problems. Of course,certain problems can be easily solved using some special techniques.

329


The commonly used methods for obtaining soliton solutions of nonlinear dynam-ical equations in nonlinear quantum mechanics are summarized in the following.

7.1.1 Inverse scattering method

The inverse scattering method was first proposed by Gardner, Green, Kruskal andMiura (GGKM) in 1967, for solving the KdV equation which connects a nonlineardynamic equation to the linear Schrodinger equation

through the GGKM transformation

where <j> satisfies the nonlinear dynamic equation. In 1968, Lax gave a more generalformulation of the inverse scattering method. The nonlinear Schrodinger equationwas solved by Zakharov and Shabat in 1971 using the inverse scattering methodwhich was discussed in Section 4.3. In 1973, the Sine-Gordon equation was solvedby Ablowitz et al..

According to Lax, given a nonlinear equation

<Pt = K(<j>), 4> = (x,t),

the inverse scattering method for solving the equation consists of the following threesteps.

(1) Find operators L and B, which depend on the solution <j>, such that iLt = BL -LB, as given in (3.20) and (3.21).

(2) The scattering operator, L, satisfies its eigenvalue equation, Lip — \<p.(3) The time dependence of the scattered wave is determined by i<pt = B(p.

Instead of directly solving the nonlinear equation to obtain 4>(x, t) for the giveninitial condition <p(x, 0), the inverse scattering method proceeds in the above threesteps and obtains solution of the linear integral equations. The major difficulty ofthe inverse scattering method is the lack of a systematic approach for finding theoperators L and B, even if they exist.

7.1.2 Bdcklund transformation

Given d<j> = Pdx + Qdt, then the integrability of the corresponding first-orderdifferential equations

j>x = P, <h = Q (7.1)

Problem Solving in Nonlinear Quantum Mechanics 331

requires Pt = Qx. The Backlund transformation consists of the following threesteps.

(1) Find P and Q, as functions of 4>.(2) Choose a trial function, <j>0.(3) Use fax = P(fa,(f>0) and fat = Q(fa,<f>o) to find a new solution fa.

The Backlund transformation method enables one to find a new solution from agiven one. When applied repeatedly, the Backlund transformation can be usedto obtain the breather and the iV-soliton solutions of the Sine-Gordon equation.It can also be applied to the nonlinear Schrodinger equation. The correspondingBacklund transformation can be (3.30) or (3.43). The difficulty for the Backlundtransformation method is to find the functions P and Q.

7.1.3 Hirota method

The Hirota method changes the independent variables and rewrites the originaldynamic equation into the form of

F(Z?™Dtn) = 0,

where D is an operator. Analytic solutions are then obtained using perturbationmethod. Here,

The Hirota method is widely used.

7.1.4 Function and variable transformations

Quite often a complicated equation can be transformed into a simpler or a standardequation which is readily solved. The forms of transformation vary and depend onthe problem and equation. The following are some commonly used transformations.

7.1.4.1 Function transformation

As an example, we consider the transformation

<j> = J-peie^t\ (7.3)

Using this transformation, the nonlinear Schrodinger equation (3.2) with V{f,t) =0 = A(<t>) can be written as two equations (3.110) and (3.111), which can be readilysolved.

(7.2)


7.1.4.2 Variable transformation and characteristic line

The following variable transformation is often used.

£ = a; — vt, r] = x + vt. (7.4)

For example, the one-dimensional linear fluid equation,

at ox

can be written as

^ = 0 , 4> = f(t) = f{x-vt)

under this transformation. The equation dx/dt = v or dx/v — dt/1 — d<p/O iscalled the characteristic equation of the fluid equation, dx/dt is the characteristicdirection, £ = x — vt = constant is referred to as the characteristic line of theequation. The value of <j> on the characteristic lines is constant, and <\> is called aRiemann invariance of the fluid equation.

For the one-dimensional nonlinear fluid equation,

-xr + 0-^- = 0 ,ot ox

its characteristic line is dx/dt = u(x,t). The variable transformation in this case is

£ = x - u{x, t)t, 7j = x + u(x, t)t. (7.5)

Using (7.5), the above equation becomes

£ = 0.dr)

Here u(x, t) = it = constant.

7.1.4.3 Other variable transformations

Let

Z = x-y, rj = x + y,

or_ x-t _ x + t

Then the Liouville equation

Ox2 dy2 ~~ '


where A is a constant, can be written as

drjdt 4 '

Under a similar transformation,

c _ x — t _ x + t

the Sine-Gordon equation (in natural unit system)

_ - _ = s l n 0 , (7.6)

becomes

d2<f> .

Many other function and variable transformations are used to solve the dynamicequations but we cannot list all of them here.

7.1.4.4 Self-similarity transformation

Assume that <j>(x, t) is a solution of the nonlinear equation, P<f> — 0. If a;, t and4>(x, i) are replaced by

x' = x + eX(x,t,(p) + O(e2),

t1 = t + eT(x,t,4>) + O{e2), (7.7)

4>' = <l> + e$(x,t,<l>) + O(e2),

respectively, where e is a small parameter. The transformation (7.7) is called aninfinitesimal transformation, where X, T and $ are coefficients of the first orderterms of x', i' and </>', respectively. If we require <j)'(x',i') to satisfy the sameequation, it can be shown that

84> d4> a . dx dt d0

Because these equations are invariant under (7.7), and X, T and $ are indepen-dent, we can assume T = 1. We then get f(x, t) = const and g(x, t, <f>) = const fromdx/dt — X and d<p/dt = $, respectively. If the similarity variables are chosen as £and </>, then £ = f(x, t) and $ = g(x, t, cj>) are called a self-similarity transformation,the solution <f>(x, t) obtained from 4> — g(%, t, 4>) is called a self-sirnilarity solution.The self-similarity transformation £ = f(x, t) and <j> = g(x, t, <f>) are often expressedas £ — a(t)x, </>(£) = (f)(x,t)//3(t), and the self-similarity solution is then given by<Kx,t) = 0(t)fa{t)x).


For the linear heat conduction equation,

cty _ d2<j>

where 7 is a constant, its self-similarity transformation is,

<-7f * = fand its solution is

Under this transformation, the equation becomes

-. y 4. Z _ L /v — n

V 2 2 d£ 7 ~For the Sine-Gordon equation

its self-resemblant transformation is

C = £ j , <£ = <£,

and the equation becomes

under this transformation. The solution of this equation can be easily obtained.

7.1.4.5 Galilei transformation

Function transformation is a general approach for solving dynamic equations. As anexample, we consider the nonlinear Schrodinger equation (3.2) with V(x) = A{4) =0, and 6 = 1 . If we let

, [2^ tX = X V ^ ' and * = n'

its solution is of the form/ A2t'\

<j>(x',t') = ^sech(Ax')exp f - t ^ " J (7-8)

Since this nonlinear Schrodinger equation is invariant under the Galilei transforma-tion (6.108), its solution can be written as

[ C,,2 _ j 2 w n-ivx' + i± 2 ~ | " ^


Let v = 2£, and A — 2T], then it reduces to the soliton solution (4.56) which wasobtained by the inverse scattering method.

7.1.4.6 Traveling-wave method

If we can write the solution of an equation of motion in the form of <j)(x, t) = </>(£)>with £ = x — vt, it is called a traveling-wave solution. This method is widely usedin wave mechanics and nonlinear quantum mechanics.

7.1.4.7 Perturbation method

In reality, due to the existence of boundaries, defects, impurities, dissipation, ex-ternal fields, etc., a system usually experiences some perturbation. As a matter offact, when the soliton behavior of microscopic particles in the system is probed,the measurement itself is bound to alter the state of the system to some extent.Thermal excitation at finite temperature is another form of perturbation that nosystem can avoid. In fact, the appearance of microscopic particles (solitons) itselfimplies some form of internal excitation has occured.

When such perturbations are relatively weak, they may be treated by pertur-bation methods. The main spirit of soliton perturbation was illustrated in Chapter3.

7.1.4.8 Variational method

The variational method will be discussed later in this Chapter.

7.1.4.9 Numerical method

When analytic solutions cannot be found and perturbation methods are inapplica-ble, one can always solve the nonlinear equation by numerical methods. Sometimesthe numerical solutions may lead to a correct guess of their analytic forms. There aremany numerical approaches for solving differential equation such as Monte-Carlo,Runge-Kutta, finite elements etc. With the rapid increase in computing power inthe last decades, numerical simulations plays a more and more important role inpractical calculations.

7.1.4.10 Experimental simulation

Some dynamic equations may be simulated by mechanical models or electric cir-cuits. For example, the Sine-Gordon equation was simulated by a chain of torsionalpendula. The overdamped Sine-Gordon equation <f>xx = +(j>u + sin </> was studiedby submerging the chain of pendula in water and the driven Sine-Gordon equationwas simulated using long Josephson junctions and using mechanical models.


7.2 Traveling-Wave Methods

7.2.1 Nonlinear Schrodinger equation

For the nonlinear Schrodinger equation given in (3.2) with V(x, t) = A(cf>) = 0, atraveling-wave solution can be assumed

<j>{x,t) = <j>{^kx'-ut'\

where

/• i J.1 i / 2 m , , t£ = * - < J = yj-Wx, t' = -.

In terms of the new variables, equation (3.2) can be written as

^ p + (w _ k>m) - if = o.The soliton solution of this equation can be easily obtained when b > 0 and 7 =w - k2 > 0,

*(M) = y^sech[V7(^ - &)]e'<fa'-wt<'. (7.10)

On the other hand, when b < 0 and 7 < 0, the solution is

4>(x,t) = y^tanh nÊ^-fc) j e ^ ' - O . (7.n)

If we assume the traveling-wave solution is of the form

<j>(x',t')=4>(OeiQt'+iQx>, (7.12)

then (3.2) with 6 = 2, V(x) = A(4>) = 0, becomes

0«(O + i(2Q - v)fa - 0(0 (n + g2) + 203(o = 0. (7.13)

If the complex coefficient of 0$(O vanishes, then <5 = u/2. Furthermore, fromA = Q2 + fi, we get that ft = -« 2 /4 + A. Finally from (7.13), we obtain

<fo - A0 + 203 = 0.

This equation can be integrated, which results in

fa)2 = D + A<?-tl>\

where D is a integral constant. The solution (j) is then obtained by inverting anelliptic integral,

r , *> =±e


Let

P{4») = (ft - 02)(02 - ft),

we have,

-±={K(k)-F(4>,k)} = ±$,

where F(<f>, k) is the incomplete elliptic integral, and

Using these and j3\t2 = 0f 2 > t n e n w e n a v e

0(O=0i{l-[(l-|)sn2(C,*)]}12.

When D -» 0, </>i -> <f>o, k -> 1, </> -> </>osechô^, where <>o = V^4, the solitonsolution of this nonlinear Schrodinger equation becomes

4>,(x,t) = VAsech [y/A(x' - vt')] exp ft (~\v2t' + vx' + AA] . (7.14)

The corresponding canonical form of the one-soliton solution has been given in (7.9)and (4.56).

7.2.2 Sine-Gordon equation

Using the traveling-wave method in natural unit system, | = x—vt, the Sine-Gordonequation,

<t>tt ~ vl<t>xx + hi sin 0 = 0, (7.15)

can be written as

(vi-vl)£g+hlsm<f> = 0. (7.16)

If v2 > VQ, (7.16) can be written as

^ + / ? 2 sin 0 = 0, (7.17)<%2

with

P ~ v2-vl


Equation (7.17) resembles the equation of motion of a undamped pendulum, andcan be separated into the following two equations

| = .. £--*>-.* «,18)This gives

d$ _ /32 sin <j)

Thus we can get

l$2 + p2{l-cos4>) = A,

where A is a constant of integration. Using

1 - cos <j> = 2 sin2 ^

and (7.18) and letting A = 2/32k2, the above equation can be written as

(|)'-"I*1--"(!)]• (719)

If 0 < k2 = A/202 < 1, the solution of (7.19) is given by

sin (£j = ±ksn[f3(£ - | 0 ) , fc]. (7.20)

li k —> 1, equation (7.20) is replaced by

sinf0=tanh[±/3(|-|o)],

or

[l-sin((A/2)J \4 4J

Thus the kink solution of the Sine-Gordon equation (7.15) is finally given by

4>± = -v + 4 tan"1 e

± / 3 («-«o ) . (7.21)

When v2 < VQ, equation (7.19) is replaced by

(3)'-«-K!M.where

oil _ hQP — ~o o-

vl-v2


The kink solution of the Sine-Gordon equation (7.15) in the case of k' -> 1 andk'2 — 1 — k2 becomes

0 = 4 tan"1 {eT/3'(«-«°>} . (7.22)

Using the following dimensionless variables, t' = hot, x' = 5x, 6 = ho/vo,equation (7.15) can be written as

<t>x'x' ~ <t>t't' = s i n ^ i . (7.23)

According to the properties of (7.21) and (7.22), the solution of (7.23) is of the form

^ = 4 t a n " 1 [ w l - (7>24)

In terms of Y and T, equation (7.23) becomes

(Y2 + T2) {^- + fr1) - 2[(Y,02 + (Tv)2} = T2 - Y2. (7.25)

Differentiating (7.25) with respect to t' and x\ and then dividing the two resultingequations by YYx* and TTf, we can get

1 (Yx.x,\ 1 (Tt.t>\, . (79R.

where 4q is a constant. Integrating (7.26), the following equation can be obtained

^ - = 2qY2+ai, T^ = -2qT2 + a2, (7.27)

and

(Yx,)2 = qY4 + aiY

2 + du (Tt,)2 = -qT* + a2T

2 + d2, (7.28)

where a\, a2 and d\, d2 are some integral constants. Inserting (7.27) and (7.28)into (7.25), we can get ai - a2 = 1, and d\ + d2 = 0.

If we choose a2 = —a, d\ = d, then ai = 1 - a and d2 = d. When q = —1, andd = 0, equation (7.28) becomes

Y2, = -Y4 + (1 - a)Y2, T2, = T4 - aT2. (7.29)

If 0 < a < 1, integrating (7.29) and taking the integral constant to be zero, wecan get

jl— sech"1 (-JLJ) = ±x' = ±Sx,Vl - a Vvl-a/

isin-'(^)=±(' = ±M.


Thus

Y = y/l - asech(\/l - a Sx), — = ±—~ sin(y/ahot). (7.30)1 yd

Substituting (7.30) into (7.24), we get the breather solution of the Sine-Gordonequation (7.15)

(<f>\ , ll — a sin(\/ahot) , . .tan ^ = ± V . ,}-—' >, ( o < o < l ) . 7.31

V4/ V a cosh(vl - o,ox)

Following the same procedure, we can also obtain the traveling-wave soliton-solution of the 04-field equation in natural unit system

(j>tt-vl(j>xx+a'cl>-pl^=Q.

Since the procedure is the same, we only give the results here,

^±^tanh[^^(f-^ ,

when v2 > ug, a' > 0, 0' > 0 or v2 <vl,a< 0, /?' < 0, and

, , /ic7 [ / -a' - -^=±y—sech^^-^(e-eo)j,

when v2 > wg, a ' < 0, /3' < 0 or w2 < ug, a ' > 0, /3' > 0.

7.3 Inverse Scattering Method

We now determine the soliton solution of the nonlinear Schrodinger equation (4.40)with an initial pulse using the inverse scattering method mentioned earlier, followingthe approach of Zakharov and Shabat, in accordance with the Lax operator equation(3.20), where L and B are some linear operators with coefficients depending on thefunction <p(x, t) and its derivatives, as discussed in Section 4.3. In other words, theequation for <j>{x, t) (nonlinear Schrodinger equation) is a condition of compatibilityof auxiliary linear equations (3.20) and (3.21), where A is a spectral parameter(generally speaking, complex), and ip in (3.21) is often called a Jost function and isrepresented by

- G Owhere L is given in (3.22) for the nonlinear Schrodinger equation (3.2) with V =A = 0. Therefore, equation (3.21) can be rewritten in the form of Zakharov-Shabat


equation (4.42), where

b is the nonlinear coefficient in the nonlinear Schrodinger equation (4.40). Here thecharacteristic value, £ = £' + it). Equation (4.42) can be further written in the form

*!*<*' ~ y *i*< + *i ^ ' 2 + \q\2 - tf y ) = 0, (7.32)

*2 = - [ * ! » ' + ^ ' * i ] . (7-33)

According to the inverse scattering method, it is necessary to find the solutions ofthe system (7.32) which have the following asymptotic behavior for a given functionq(x') and real £ = £'

l(l)e-**', ifV->-oo,l i m b s ' ) ={ VVJ , (7.34)

[ o(O (JJ e - ^ ' - 6(n ( J J e^'-', if x' -> oo.The coefficients a(^') and b(£') are complex and satisfy

\a(a2 + Ha2 = i.

The quantity, l/a(^'), determines the transmission coefficient at a;' of a planewave incident on the potential q(x') except at x' w oo, while the ratio, R(£') =b(£')/a(£') determines its reflection coefficient. The functions in (7.34) continue tobe analytic into the upper half (rj > 0) of the complex plane of £. With the inversescattering method, we can obtain the values of a(£') and &(£') at time t' ^ 0 fromtheir values at t' = 0 using the relations

a(U')=a(0, b(C,t') = b(0e^2t'. (7.35)

If for certain values £j, a(£j) vanishes in the upper half of the complex £ plane,then the asymptotic behavior of (Q,x') is given by

((jW«i>'\ x' -oo,*&,*') = { V , n x (j = l,---,N). (7.36)

For r)j 7 0, *l!(Q,x') decreases asymptotically when |a;'| —> oo and therefore de-scribes bound states corresponding to complex Q given in (4.42) or in (7.32) -(7.33). Values of Q and Cj(t'), (j = 1,2, • • • , N), where

c,(O^(C^')[^]^=fe4^', (7.37)


and

form a set of "scattering data". Prom these data, Davydov obtained the followingauxiliary function

Pi*',?) = ± deRtf,t)e*'' + Y,cj(t')ei^'x>. (7.38)

z n J-°° j=i

Inserting (7.38) into the Gelfand-Levitan-Marchenko (GLM) equation

rOO /-OO

K(x',y') = F*(x'+y',t')- / dp / dzF*(p + y'1t!)F(p + z^)K(xf,e). (7.39)J-oo J-x'

the undetermined function

q(x',t') = -2K(x',x') (7.40)

and the wave function satisfying the nonlinear Schrodinger equation (4.40), withthe initial value <f){x' ,0), can be expressed in terms of the solution of (7.39), withthe help of the following

Vb

Using the inverse scattering method, Davydov obtained a soliton solution of thenonlinear Schrodinger equation (4.40) with the following initial data

4>(x', t' = 0) = l^be™*1 sech (^pj , (7.42)

where 2k = mexrov/h. Here ro is the lattice constant, mex and v are effective massand velocity of the particle, respectively.

To obtain the scattering data, it is sufficient to find the asymptotic solutions of(7.32) and (7.33) with

<?(*', 0) = iy/l<Kx>, 0) = je™*' sech ( ^ ) . (7.43)

If \x'\ -> oo, equation (7.32) becomes

*i' - (2ik ± | ) * i + C (C - * =F ^ ) * i = 0, (7.44)

where

~, dVdx'

(7.41)


The upper and lower signs in (7.44) correspond to x' -> —oo and x' -> oo, respec-tively. It can be shown that (7.44) has the following solution

with

mi=C + k-i~, m2 = C + k + i-.o 8

lib ^0, (7.45) coincides with the asymptotic function (7.34) with

<i= £+1771, £ = - f c , m = - .o

Prom (7.34), we found that as x' -»• 00, #2(£') = 6(^')e^'*'. However, (7.45) and(7.33) imply that *2(oo) = 0. Therefore, for the potential (7.43), b(f') and R(£)must equal to zero. Such potentials are called non-reflective potentials and cannotbe observed by scattering of plane waves coming from infinity. In such a case, thespectral data (7.37) include

R(t',t') = 0, Ci = - « + f | , d(t') = coe^',o

where Co = ib/4. Thus,

F(x',t')=cl(t')ei^'.

Inserting this expression into (7.39), we get

K(x',y') = f(x')e-<y'.

We can then obtain

/(*') = cUt')e-^' [l + !6i£ip!e-6*72p .

The square of the modulus of the wave function, including q(x',t') in (7.40) is givenby

\<Kx',t')\* = l\q(x',t')\2 = \ (7.46)0 8 cosh [b(x' - x'o - 4/ct')/4]

where

This indicates that given an arbitrary nonzero value of b and the initial pulse (7.42),the microscopic particle propagates with velocity v in the form of a single soliton.

(7.45)


Davydov further obtained solution of the nonlinear Schrodinger equation (4.40)with a rectangular step initial pulse

{ J_ 2ikx' if 0 < r' < /V i 6 ' li0^X^1 (7.47)

0, if x' < 0 and x' > I.In this case the solution of (4.42) or (7.32) - (7.33) can be written as

*=r j )e-«*\ forz'<0,

* = "(0 (I) e~Kx' + KCKCl\ ^ x' > i,

while in the region of 0 < x < £,

*i = [Ai sinnx' + A2 cos nx']elkx>,

* 2 = 7^-{[(C + ^)î + inA2]smnx' + [(( - K)A2 - inA±] sinnx'}e~ikx',Qo

with

^0 = ^ 1 , n2 = Q 2 _ ( c _ K ) 2 ( 7 4 8 )

It follows from the continuity of the solutions that

4i(C) = - » — , A2 = l, (7.49)

n

a(0 = ^ « , k)eiiK+Ot', 6(0 = ^Qoc-^*+c) i ' ,

where

S(C, k)=n cos(nl) - i{k + Q sin(nl).

Using (7.49), we can find the reflection coefficient at time t' = 0

R(^') = iQoS-1^1 ,k)e-2i^k+^t.

According to (7.36), the reflection coefficient at time t' is given byR(?,t!) = R(?)eiit"t'. (7.50)

The parameters of the bound states are obtained from the condition 5(C, k) — 0.This equation has the following solution, £0 = —k + irfo, where 770 is determinedfrom

Ibl 2 [bi ~1 & 1 &


If ?7o ^ 0, according to (7.35), (7.37) and (7.48), Davydov obtained that

co(0 = «e4*o'\ (7.51)

wheren2P2lrlo

„ Hit- 2 /^2 2

Q " g0[niIcos(ni0-sin(niZ)]' " i - ^ o " ^ -

Thus, the auxiliary function F(x', t') defined in (7.38) can be written as

F(x',t') = coit'y^' - ~ r dt'RtfJ)e**\ (7.52)27r7_oo

where Co(i') and R(£',t') have been given in (7.51) and (7.50), respectively. Thefirst term in (7.52) represents the soliton solution, while the second term determinesthe "tail" accompanying the soliton.

For a sufficiently long time £', the integral in (7.52) can be approximated by

2TTJ ^' ' s 4v^r75(0,fc) I [ 16*' V 8/JJ

Then, the "tail" decreases with time according to l/y/¥. Keeping only the firstterm in (7.52) in the long time limit, obtained K(x\y') and the wave function, bysolving (7.39),

_ i2>/2tt,exp{2t[fts' -2(fc2 -qg)f]}

^ ( a r ' ° " VSoo8h[2»(x'-x'0-4*f)] ' ( }

where

2% 2rjo

Thus, given the initial excitation in the form of a pulse (7.47), soliton solutions existif the nonlinearity parameter b is greater than the critical value bCT = TT2 /4£. Theamplitude of the soliton increases and its width decreases with increasing b.

7.4 Perturbation Theory Based on the Inverse Scattering Trans-formation for the Nonlinear Schrodinger Equation

In this section, we establish the perturbation theory for the nonlinear Schrodingerequation based on the inverse scattering method. According to the basic idea of theinverse scattering method described in (3.21), we seek a pair of Lax operators L andB and the corresponding scattering data. We will first build the general theoreticalframework.

The first step in using the inverse scattering method is to solve directly thescattering problem, i.e., to find the eigenfunctions of the spectral equation (3.21)


(or scattering data). Since the coefficients of L depend on <f>(x,t), we can map thisfunction into the scattering data. In general, the scattering data consist of twocomponents, S(X) and Sn, representing the continuous and discrete spectra. Heren is the number of discrete eigenvalues. The evolution of <t>{x, t) gives evolution ofthe scattering data \j)t = Bip. The evolution equations for S(X) and Sn have thefollowing form

———-til(X)b{X,t), ••- = ilnbn{t). (7.54)

To solve the Cauchy problem for the nonlinear Schrodinger equation, we firstdetermine the initial scattering data, S(X,t = 0) and Sn(t = 0), corresponding toan initial condition </>(x, t = 0) (direct scattering problem). Next we find S(X, t) andSn(t) from (7.54). And finally reconstruct <f>(x,t) on the basis of S(X, t) and Sn{t),i.e., solve the inverse scattering problem. The discrete-spectrum scattering datacorrespond to solitons, and the continuous-spectrum scattering data correspond toradiation. In this way, the variational derivatives 6S/5cp(x) can be written in termsof the Jost function and the scattering data. Making use of these quantities, we canobtain a generalization of the evolution equations (7.54) for the perturbed nonlinearSchrodinger equation,

i<h + « W + 2|<6|2^ = eP[<t>], (7.55)

where P[</>] represents the perturbation, x' — yp^mfW-x and t' = t/h. The followingwas obtained by Kivshar et al.,

df - ; _„ dx mx',t')n<p]+€]_„ sct>(x',t')rm

= in(X)S(X,t>) + e£y^P[<t>}. (7.56)

dSn(t')/dt' can be obtained similarly,

where

F[<f>] = -4>x.x, - 2 | 0 | V ,

is the unperturbed part in the nonlinear Schrodinger equation. If e is small, we cansubstitute the unperturbed 4>(x',t') and the Jost functions into the right-hand sideof (7.56) and obtain the lowest order approximation for the perturbed evolutionequations for the scattering data. Obviously, this procedure can be iterated toobtain higher orders of approximation. These ideas were independently proposedby Kaup, and by Karpman and Maslov. Their work were further generalized byKivshar and Malomed.

(7.57)


Following the approach of Zakharov and Shabat discussed in Chapter 4and (3.22), the operators L and B corresponding to the unperturbed nonlinearSchrodinger equation (7.55) can now be written as

- i £ - <S>*{x')L= & e , (7-58)

A [ - 4 U 3 + 2A|</>|2 + 4>*4>x. - # ; , - 4 i A V - 2i\ij>*x, + i<t>x,x, + 2 i0- 2 0 l~ [ -4»AV + 2iA0,. + t0,»a. + 2i020* 4iA3 - 2A \<j>\2 - 0a,0* + 00*, J '

(7.59)The Jost functions for real spectral parameter A are defined by the boundary con-ditions

*±(z', A) = eiXcr3X> + 0(1), when x' ->• ±oo, (7.60)

where 0-3 is the Pauli matrix. This matrix form of the Jost functions can be ex-pressed as \P+ = (tp1,^), iff- = (-ip,ip'). Here ip' and <// are independent vectorcolumns. The linear involution operation acting on

" [ S Itransforms it into

The monodromy matrix T relates the two fundamental solutions * + and *_,

¥ - 0 0 ) = ¥+(! ' , A)T(A).

Kivshar, et ai. gave this matrix in the following form

[a*(A) 6(A)1 f 7 g n

where the two Jost coefficients a(A) and b(X) satisfy

|o"(A)|2 + |6(A)|2 = 1. (7.62)

The vector Jost functions ij}'(x',\), ip'(x',X) and the Jost coefficient a(\) areanalytic and continuous in the upper half of the complex A-plane. The zeros,An = £„ + ir)n (n = 1,2, • • •), of the functions a(X) in the upper half-plane givethe discrete spectrum of the corresponding linear problem (4.41). The Jost func-tions ip'(x',\n) and ip'(x',Xn) are linearly dependent, cp'(x',\n) = bnip'(x', Xn).They decay exponentially as |x'| —> co. a(A) and 6(A) with real A constitute thecontinuous-spectrum scattering data, and the set of complex numbers An and bn


constitutes the discrete-spectrum scattering data of the corresponding scatteringproblem. Their time evolution gives the linear equation (4.41) of the operator B in(7.59),

o(A, t') = a(X, 0), b(X, t') = 6(A, 0)e4 i A V,

Xn(t') = AB(0), bn(t') = M0)e4a»*' • (7.63)

The inverse scattering problem for the operator L in (7.58) now reduces to asystem of singular integral equations

+(x,\)e -{1)+L[ {X-Xn)a'(Xn) '

2 m J_0O X' - X + IO v ;

n*,Xn)e {l) + 2î (K-Xm)a'(Xm)

1 A00 i?(A)^(x',A)e^'+ s s y _ 0 0 — > r ^ — d A ' (7-65)

where

a'(Xn)=^ , / ? ( A ) H M . ( 7 . 6 6 )

5A A=Xn a(A)

Finally, </>(a;', i) is given in terms of the scattering data and the solutions (7.64)and (7.65),

tfV.f) = -2 E ^rî(*',An)eiA»*' + 1 /°° ^(AJ^Car'.AJc^W. (7.67)n = 1

a \*n) Kl J-oo

The reflectionless potentials (f>(x',t'), for which b(X) = 0, are soliton solutions ofthe unperturbed nonlinear Schrodinger equation. The refectionless scattering datawith the single zero Ai = £' + if] of the function a(X) corresponds to the case of onesoliton (4.56) or (7.53).

When a perturbation is present as in (7.55), in the lowest order approximation,Kivshar et al. obtained the following from the evolution equations (7.56) and (7.57)

^%P- =e /+°°dx'P[î(x'>A)V,2(x'>A)

r+oo+e* / ds'P'[M>a(x',A)<M*'lA)> (7-68)

J — oo


^%P- =4iA26 + e [+0O<&P[4>]MJ,*)<PW,X)

m J-oof+OO

-e* dxT'iîxWteKxW), (7.69)J — oo

ri\ 1 r+o°^ T = - ^TTT / dx'{eP[<t>]Mx'An)V2(x',K)at a \<\n) j -oo

+e*P*[<A]t/>2(x',AnV1(x', An)}, (7.70)

^ = 4<A»6n + - ^ T /+ 0°dx'{ePMg1(x',AB) + e*P»Mg3(*l>AB)}. (7.71)

Here

Qi(x', A,,) = ^ foOc', An)^(x', A) - ^ (x ' , A)^(x', An)]A=An , i = 1,2, (7.72)

and a'(Xn) has been denned in (7.66). The general evolution equations for theparameters for the one soliton solution of the nonlinear Schrodinger equation (4.56)are

where C' = — 4f'i + x{,, 0 = 4(^'2 - T?2)* + 0O- Applying these formulae, we canobtain the time-evolutions of the scattering data and the corresponding solutionsof the equation (7.55) only if the perturbation P[<f>] is known. For the dissipativeperturbations,

eP[<f>\ = ial(f>, eP[<f>] = ia2<f>x.x. - ia3\ct>\2<f>. (7.77)

(7.73)

(7.74)

(7.75)

(7.76)


Ott et al., Karpman et al., and Nozaki et al. obtained the evolution equations forthe amplitude r\ and velocity v = —4f' of the soliton

dV n ( 4 , 8 , \ 1 o ,_ = _27? + rir]2 + _a3T]2j _ _a2r?2?;2;

* - -y W- (7-78)

In contrast to the unperturbed case, in which the soliton's amplitude 77 and ve-locity -4£' may take arbitrary values, the stationary values in the presence of thisperturbation are uniquely determined from if = 3ai/4(a2 +2a3) and £' = 0. How-ever, it is clear that the corresponding soliton is unstable, since as |a;'| -> 00, <f>(x', t')has the same asymptotic behavior as the unstable trivial solution ) = -iaicj) + ia2<t>x<x> + ia3\<j>\2 - i a 4 | ^ | 4 ^ , (7.79)

where all the coefficients ai to 04 are positive. Petviashvili and Sergeev provedthat a hard excitation does occur under the condition a3 > 2^/aâl, the solution(4.53) or (7.53) may exist with zero velocity. The last term in (7.79) is necessary toprovide global stability. The evolution equation for its amplitude can be obtainedusing the first order perturbation theory as

dv n ( 4 2 8 o 128 4 \_ = 2 , (-ai - -cW + -a3r? - —a^) •

From this equation, Malomed showed that the stationary amplitudes are given by

7)2 = d b {(2as ~az) ± [(2a3 ~a2)2 ~ ? a i a 4 ] }where the upper and lower signs correspond, respectively, to stable and unsta-ble solitons. The condition, «3 > 0:2/2 + ^(6/5)0:10:4, for the existence of thesoliton is more restrictive than the above condition for hard excitation. The coef-ficient a2 in (7.77) may be negative. In this case, cti = 0 is required. Thus it isnecessary to supplement the perturbation (7.77) by an additional stabilizing term—ia<i<t>x'x'x'x'{oi-i > 0). Pismen demonstrated that in this case the perturbationmay support a stationary soliton solution with a nonzero velocity. However, if OCJis changed, then bifurcation between the quiescent and steadily moving solitonsoccurs.

For the following nonlinear Schrodinger equation with a dissipative perturbation

r+00 1^/^12

*# + 4>x-*> + 2|</f4> = # / ^r^rdx, (7.80)

where e is a small real parameter that describes the nonlinear Landau damping ofthe microscopic particle. Ichikawa derived the following evolution equation for the


soliton velocity and amplitude using the adiabatic approximation in such a case

dv 2 dn

whereiqo

9 = i^C(3)«7.44.

Thus the nonlinear Landau damping acts upon a microscopic particle (soliton) asa constant force. Similar results were obtained by Kodama and Hasegawa for theequation

i&' + 4Vz< + 2|</f</> = - i e | 0 B , | V (7.81)

This equation resembles that for the evolution of the envelope of an electromagneticwave in a nonlinear one-mode waveguide with regard to nonlinear dissipation due toinduced Raman scattering. The corresponding evolution equations for the soliton'sparameters are

In the presence of both dissipation and an external periodic force, P(<j>) =—ia<j> + eexp(int'), where fi is the frequency of the external force, a soliton may beable to survive. Kaup et al. obtained a perturbed solution for a quiescent solitonin the form of

where 6'{t') is a real phase. Equations (7.73) and (7.75) reduce to the autonomousdynamical system

^ = -2ar, + ^Tresin^, ^ = ft - 4r?2, (7.84)

in such a case. They possess two stable stationary points which are actually equiv-alent to each other

Vn = ^ ( - l ) V n , 6'n=nir + e'0, (n = 0,l), (7.85)

where sin#o = 2ay/U/Tre. It follows from the expression for 8'0 that a stable soliton,with its frequency-locked to the external ac force, exists provided that the amplitudeof the force exceeds the threshold value /thr = 2aVTi/n.

Kivshar et al. investigated the effect of the perturbation

eP(4) = e<t>*eiuot' - ia<j>. (7.86)

(7.82)

(7.83)


In this case, the soliton is taken to be the same form as (7.83), and the equationsanalogous to (7.84) are

-jt = -2ar} - 2er)sin(26'), ^ - = w0 - 4T?2 - e-q cos(26>'). (7.87)

They have the following stable stationary points

1 / i \ 1/2

Vn = (-l)n[uJo + Ve2-a2j , d'n = mr + 6'0 (7.88)

where n = 0, 1, and sin(2#o) = —a/t. Clearly, a stable soliton exists for a < e <\/ti>o + a2. In addition to the stationary points (7.88), equations in (7.87) possessesthe trivial stationary point 77 = 0, 0' = constant, which is also stable under thecondition e < s/u/2, + a2. This means that there is a separatrix on the phase plane(T],6'), which is a boundary between attraction basins of the two types of stationarypoints.

Karpman and Maslov demonstrated how the perturbative approach can be ap-plied to the following nonlinear Schrodinger equation with slowly varying coefficients

ifo + a(et')\4>\24> + P{et')4>x,xl = 0. (7.89)

The substitution

T = j T 0(et')dt', Mf ,T) = yjlfa ,*), (a = aj3)

transforms the above equation into the standard form (7.55) with the perturbation

PW = %*.

Grimshaw investigated the one-soliton dynamics governed by the same equation(7.89) with slowly varying coefficients by using a multiscale expansion technique.

7.5 Direct Perturbation Theory in Nonlinear Quantum Mechanics

There are many ways to solve the perturbed dynamic-equations in nonlinear quan-tum mechanics besides the inverse scattering transformation discussed above. Inthis section we will introduce a direct perturbation theory which is applicable toboth perturbed integrable and nonintegrable systems.

7.5.1 Method of Gorshkov and Ostrovsky

We now consider a system described by the following perturbed nonlinear Klein-Gordon equation in natural unit system

<t>tt-<t>xX + F{<t>)=eR{<l>). (7.90)


Clearly, at e = 0 and F{<j>) = — <j> + cj)3, it is a nonintegrable <^4-field equation whichhas kink (a — 1) and antikink (<r = —1) solutions. We now solve the perturbedequations (7.90) by an asymptotic method based on a small-parameter expansionproposed by Gorshkov and Ostrovsky (see book by Abdullaev). This method yieldsanalytic forms of evolution of single solitons under weak perturbations.

Consider a set of general differential field-equations in first order in time andspace,

M(4>,<t>r,V4>,X) = O, (7-91)

where 0 is a vector function, (j> = {0i,• • • , 0 J V } , T = et and X = ex (e -C 1) aretime and coordinates, respectively. We now consider the evolution of a particularsteady-state solution of (7.91) which for e = 0 has the form.

4> = 0W( | ,A) , (7.92)

where A = {Ai, • • • , Am} are arbitrary constants, £ = x — vt for (7.90). We expectthat the solutions have the following forms for large |£|

00) (|, A) = ^ W ) , | -»• ±oo. (7.93)

Following the procedure of the small-parameter method, we seek the solution of thefollowing form

TV

4>{x,t) = ^0\lA,X,r) + £ £"</><")(!, X,T), (7.94)n=l

where n is the order of expansion. Note that A is a slowly varying function of Xand r . Inserting (7.94) into (7.91) and equating the terms of the same order in eyield a set of linear equations in <j>^,

K(j>^ = G<">, (7.95)

where

k _ 0M<°> d dM<® (1) dM dM (0) dM (0)

and

iW(0> = M(e = O,0 = (£(o)).

We can infer from (7.95) that its solution, (j^n\ is of the form

0<n> = W ( D<") + I di'W+G^ ] , (7.96)


where D^n> = constant, W is a matrix corresponding to the solution of the equationKW = 0, and

—-pepThe Wi's are found from the known <^0' by varying it with respect to £ and A, i.e.,

W!=(l>f\ Wi = $l i = l , 2 ,3 , - - - ,m + l. (7.98)For the expansion (7.94) to hold we have to impose the condition of boundedness

in terms of this series that are derived from (7.96). Among the terms in (7.96), onecan see an exponentially growing W{. Thus the finiteness of the solutions <j>^ canbe achieved by two ways.

(1) For an odd function W+G^n\ the solution <j>^ is finite if the constants D^hsatisfy the relation,

/•OO

DjW = / d£wfG(n). (7.99)Jo

(2) For an even function W+G^, the condition/•OO

/ dJ£w?G(n) = 0Jo

must be satisfied. A secular growth of </>("', due to finite solutions of (7.98) for£ —> ±oo, is ruled out by the relations

l i m M ^ G ( n ) = 0 , i = l + l,l + 2,---,m + l.£->±oo

The above is the direct perturbation theory of nonlinear equation (7.91).We now consider the solutions of the perturbed nonlinear Klein-Gordon equation

(7.90) following the same procedure. In (7.90), R(<f>) is a nonlinear operator, and_R(</>(°)) -»• 0 when | ->• ±oo. The soliton solution of (7.90) for e = 0 is of the form

4ff»=J-JL=\, i=x-vt.yvi -v2j

By means of the expansion (7.94), the linear operator k in (7.95) is found to be

k = (v*-l)£;-F'(4>{0)).

There exists a solution, $i = < °Hf)> °f k$ = 0, that decreases for | -> ±oo.We deduce the solution for each order of the approximation

0<n> = *i (D[n) + f de'*2G(n) ) + *2 ( D(2

n) + f d|'$2G(") j ,

(7.97)


where

For <f>^ to be finite, it is necessary and sufficient that the orthogonality condition

/•OO

/ rfM G(n) = ° (7-10°)J — OO

be satisfied. Thus, we have

GW=ct>f^+2v<t>f+eR(cl>W)4 at «T

for n = 1. Substitution of this into (7.100) results in the following equation for asoliton with velocity v

*{w*s) ~4> /*'*-wm)>

where

In a similar way one can derive equations describing soliton evolution in thefollowing two-component perturbed nonlinear Schrodinger equation system

i<Pf + <Px<x> -u<j> = eR{4>), (7.101)

uvv -ux.x, = (\4>\2)x,x,. (7.102)

If we assume that £ = x' - vt', u = u(|), and <j>(x',t') = <p(|) exp(iax' + if3t'),then from (7.102), we get

where u is an integral constant. In the presence of the perturbation, we let u =u(x',t'). According to the direct perturbation theory discussed above, solution of(7.101) and (7.102) can be written in the form of a series

4>(x,t) = <p0(I,X,T) + J2eVn)(I,X,T)\ ci^"/2+«),n=l J

u = u0 H, X, T) + ^ enu^ (|, X, T), (7.103)n=l

where <>o, Uo are soliton solutions in the unperturbed case, v = v(t) are slowlyvarying function of space and time, X = ex', T = st', and e is a small parameterwhich is proportional to the ratio of the soliton scale to the given low-frequency


wave scale. Inserting (7.103) into (7.97), (7.95), (7.101) and (7.102), and equatingcoefficients with the same powers of e, we obtain a set of linear equations

Ki&ipW = f i - (A'2 + 3tio)l 3fy(n) = KG(n),Uf2 J

K2$*vln) = I X - (A'2 + uo)l %>(n) = SG(n).\_d£,2 J

In accordance with the method of Gorshkov and Ostrovsky, it is necessary forG^") to be orthogonal to the eigenfunctions of K\ and K% which decrease to zeroas xi -» ±oo in order to stop the growth of the corrections, i.e.,

<¥>0|!RG(">> = 0, (<PoiS*&n)) = 0,

where

(•••) = f(---)di.

Prom this, we obtain the following for the soliton parameters,

Thus, this system may be rewritten as a second-order differential equation

d?x' ., , T 8 o l + 5a;'2 I " 1

where x' = dx'/dt', m = constant is the "quanta number".

7.5.2 Perturbation technique of Bishop

For small perturbations, Bishop developed other direct perturbation techniques. Forexample, for the following perturbed Sine-Gordon equation in natural unit system,

<j>tt ~ 4>xx + sin0 = £#(</>), (7.104)

where s is a small positive parameter, eR((j>) is a perturbation action. Bishoptransformed this equation into a set of two first-order differential equations withrespect to the time-derivative

ipt - <t>xx + sin<j) = eR(<j>),

xp = <pt- (7.105)

Equations (7.105) may be obtained from the variation principle

SL _ SL _

8<j>~ 8il>~ '


from

L(0, t/0 = Lo(0, ^) - £<W) , (7.106)

where

Lo{<t>, 1>) = l + 1 ~ cos 4>-M+ 2^ 2 .

(j> and ^ are considered to be independent.Introducing the matrices

equation (7.105) can be written as

^ + M * = eiZ'(*). (7.107)

Obviously, solution of the corresponding unperturbed equation, (7.107) with e = 0,is given by

*o=(\M, (7.108)

where

J. ( i\ At. -1 / [^(Z-ZO - U t ) ] !</>o(x, t) = 4tan M exp - ^ ,

<£o,t(z, t) = -2f]v sech [ X ~ X ° ~ ^ 1 . (7.109)L vi-v2 J

When perturbed, the solution of (7.107) can be expressed as

* = *o + e*i, * i = C ? 1 V (7.U0)

Inserting (7.110) into (7.107) and retaining only the first-order correction termsfor £ and in time-dependence of the parameters v and Xo', we obtain

[ ( i J ) J ; + w ]* i = Wo>to.t). (7.111)

where

Q(ô,t) = # ( * ) — £ ^ 7 ^ ' (i = l,2) (7.112)

is the effective perturbation and pi = v, gi = x0. In order for * i not to containsecular terms which increase linearly with time, we have to require that the effective


perturbation (7.112) to be orthogonal to the functions Idô/dgi, where

^ ( l V ) ' 9i = v,x0.

Prom the orthogonality condition we obtain a set of two equations

Because the energy corresponding to the unperturbed Sine-Gordon equation (7.105)with e = 0 is

1 f°° 8E = o / N K + <#L + 2 ( ! - cos 4>o)] dx = = constant,

* ./-oo V 1 - ^where <fo is given in (7.108), we assume that the small perturbation leads only toa change of the velocity v with time and that the general form of the excitation(7.108) is conserved. Then to first order approximation, the following relation holds

^ = j eR(<j>)<j>tdx.

Using (7.108) in the above equation, the direct forms of the equations for dv/dt anddxo/dt were obtained by Bishop and are given here,

^ = |ei,(l -v2)jR(<t>0)sech[n(x,t)]dx,

^ = -leri(l-v2) J R(4>o)n(x,t)secb[n(x,t)]dx, (7.113)

where

n(a;, t) = r)y/l-v2(t)[x - xo(t) - v(t)t].

If R(4>o) is not related to 0O, but is only related to coordinate x, i.e., R((j)o) = R(x),then (7.113) becomes

± = I«,(i - v2) I R(x) sech[fi(x, t)]dx.

7.6 Linear Perturbation Theory in Nonlinear Quantum Mechanics

Linear perturbation theory allows one to study the influence of small externalperturbations on behaviors of microscopic particles described by the nonlinearSchrodinger equation and the Sine-Gordon equation. Many scientists such as Pang,Fogel, Trullinger, Bishop, and Krumhansl, contributed to the development of thelinear perturbation theory. The theory is based on an expansion of the solitonwave function in terms of the complete set of eigenfunctions of the self-conjugate


differential operator. Such an operator has, among its eigenfunctions, a single func-tion representing the stationary wave propagating with the same velocity as thefree microscopic particle (soliton). In this section, we describe the linear perturba-tion technique for the nonlinear Schrodinger equation (3.2) with A(<j>) = 0, as putforward by Pang and by Fogel et al., respectively.

7.6.1 Nonlinear Schrodinger equation

We introduce a small quantity e, to denote the small external perturbation potentialV(x). In terms of this, equation (3.2) can be written as

^ + 0 + 6|^ = eVVM (7-U4)

where t' — t/h, x' — yj2m/h2x. Pang assumed the following form for the solutionof (7.114)

<f> = fa + <j>' = (/ + eF)etf(*'.*'). (7.115)

Here </>0 = f{x',t')ete(-x •*) is an unperturbed solution of (7.114) with e = 0, andit is the same as (4.8). Inserting (7.115) into (7.114) and neglecting terms higherthan the second order in e, we obtain the following equation for F.

ij- + f—i - v2[l - 4sech2(vX')]F + 2v2 sech2(vX')F*

= vJl secHvX')V(X' + wet'), (7.116)

where

X' = x'-vet', v* = ±{vl-2veve) = (^)26, S = l-^. (7.117)

Performing the transformations vX' = y and v2t' = T, equation (7.116) becomes

i-frF{y,r) + g^F(y,T) - (1 - 4sech2y)F(y,T)+2sech2yF*(y,T)

Now let

F(y,T)=F1(y,T)+iF2(y,r). (7.119)

Substituting (7.119) into (7.118), we get

f^ + ~ - ~ (1 - 2 sech2j/)F2 = 0, (7.120)

~ f r + W~ ~ (1 ~ 6sech^)Fi = l\fl^chyV(y,r). (7.121)

(7.118)


Differentiating (7.120) and (7.121) with respect to r results in the following equa-tions for Fi and F2.

d2Fi- ^ + M2F2 = A2(y,r),

d2F,-^-2- + M1F1ÂX{V,T),

where

, v2 [2 , dVA2 = W-rsechj/—,

ve V b dy

Al = ~vib [z^y^y-^ - sech2/^J.M I = -r-7 ~ 2(1 - 4sech22/)—- - 24sech22/tanh?/— + (1 + 16sech2w - 24sech4y),

dj/4 dy2 dy

M2 = -r-7- 2(1 - 4sech2j/)—- - Ssech^tanhy— + 1.dy* dy2 dy

Clearly, A2 possesses the property of a force. The operators Mi and M2 satisfy thefollowing eigenequations,

M2g2(vX') = ^yg2(vX'), (7.122)

Ml9l(vX') = ^ygi(vX'), (7.123)

where y = vX'. We can show easily that Mi and M2 are not Hermitian operators,but are Hermite conjugate to each other, i.e., M+ — M2, and M^ = Mi. Theireigenfunctions are also, orthogonal to each other, i.e.,

The eigenfunctions belong to the eigenvalues u — ±(vk2 + v2) and u — 0 in(7.122) and (7.123) can be obtained easily, k can take any value, —oo < K < oo,when cj = 0. The eigenfunctions are

gi(x') = Ci sech(uX')tanh(i;X'),

g2{x') = c2sech(vX'), (7.124)

where

C1 — C2 — V2v.

Pang showed that (7.124) are only local solutions of (7.122) and (7.123) when w = 0.


We now seek for a power-series solution of M2- We consider first the asymptoticbehavior of M2 when y -> oo, i.e.,

d4 d2

y^<x, dy* dyz

The characteristic equation of M% is

or

u>2 = (v2k2+v2)2.

Assume the following trial solution

92 (y, k) — ha cos(ky) + hb s'm(ky). (7.125)

Substituting (7.125) into (7.123), and using the fact that cos(ky) and sin(fcy) arelinearly independent, we can get

^ % - [6A;2 + 2(1 - 4sech22/)] ^ % - S s e c h ^ t a n h y ^ - 8A;2 sech2yhady4 dyl dy

+4fc f^% - (fc2 + 1 - 4 s e c h 2 y ) ^ - 2sech2y tanhy/iJ = 0,L dy3 dy J

^ - [6fc2 + 2(1 -4sech22/)l ^ - 8 s e c h 2 y t a n h » ^ - 8K2 s e c h 2 ^dy4 dy2 dy

-4k [ 4 % - (fc2 + 1 - 4sech22/)^ - 2sech2y tanhj/ZiJ = 0.L dy dy J

A further transformation, z — tany, was made by Pang. Under this transformation,equation (7.123) becomes

/ Ni N2\ fha\

{-N2Nj{hb)=°> (7-126)

where

* =(1 - z2fit ~Uz^ -z2)it-2^ - z2)^k2+' ~ Uz2)t2

-4z(3fc2-l)^-8fc2

AT2 = Ak [(1 - , 2 ) 2 ^ - 6,(1 - , 2 ) ^ - ( * » - ! - 2 , 2 ) ± - 2Z] .

With the following unitary transformation

S-J-f11] 5-1 - J - T 1 ~{\



(Nt-iN, 0 \ ( h a + i h b \ _

Equation (7.127) is equivalent to two equations. Their solutions can be assumedto be of the form

oo oo

ha=ZPJ2 CnZn, hb = Z^Y^ ftm2m. (7.128)n=0 m=0

Solution of (7.127) can be obtained by substituting (7.128) into it. Since we desirea finite solution in the complete range of interest, the series in (7.128) must betruncated. It can be shown that this truncation is unique. Besides a constantcoefficient, it has a group of solutions of the following form in such a case

ha(z) = {k2-l) + 2kz,

hb(z) = {k2 - 1) - 2kz.

The solutions gi and g2 can be then be expressed as

9i(x') = ci sech(i>X') tanh(uX'),g2(x') =c2sech(vX'),

when u — 0, and

gi(X',k) = —l—<rf{[k2-l + 2kta>nh{vX') + 2sech2{vXl)]cos{kvX')

+[k2 - 1 + 2A;tanh(wX') + 2 sech2 (vX1)} sin(kvX')} ,

g2(X',k)1-^ J^{[k2 -l + 2ktanh(vX')}cos(kvX')

+[k2 - 1 + 2kta.nh(vX')]sin(kvX')} ,

when (j = ±(v2k2 + v2), where <?i and g2 satisfy the following orthogonality and

(7.127)


normalization conditions,/•OO

/ 9l(X')g2(X',K)dX' = 0,J-OO

/•CO

/ 9l (X1, K)92{X', K)dX' = 8(K - K'),J —OO

/ 9\{X')dX' = \ ,J-oo J

/•OO

/ 92(X')gi(X',K)dX' = 0, (7.129)J-oo

| ~ 9l(X', K)g,{X')dX' = ^ sech ( ^ ) ,

f" fl22(X')dX' = 2,

•/—oo

|_°° niX'^toWdX1 = -^fsech (^) .

Thus, F\ and F2 can be expanded using the eigenfunctions gi and g2, i-e.,/•OO

Fi(y,r) = <Pw(r)gi(y) + / dkipik{T)gi(y,k),J — OO

/.OO

F2(y,r) = <p20(T)g2(y) + dk<p2k(r)g2(y,k). (7.130)./-OO

Applying the conditions in (7.129), we get

h ^ d ^ L + (fc2 + 1 ) 2 ' ^ r ) = ^ 2 ( T ' fc)> (7-131)

and

% ^ = 2 ( T ) + \ / l jykMr,k)ksech ( ^ ) = ^(r), (7.132)

where/•OO /-OO

M(T)= I A1(T,y)gi(y)dy, A2(T) = A2(T,y)g2(y)dy,J— OO «/— OO

/•OO /»OO

A1(T,K)= A1(r,y)gi(y)dy, A2(r,k) = / A2(T,y)g1(y,k)dy.J-co J—00

Prom (7.131) and (7.132), we can see that the amplitudes of the "classical modes"^10(T") and <p2o(r) in (7.132) which possess discrete eigenvalues, satisfy Newton-type


of equations of motion, and the amplitudes of the "quantum modes", yu ( r ) and¥>2*(T) in (7.131) which possess continuous eigenvalues, satisfy the wave equations.There is a gap, 6, between the "classical mode" and the "quantum mode", of sizeAE — v2. When the system is quantized, i.e., 6 -t 0, the "quantum mode" isnear the "classical mode", and the gap approaches zero. When 8 = 0, the "clas-sical mode" disappears and only the "quantum mode" remains. However, whenthe classical condition is satisfied, i.e., ve -t oo, the gap approaches infinity. Thetwo "quantum modes" are separated at ±oo. Now there is only a "classical mode".In this case the system shows pure classical behavior, a Newton-type equation ofmotion is sufficient. Therefore, for a potential field V(x'), the solitons of the nonlin-ear Schrodinger equation possess not only classical mechanical properties, but alsoquantum mechanical properties.

Once the solutions in (7.131) and (7.132) are known, we can determine tpik, <P2k,ipio and <p2o- Inserting them into (7.130) gives us F\ and i*2- From (7.119) and(7.115) we can then obtain </>, which clearly has the soliton characters.

7.6.2 Sine-Gordon equation

We now consider the following Sine-Gordon equation with a small perturbationR(x) in natural unit system,

4>tt-<l>xx+sin (j) = R[x]. (7.133)

Fogel et al. showed that its solutions can be expressed in the following form (seebook by Davydoy)

(X,T) = M±*) + M*,t), (7-!34)

where <f>\ (x, r) is a small change from the unperturbed solution (7.22), £' = 'j(x-vt),and 7 = (1 — v2)~x.

Inserting (7.134) into (7.133) and retaining only first-order terms in cj>i(x,r),one obtains

[W ~ & + 1 ~ 2 S e C h 2 ?] ^ ( '! t] = R[X]- (?-135)In the coordinate system (£',i) which moves with velocity v, equation (7.135) be-comes

[w " W + *"2sech2^'] 0l(?'*} = R[l{?+vt)l (7-136)To solve this inhomogeneous equation, we make use of the eigenfunctions and eigen-values of the Schrodinger equation

K'fkm = n2(k)fk(a (7.137)


with the self-conjugate operator

K' = -^+V(?), V(?) = l -2sech 2 ? . (7.138)

The eigenvalues of (7.137) contain a zero level Oo = 0 and a continuum spectrumft2 (k) = 1 + k2. The corresponding eigenfunctions are

/0(f) = 2sechf, and /„(?) = -±=[k + ftanh f ]el'*«', (7.139)VLit

respectively. Prom (7.22) and (7.139), we can get

±/o(?) = ^>0o(±?). (7-140)

The small displacement of £' in the function (7.22) is consistent with result of thefirst-order perturbation treatment, as shown in (4.39)

0o[±(?+a)] = to(±|')±a/o(|').

The term a/o(£') may be considered as an operator for a displacement a of thesoliton coordinate £'. Thus the function in (7.137) may be referred to as a translationmode.

The eigenfunctions (7.139) of the self-conjugate operator (7.138) satisfy the con-ditions of orthogonality

//o2(M' = 8,

//otmiM'-o,

and completeness

g/o(£')/o(6) + / fk{?)ft(ti)dk = 6(? - &).

Similar to (7.130), we expand the unknown function <j>\ (f, t) in (7.136) using thecomplete set of basis functions given in (7.139)

M?,*) = |^(0,t)/o(f) + J*Kk,t)fk tf')dk. (7.141)

Here the factor ^»(0, t)/8 in front of the translation mode describes the motion of thecenter-of-mass of the microscopic particle, and the coefficient ip(k, t) defines a change


in its form. Inserting (7.141) into (7.136) and making use of the orthogonalityconditions given above, we obtain

&gp = /*[7(?+ »W.(i'K',

^ & S + n^flfc, <) = | FU? + «')]/;(? )<if. P-I42)

which determine the first-order correction to the motion of the microscopic parti-cle (soliton) due to the perturbation R(x). Once R[7(f' + vt)] is known, we candetermine ifi{0,t) and ip(k,t). Thus, <î(£',t) can also be determined.

7.7 Nonlinearly Variational Method for the Nonlinear SchrodingerEquation

The time-dependent variational principle proposed by Dirac is of great use in nonlin-ear quantum mechanics. Since the nonlinear Schrodinger equation is a Hamiltonianintegrable system, the same variational principle can be applied to the nonlinearSchrodinger equation to obtain time-dependent approximate variational solutions.In this section, we introduce the Dirac variational method and apply it to the non-linear Schrodinger equation, which were performed by Cooper et al.

As mentioned in Chapter 3, the action for the nonlinear Schrodinger equation,in natural unit system, is denned as

S=fdtL, (7.143)

where

L=l-jddx{^cj)t-^t(j>)-H.

The Hamiltonian of the nonlinear Schrodinger equation in d-dimensions with non-linearity parameter a is of the form

The nonlinear Schrodinger equation with a given nonlinearity follows from Dirac'svariational principle. If 5 is stationary against variations in c\> and (f>*(x, t), i.e.,

SS _ 6S6<f> ~ 6<l>* ~ '

we can get

i(t>t + V 2 0 + b{<t>*<f>Y<t) = 0,i4>\ - V V - b{4>*<t>Y<l)* = 0- (7.144)


The variational principle is a form of Hamilton's least-action principle. In thevariational principle, 0 is an arbitrary square integrable function subject to

| [ /ddz(^)] =0. (7.145)

To obtain an approximate solution of the nonlinear Schrodinger equation,Cooper, Lucheron and Shepard considered a restricted class of (p(x,t) = <j>v{x,t),here <j>u(x,t) is constrained but is able to capture the known behavior of <fi for theproblem at hand. They constructed only one simplest trial wave function which issuitable for Gaussian initial data and leads to a very simple Lagrangian dynamicsfor the variational parameters. For a one-dimensional system (d = 1), the trial wavefunction is written as

which automatically satisfies the constraint (7.144). The normalized probabilityfunction is p(x,t) = <f>*(j)/N, here N plays the role of the conserved "mass", M,of the nonlinear Schrodinger equation. The physical meaning of the variationalparameters given above is very clear, i.e.,

(x) = I dxxp(x) = q(t), ([x - q{t)}2) = G(t),

(-idx)=p(t), (idt)=pq-GQ.

Thus G is the variance and p and Q are the canonical conjugates to q and G. Theaction for the trial wave function can be expressed as

S(q,p,G,Q) = N J dt(pq + QG - tfeff) (7.147)

/

[ 1 hN" 1

Then, p(t), the momentum conjugate to position q(t), is conserved. The variationalequations dS/Sqi = 6S/6pi = 0 yield q — 2p, p = 0, so that the Gaussian wavefunction moves with a constant velocity q(t) = vt, v = 2po, and

G = SQG, Q - 4 ^ - 4 Q 2 - W 2 ( 2 G ) ^ ; ( 1 + ( T )3 / 2- (7-148)

The above equation has a first integral which gives the conservation of energy

<f> = E = Pl + 4QGQ + ± - ( 2 7 r G ) g /^+ g)3/2 •

Using (7.148), this can be rewritten as

° ==16GjB+(27rG)*/*(l + a ) 3 / 2 - 4 '

(7.146)


where E = E — pi is the energy available to the generalized coordinate G. Whena — 0, which corresponds to the linear Schrodinger equation, we have

G2 = 16G(E + b) - 4.

Thus

G(t) = Go + 4(E + b)(t - to)2 + (t - io)[16Go(£ + b) - A]1'2,

which shows the spreading of the Gaussian wave packet with time for an initial dataGo in linear quantum mechanics.

When a = 1 which corresponds to the general nonlinear Schrodinger equation,one has

G2 = 16GE + AbN\ — - 4. (7.149)V 7T

Prom (7.148), we see that there is an energy regime, -627V2/16TT < E < 0, for whichG, the width of the Gaussian, does not spread but oscillates between two bounds

_62iV2 L f . 164B|\1/2]2

The lower extreme E = —&27V2/167r is a fixed point of the equation where thewidth of the Gaussian is 2I/TT/&-/V. Thus in this approximation there is a range ofinitial data where the Gaussian does not spread in time. This is a feature of themicroscopic particle described by the nonlinear Schrodinger equation in nonlinearquantum mechanics. It demonstrates again that the microscopic particle has exactlythe property of classical particle, i.e., it can always retain the shape during itsmotion and cannot be dispersed with time.

Cooper et al. obtained also the following expression for G(t) for a = 2 whichcorresponds to the 06-nonlinear Schrodinger equation.

G(^L/G0 + ^ 2 - 2 ^ - J 2 - ^ ,V 16E 16E

where

For d ^ 1, it is sufficient to consider a spherically symmetric trial wave functionthat is centered at all times at the origin (i.e., p(t) = q(t) — 0). We parameterizethe trial wave function as follows,

4>(r,t) = VN ( - ) e-r2l°(t)-iQW]. (7.150)

V 7T J


The effective "mass" is given by

N = nd /</>*(rMr)rd-1dr,

where

nd~r(d/2y

With p(r) = (j)*(j)/N, we have

(r2) = G(t) = ^ y , <«*> = -GQ,

so that Q is still canonically conjugate to G. Thus the action for the trial wavefunction is of the form

S(G, Q) = N f dt(QG - #eff) (7.151)

/

d2 hN° ( d \ crd/2~

dt[QG-iQGQ-~ + (1 + a)(d+2)/2^) .

The variational condition SS = 0 leads to the following equations of motion

r-znr o- d' AQ* hadN° ( d Yd/2 1

G-8QG, Q-—-4Q --££-{—) ___I_W_.

The first integral of the first equation gives the energy E

iEt-E-— * bNa / d yd/2

N ~ 1 6 G + 4G (i + CT)(d+2)/2 \2nGjThe key equation is

&-16GE l j 2 . 16&GJV- ( d yd/2

G -16GE-4d + ( 1 + ff)(d+2)/2^J •Using the same arguments as in the case of d = 1, we can find that G -> 0

as long as crd > 2. At the critical point ad = 2, we can obtain the "mass" Ncorresponding to the singularity. Prom the criterion and G < 0 for small G we candetermine the parameter which is given by

1 J 2 < d 16bN*- 27r(l + a)(d+2)/2 '

In terms of d = 2/cr, we obtain

f-ndV2 ( 2\d{d+2)/4K-^=(10 ii+l) <"52)


for the critical "mass". Thus for d = 1, b = 1, Nc = y/nj233/4 = 2.8569; for b = 1,d = 2, iVc = 47r = 12.5664; for b= 1, d = 3, Nc = 69.4696. These can be comparedwith the exact numerical results in one and two dimensions, d=l,b=l, Nc — 2.7;d = 2, b = 1, Nc = 11.73, obtained by Rose, Weinstein and Schochet. Therefore,equation (7.152) gives quite reasonable results.

Cooper, Shepard and Simmons solved approximately the nonlinear Schrodingerequation (7.144) by the perturbation (5-expansion proposed by Bender et al. which isan expansion in terms of the degree of nonlinearity of the equation. This techniqueconsists of first replacing a by a parameter 8 and treating 8 as a small perturbationparameter.

To improve the dependence of the above description on the coupling constant b,Cooper et al. further introduced two parameters, M and c, in (7.144), and rewrite(7.144) as

which is equivalent to (7.144) when c — 8. The J-expansion is obtained by firstexpanding this equation as a power series in 8.

^ + g + 6^£(£i^W=0. (,153)71=0

Assuming a power series solution for 4> in terms of

8<t> = Yj4>n{x,t,M)6N,n=0

one obtains a system of linear equations for the <f>n with known driving terms. Forthe exact solution, <f> is independent of M when c = 8. However, when we expand has an expansion up to the order N in 8,

N

(f,(N)^Y,<f'n(x,t,M)8n, (7.154)

n=0

and M is a function of x and t by the requirement of dcj)^/dM |c=o= 0, which isknown as a scaling relation.

The linear delta expansion, proposed by Okopinska et al, Duncan et al. andTones et al, consists of replacing (7.144) by

i^- + ^ + b\<j} + 8{b\U\-b\)4> = Q, (7.155)at ox2

which contains two new parameters 8 and A. When 8 = 1, it reduces to (7.143).At any finite order in S, however, the solution depends on A and this dependence


can be minimized by imposing the "principle of minimal sensitivity", which is toimpose the condition d(j>^N' /dX 1,5=1= 0, in the finite order TV expansion

N

4iN) = J2<l>n(x,t,\)6n (7.156)n=0

where A is a function of x and t.

To solve for <j> in the rf-expansion, we assume that <f> can be written as

<t> = 0 O + S<fn + 52<f>2 + ••• (7.157)

and 4>n = Vnei9t, where g = bMc. Substituting (7.157) into (7.153), we get

«$ + £i-A--.B*±B*-».ll.(S*)-

The corresponding boundary conditions are,

(j>[x) = *(x) = h{x) = V0{x), and * n ( x ) = 0, for n > 1, (7.159)

at i = 0. For n = 0, the solution of (7.158) is given by

•$Q(x,t) = ^L=- [ dxe^x-i)2'Ath{x). (7.160)

For n > 1 the solution can be expressed in terms of the Green function as follows

9n(x,t) = f di f dx'G{x -x,t- i)fn(Z,i), (7.161)J0 J-oo

with

G(x,t) = e-^-e(t)ei*2/it,

and fn given by the right hand side of (7.158).For the linear <5-expansion, equation (7.157) still holds and 4>n = *ne1 9*, where

g — bX. The structure of the $ n equation is exactly the same as that for theJ-expansion except for the right-hand side, / „ , which are now given by

A = b[A*0 - (*5*o)CT*o],f2 = 6[A*i - (a + l)(*S*o)CT*i - aWVoV^WVoWo]. (7-162)

(7.158)


In order to show how the variational approximation works in a simple case, Cooperet al. considered the initial condition

(f>(x, t = 0) = h(x) = CS(x). (7.163)

Then from (7.160), we get

<bo(x,t) = -¥=eix2'it-inli, (7.164)\l\-nt

so that

V4TT<

For the functions fn corresponding to the rest of the terms in the two perturba-tion expansions, we can obtain

fn(x,t) = 90(x,t)Tn(t), (7.166)

after some iterations, for the ^-function initial data. Using the properties of theGreen function in (7.161), we can obtain

*„(!,*) = (-i)*o(a:,*)Vn(t)» ¥>»(*) = f Tn(t')dt'. (7.167)Jo

These perturbation expansions lead to the following form for solutions of all ordersin S.

<j>{x,t) = (j>0{x,t)a{t). (7.168)

Inserting (7.168) into (7.144), the following can be obtained for a(t)

For the linear 5-expansion substituting (7.168) into (7.155), we can obtain

The boundary conditions for (7.169) and (7.170) are a(t = 0) = 1. Let

a(t) = fl(t)ei7W, (7.171)then R(t) satisfies

Using the boundary condition at t = 0, one gets

R(t) = l, rtt) = -gt + b[—) J^¥. (7.173)

(7.165)

(7.169)

(7.170)

(7.172)


Substituting (7.173) into (7.172) and (7.168), an exact solution of (7.144) for all acan be obtained. For the linear ( -expansion

^-'hossFl- (7-174)where a is a continuous variable. Thus <pi{t) in (7.167) has different analytic be-havior for a < 1, a = 1 and cr > 1, which are

Mt) = b[kt + X{<T,t)], x M = - ( j y J \y- (7'175)

Thus to the Jth order, one has

^ (x, t) = eibXt*0 {1 - iSb[Xt + x(v, *)]} •

By determining the variational parameter via

d± = 0

Cooper et al. obtained X(a, t)t = -x(<r,t), so that to this order

<l>{x,t) = yoe-bx^'t\

This is the exact solution for the boundary conditions given above. Extending thecalculation to the order of S2, they obtained

4>{x,t) = eibXtV0 | l - i5b[Xt + x(<M)] - \s2b2[\t + x(<7,i)]2 + • • • j

for a < 1. This series in 6 can be written as

4>(x, t) = ôe*^1-')-*****'*)!. (7.176)

This solution is independent of the variational parameter A at 6 = 1 and by directsubstitution it can be shown that it is a solution of (7.155). Thus the solution tothe original nonlinear Schrodinger equation is

^(x,i) = *oe-i i )x ( f f l t ) .

This solution satisfies the boundary condition t — 0 for a < 1. For a = 1 and a > 1,one gets an additional infinite phase as t —¥ 0, unless we allow t —> to first and thenlet t = t0 -> 0.

For the J-expansion, instead of (7.174), we have


In accordance with the above method, one can obtain

<Pi{t) = -9t ll - In ( - ^ T " ) I .

so that

[ \47rtM ) J

Thus to the (5th order,

• -•^{i+wfi+i-^)]}.By taking the exponential of both sides of the above equation, Cooper et al. ob-tained

^ e x p ^ l + jfl + l n ^ ) ] } )

Using the above scaling law, they obtained

1 + lnf-^-Uo.\4irtM J

Thus, M = C2e/47rt and the final result is

f fC2e\S0 = *oexp t 6 t ( — J . (7.177)

Comparing the above result with the exact solution, we find that the t-dependence is correct except at S = 1. However, the coefficient of the ^-dependenceshould have a factor (1 — (5)"1 instead of es. That is,

*(*,*) = *(*,*)exp ^-8{~) ,

which can be only obtained by (7.177) at <5 < 1.Therefore, Cooper et al. obtained a series of variational approximations to the

initial value problems for the nonlinear Schrodinger equation with arbitrary non-linearity a. For the Dirac delta-function initial conditions, the variational principleassociated with the linear 5-expansion gives the exact result for arbitrary a whenkeeping the first order terms in the expansion. The exact result changed its analyticstructure at a = 1, which is the integrable case. Study showed that the ( -expansionresults are less accurate for this problem. The solution converges to the exact an-swer when a — S < 1. In the 5-expansion we do not seem to be able to handle thecase of 6 > 1.


7.8 D Operator and Hirota Method

Solutions of dynamic equations in the nonlinear quantum mechanics can also beobtained by the Hirota method of function transformation. In this method, a Doperator is defined by the relation (7.2). Let's further define an operator Dz and adifferential operator d/dz as follows

Dz = 6Dt + eDx,

dz at ox

where 6 and e are constants. These operators have the following properties:

(l)D?f-g=(-irD?g.f;

(3)i>™/-ff = D™-1(/.-s-/-fo);(4) D™epixep*x = (pi -p2 ) m e ( p i ~ P 2 ) x ;

(5) eeD'f(x) • g(x) = f(x + e)g{x - e) = [exp ( e A ) ] f(x) [exp ( ~ ^ ) ] 9(x);

(6)e*D>fg-hq=[e*D>f-9]-[eeD>g-h];( o\ If\ eeD'f -a

(7) exp Ve-*) Kl) = cosh(^) g-g;

(8) 2cosh (e^-\ log/ = log[cosh(eDz)/ • / ] ;

(9) f>'f • g = exp [2cosh ( £ | ) log,] [exp ( e | ) • ( 0 ] ;

(10) e£D'/ • 9 = exp [sinh ê£^ log (£j + cosh ( e £ ) log(/fl)J ;(11) i ? ^ - • • • D « f . g = ( - i ) ^ + - + « D « D ™ ••-DSg-f;

(12) (2coshe | - ) In/ = ln(/ + e) + ln(/ - e) = ln[cosh(eDx)/],

where / , g, and # are functions of x and i.For the nonlinear Schrodinger equation,

i<t>t' + P^x-x' + b\<!>\2cf> = 0 , (7.179)

we assume + e)g{x' - e) exp(eDx,)f • f

(7.178)


where t' = t/h, x' — x^j2m/h2, we find that g and / must satisfy

(iDt, + Dpg • / _ g (PDl,f-f-bgg*\ _

P A P ) ~Substituting <j> = g/f into (7.179), we have

{iDt, + pD2x,)g • f = 0,

pDlf • f - bgg* = 0, (7.180)

where / and g satisfy (7.2). The one-envelope-soliton solution of the nonlinearSchrodinger equation is given by

g = Aeê71, f = 1 + ae"+r1', (7.181)

with

and the dispersion relation is

-i(n + iui) + p{K + ik)2 = 0.

Here we see that the decoupling of (7.180) does not result in a simple relationbetween g and / , and the Hirota form (7.179) is a coupled system (7.180) in g and/ . To obtain solution of (7.180), we assume that

g = e g x + e 3 g 3 + ••• ,

f = 1 + e2f2 + • • • . (7.182)

We then find that

igw + gix'x- = 0,

Sixx = 2#101>

ig3t + 93x'x' = -{iDf +pDl,)gx • f2,

Thus these equations may be solved recursively until an inhomogeneity is found tovanish where upon the term defined by that equation and all subsequent ones maybe take to be zero. Substituting (7.181) into (j) = g/f, we obtain the one-solitonsolution, as given in (7.53) or (4.56). Continuing in this way, the iV-soliton solutionof the nonlinear Schrodinger equation can be constructed.

For the Sine-Gordon equation, (7.6) in natural unit system, if we required(f>/dx —> 0 when \x\ —> oo, we can assume that

^ • " ^ " ' [ T I M ] ' (7-I83)


where

N/2

/(*,*) = EEa( i i ' i 3"-' i») c I h l + I K a +"'+ l h a"'n=0 Nn

|(JV-1)/2|9(z, *) = E E a ^ 2 • • • ,i2m+i)e"'i+1?>»+-+f''*»+>. (7.184)

m=0 N2m+i

Here

{ («)

1, n = O,l

^ ' ^ " ( P i f c + p ^ - ^ f c + nw)2'

Inserting (7.183) into (7.6), we get,

/&* - 2/«Sx + fxxg - (fgtt - 2ftgt + fug) = fg, (7.185)

fxxf - Vl - ffxx - (fttf - 2/f + fftt) = gXx9 - 2g2x - (gag ~ 2g\ + ggtt).

Substituting (7.184) into (7.185), we can determine a{ik,i[) and t)i2n, r/j2m+1. Wecan thus finally obtain the solution of (7.183), and find out the iV-soliton solutionof (7.6). To obtain the one-soliton solution, we insert (7.183) into (7.6) to get

{Dl-Di)(f.g)=fg,

(Dl-D2t)(f-f-g-g)=0. (7.186)

Obviously, g = 0 and / = 1 is a solution. Let

/ = l + e 2 / ( l ) + £ 4 / ( 2 ) + . . . )

g = e g W + e 3 g W + ---. (7.187)

Substituting (7.187) into (7.186) and equating the coefficients of the e terms, weget the following linear equation

g£-gV = g(1)-

The solution is given by g^ = exp(fcx — ut + S), where k2 = u2 - 1, or equivalently,<f> = 4tan~1[exp(fca; -ut + 5)] which is similar to (6.14) or (7.22).


For the 04-field equation (3.5) with A((t>) = 0 in natural unit system, we canassume cj> = g/f, and insert it into (3.5). Tajiri et al. obtained that

{D*-Dl-a)g-f = 0,

(D2-D2x)f.f-/3\g\2=0,

where

S = e", f = l + ae"+r>\

1 P

2(fi + n*)±(p + n*)'Here j3 is a nonlinear coefficient and a is a linear coefficient in this equation, 77 =Px — £lt±r)0, and P2 — fi2 = —a. Its one-soliton solution can be finally written as

<l> = ^v/2(u 2 ±l)exp U \J^TZi ~ pr (* - vt) + %r| I sech[Pr(a; - vt + 6r)),

where Pr, fl r , and 7jor are real parts of P , Q and 770,

and w2 > 1.Tajiri et al. generalized the Hirota's method to two- and three-dimensional

systems and to solve dynamic equations of the microscopic particle in these systems.For the three-dimensional nonlinear Schrodinger equation,

i(f>t> + p(j>x'X' + q'<f>y'V' + r'<j>z>z> + b\<j)\2(j) = 0 , (7.188)

with 6 > 0 (for p, q' > 0 and r' < 0), we introduce the dependent variable transfor-mation,

g{x',y',z\t')9 f(x',y',z',t')'

where / = /*. Substituting it into (7.188), we have

f {iDv + pDl + q'D2y, + r'D2

z,)g • f = 0,\(pDl, +q'D2

yl +r'D2z,)f • f -b\g\2 = 0.

Here, the bilinear operators are defined by

DZlD1y,DyD?,a(x',y',z',0 • &(*',v\*\0

= f_d d_\K fd_ d_\l f_d d_\m (_d__d_Y~ \dx' dx'J \dy' dy'J \dz' dz<) \dt' di')

•a(x',y',z',t')b(x,y,z,i)\ , . , . , . , .\x'=x,y'=y,z'=z,t'=t


The one-envelope-soliton solution of the 3D nonlinear Schrodinger equation is given

by

g = Aeê", / = 1 + ae"+7)*

where

9 = kx' + ly' + mz' - ut',

r] = Kx' + Ly' + Mz' - nt1 - r?°,

bA2

° ~ 2 \p(K' + K*)2 + q'{L + L*)2 + r'{M + M*)2]'

and the dispersion relation is given as follows

-i(fi + iw) + p(K + ik)2 + q'(L + ilf + r'(M + im)2 = 0.

7.9 Backlund Transformation Method

Applying the Backlund transformation (BT), we can obtain new soliton solutionsfrom a given solution of the nonlinearly dynamic equations in nonlinear quantummechanics. However, there are many different types of Backlund transformations innonlinear quantum systems as described in Section 3.3. Therefore, it is importantto choose an appropriate Backlund transformation in order to obtain the solutionsof a given equation.

7.9.1 Auto-Backlund transformation method

The auto-Backlund transformation method was studied by Rao and Rangwala, andRogers et al. For a 2 x 2 eigenvalue problem

A = H, (7.189)

with

*=(*\ Y=( X q{-X^\ f_(A{x,t,X) B'(x,t,X)\

where ipi and «/>2 are eigenfunctions, while A is an eigenvalue, the compatibilitycondition ipxt = iptx yields

Yt-fx = [Y, f] = YT- TY. (7.191)

Equation (7.191) results in the following set of conditions on A, B' and C,

Ax=qC-rB', B'x - 2XB' = qt - 2Aq, Cx + 2XC = rt + 2Ar. (7.192)

(7.190)


If we set

\ r -to)

then the Lax system in (3.20) - (3.22), where B and L are linear operators subjectto the compatibility condition Lt + [L, B] = 0, can result in the Ablowitz-Kaup-Newell-Segur (AKNS) system. Specializations of A, B' and C in the AKNS systemlead to nonlinear evolution equations amenable to the inverse scattering methodfor the solution of privileged initial value problems. For the Sine-Gordon equation(7.6), there are r — -q = -<j)x/2 and

A=l-cos<t>, B' = C =^r sin <j). (7.193)

For the nonlinear Schrodinger equation (4.40) with y/2m/h2x -t x', t/H ->• t',there are r = — q* = -<j>*, and

A = 2i\2 +i\4>\2, B' = i<j>x, + 2i\<f>, C = i<t>*x, - 2i\<j)* (7.194)

Auto-Backlund transformations for the above equations are constructed via theAKNS formalism. The key step in the derivation involves the introduction of F =* i / * 2 , so that the AKNS system (7.189) - (7.194) results in a pair of Riccatiequations

rx = 2\r + q-rr2, (7.195)

rt = 2 r + s'-cr2.

If r = —q in the Sine-Gordon equation, the AKNS system yields

ipxx - AV = H>, (7-196)

where tfi = tpi + iip2 arid <)> = — iqx — q2.

The system (7.196) is invariant under the Crum-type transformation

rlt = \ t <j>' = + 2(lnrp')xx,

r' = \ , ^ ' = ^+2(lnV»*')x«. (7.197)

Whence

^—«ta(i)-.+.«.-(fe),


where w and w' are potentials given by q = wx, q' — w'x. Thus, under the transfor-mation (7.197),

T = cot I ^ (w' - to) j . (7.198)

Substitution of (7.198) in the Riccati equations (7.195) provides a generic Backlundtransformation for the present subclass of the AKNS system with r = —q, namely

wt-w't = 2Asin(w' - w) - (B' + C) cos(u/ - w) + B' - C,

wx+wx=2\sin(w-w'), x = x, i=t. (7.199)

If we set a' = A/2, w = —0/2, substitution of the specialization (7.193) in(7.199) produces the following auto-Backlund transformation for the Sine-Gordonequation (7.6)

4fx = 4>x- 2a' sin ( ^ p - ) = -BTiOM^' ) ,

0t = -<!* + •£sin {~-^-\ = BT^> &> ^')' (7-20°)

where x = x', t = t', and a' is a non-zero Backlund parameter. The invariance canbe easily proved. In fact, applying the integrability condition

dBTi 8BT2

dt dx ~ '

we can get <j>xt = sin cj>. However, if we write (7.200) in the following form,

4>x = 4>'x ~ 2a'sin ( ^ ) = 2?ri(0\&,0),

4>t = ft + 1 sin {^Y^J = BT^'A't, <A),from the integrability condition,

dBTj dBT±dt dx ~ '

we can also get <f>'xt = sin^'. Thus the invariance of the Sine-Gordon equation isverified. In fact, the Backlund transformation (7.200) is the same as (3.45).

Setting $ = 0 in (7.200), we get a pair of equations

A. =20* am ( | ) , ^ = | ; sin ( | ) ,

which yield a soliton solution (6.14), i.e.,

A A. - i / [, x-xo-vt]\ I-a12

(£ = 4tan Wexp ± — V, « = ,2-L L V l - v2 J J 1 + "


For the nonlinear Schrodinger equation with r = — q*, generic auto-Backlundtransformations were generated by Konno and Wadati via I" and q' which leavethe pair of Riccati equations (7.195) invariant. In fact, (7.194) can result in theauto-Backlund transformation

<j>x + <S>'x = {4>-<j>'W^-\<t> + l\2, x = x, i = t, (7.201)

& + # = »(*. - 0 ' X ) V 4 A 2 - | 0 + 0 T + ±{4> + <t>') [\<t> + <t>'\2 +10 - <t>'\2} •

The corresponding nonlinear superposition principle can also be obtained as follows

(0o - (A0V4A2 - \(t>0 + <f>i\2- (<f>0 - 02)\/4A2 - \4>o + h\2+ (7-202)

(02 - 0i2)\/4A2-[02 + 01212 ~(0i - 0 i 2 ) ^4A 2 - | 0 1 + cM2 = 0,

where & = 5A</>0, (i = 1,2), 4>12 = BXlBX2<t>o = BX2BXlô and 0O is a startingsolution. Using (7.201) or (7.202), we can get the soliton solution of the nonlinearSchrodinger equation (4.40) as (4.8) or (4.56).

7.9.2 Backlund transform of Hirota

The Backlund transformation provides a mean of constructing new solution fromknown solutions of the dynamic equations. In the context of bilinear equations,the Backlund transformation method was introduced by Hirota to solve nonlineardynamic equations in nonlinear quantum mechanics based on the D operator inSection 7.8.

For the nonlinear Schrodinger equation (7.179) and

/2m , t , •

Hirota assumed that

n = [(iDt, + D2X,) (g • /)] / ' 2 - f [{iDt, + Dl) (g' • / ')] , (7.203)

which vanishes provided that (g, f) and (g1, / ' ) satisfy the first of (7.180). By meansof a pair of identities he rewrote (7.203) as

n = iDf,(g-f' + f-g')ff - (gf + fg')iDt,f • f'+

22?s.[Z?s(ff • / ' + / • 9')} • / / ' + (5 • / ' - / • 9')Dlf -f'-DUg-f'-f- gf) • / / ' +

\ [(gf1 + fg1) (gsT - g*g') - (gf - fsf)(gsT + g*g')}, (7-204)

where it is also assumed that (g, f) and (g1, / ' ) satisfy the second equation in (7.180).Considering the relations between successive "rungs of the soliton ladder", Hirotaobtained

Di.(g-f + f-g') = n(gf-f91)-


Thus (7.204) decouples to give

iD-t,(9 -f' + f - g 1 ) - (D% + X)(g •f'-f-g') = 0,

[iD-t, + itiDv + D\, - (2/z2 - A)] / • / ' = gg'*. (7.205)

Equation (7.205) constitutes a Backlund transformation between (g, / ) and (</, / ' )satisfying (7.180). We can find (g',f) and the corresponding solution if (g, / ) orinitial value of <j>(x, t) is given. In such a case we must choose \i = KN — K^,,A = 2{Kjf + K$) + (K% — KH)2, where KN is a constant introduced by integrationof the first of (7.182).

Applying the Hirota's transformation, <f> = 2i log(g//), to the Sine-Gordon equa-tion, 4>xt = sin <fi, it becomes

DxDtf • f = \{f2 - p2), DtDxg • g = \{g2 - ?).

The Backlund transformation of the Sine-Gordon equation is

Dxf • f = jg' • g, Dtf-g = ±g'-f, (7.206)

as well as their complex conjugates, where if is a real constant. In order to obtaina superposition formula, we rewrite (7.206) as

Dtfigo = -j j^Si • /o, Dtfi • go = -^-gifo,

Dtf2-gi = -^g2-fi, Dth-gi = -^-g2f\, (7.207)

where f\ and g\ are one-soliton solution with parameter K%, K\ and Ki are thecorresponding parameters of soliton solutions. Using the properties of the D oper-ator,

Dl(g • f)hq - gf(D2xh • q) = Dx{(Dxg •q)-hf + gq. (Dxh • / ) } ,

Hirota obtained the following superposition formula for the soliton solution of theSine-Gordon equation,

V / o = * 2 (d / / i ) - -g i (g i / / i ) (? 2 m

Inserting <\> = 2i \og{g/f) into (7.208) we obtain,

e*(*-*o)/a = K2-K^-**l*-Kx+Kie1^-^)/2'

which gives

tan ^(fo - *,)] = f ^ ^ tan ^ (0 ! - ^ ) ] . (7.209)


This is the same as (3.44) with a.\ = K\, a-i = K2, i.e., it is the superpositionformula of the Sine-Gordon equation. Applying (7.206), or (7.208), or (7.209) wecan get a new solution of the Sine-Gordon equation from a known solution.

7.10 Method of Separation of Variables

Lamb proposed a method of separation of the variables to solve nonlinearly dynamicequations. For the Sine-Gordon equation given in (6.13) in natural unit system,Lamb expressed its solution in the form

tanfi#M)| = g(x)F(t). (7.210)

Inserting (7.210) into (6.13) results in two first order differential equations as follows

gl{x) = -C'g\x)+lg2(x) + p',

F?(t) = -p'F4(t) + {£- l)F2(t) + C", (7.211)

where C", /?' and £ are integral constants. When C" = /?' — 0, equation (7.211)become

gx = nyflg, Ft = nV£ - IF, n = ±1.

The solutions of this set of equations can be easily obtained. Equation (7.210) cannow be written as

tan [ ^ ( M ) | = 3?exp ln>Tt \x - x0 - y'l~ 1fl | . (7.212)

Thus there are three kinds of solutions corresponding t o £ > l , 0 < £ < l and I < 0in (7.212) respectively.

(1) In the case of £ > 1, we assume that y/l - l/£ = v, •/£ = 7 = 1/Vl - v2

with 0 < v2 < 1. In such a case, equation (7.212) is replaced by

<P(x, t) = 4 tan"1 [en^-I0-"f>] . (7.213)

This is consistent with results given in (7.22).(2) In the case of 0 < £ < 1, we assume that \J£ - l/£ = iu, \f£ = 1/Vl + w2

with 0 < u2 < 1. Then (7.212) becomes

<t>{x,t) = 4tan"1 [exp ( j ^ «» (jT^)] "

This represents a stable solitary wave with frequency ui/VT+u2.(3) In the case of £ < 0, assuming yj{£ - l)/£ = v, 7 = 1/Vi>2 - 1, we get

4>{x,t) = 4tan"1 {cosftOc - vt)]} .


This solution depicts a periodic wave with a velocity v > 1 and wavelength A =2TT/7 = 2ny/v2 — 1.

When C" = 0, fi = 1, (7.211) becomes

g2x = £g2 + l, F? = -F4 + (£-l)F2. (7.214)

In the case of £ > 1, there are two types of solutions. The first one is given by

g(x) = sinh(7z)/v^, F(t) = s/£-lcsch(jvt).

Equation (7.210) can now be written as

This solution describes two interacting kinks moving with velocities v and —v,respectively, as described in Section 6.2.

Equation (7.214) also has the solution

g(x) - iVtcoshes), F(t) - -iV£ - 1 sech(^vt).

Thus

It describes the collision between a kink and an antikink in which they pass througheach other, as described in Section 6.2.

When C = 0 and /?' = 1 and 0 < £ < 1, equation (7.214) has the followingsolutions

g(x) = i cosh(vâ;)/v^, F(t) = n / l - £sech(Vl - It).

Thus

,, . . _! [ sin(wt/y/l + w2) 1 / l - ^ /-O1-x<A(x,*)=4tan x v / y , w = J—r. (7.217)

wcosh(x/vl +w J )J V tThis solution denotes a localized pulsated entity - the bound state of a kink

and an antikink which is a breather or bion. It is also called O7r-pulse becauseit satisfies the boundary condition <f>(x,t) —¥ 0 as |x| -> oo. It has an internalfrequency w — w/y/l + w2, spread over a region which is inversely proportional to1/Vl + w2.

The corresponding traveling breather can be obtained by the Lorentz transfor-mation

t —> j(t — x/v), x —> 7(0; — vi),

(7.215)

(7.216)


where 7 = l / \ / l - v2. The solution can be written as

*(*,t) = 4 tan"1 (tan^siny-x/.)Tcos^]|^ cosh [7(x - vt)} smfi' j

with î' = ta,n~1(l/w). The frequency of the pulsation in the moving breather isu/ = 7cos/x' = 710'/Vl + w2. According to

E = \ f dx [<j)l + cj>2 + 2 ( 1 - c o s <f>)], a n d P = - [ dx<j>t(j>x,* J-00 J-00

we can obtain the energy and momentum of the moving breather

_ 16 sin u! , „ \&vs\nu!Ebr= y-—'-i, and Pbr = ftm (7.219)

VI — v* VI — vl

We then obtain the relation

El = Pi + Ml,

where Mbr = 16 sin/x' is the mass of a motionless breather. Since the total energyof a free kink or an antikink is equal to 167, the binding energy of the breather isEB = 167(1 -s in fi').

In the case of C" ^ 0 and /?' ^ 0, Lamb and Davydov assumed that

a' = v W , 9{x)F{t) = g(x)F{t).


g2x = -a'g* + lg2 + a1, F? = -oclFi + (£- 1)F2 + a1.

When a' > 0, we can denote

gl=a'(g20-g

2)(g2-gl),

where

Thus

^(ar) = 50 cn(u, K), F(t) = Fo en | 2 * , j ,

(7.218)


where

u = aK lgo{x-Xo), ^ = _ _ _ _ _ _

o = ^ 7 ^ - l + V^-l)2+4a'},

j2 _ i - 1 + y/(£ - 1)2 + 4a'

2^/(1 - I)2 + 4a'

The solution of the Sine-Gordon equation in such a case is

t_ [*(£!_] = <j0F0 en [*^(« - z0),if] en [ ^ t , j \ . (7.220)

This solution represents "plasma vibrations" of standing periodic waves.If /?' in (7.210) is replaced by -/?' and C" 7 0, /3' < 0, ^" = y/-pO > 0, in

accordance with the above method, Lamb and Davydov obtained solution of theSine-Gordon equation

tan i,/>(z,t) I = 30^0 dn(u, K) sn(V, J'),

with

u=~goyrPJ(x-xo), V = ^Fot,

(_ji\2 _ x <- v i,1 <-; ^p

~ l -z+VCi-^) 2 -^ 1 ' '.2 _ t + y/p-y -2 _ l - z + y / a - Q g - ^ "5o - 2/?" ' ° ~ 4/9"

This solution describes breather oscillations.

7.11 Solving Higher-Dimensional Equations by Reduction

In the previous section, we solved dynamic equations in one-dimensional systems.In this section, we discuss methods for solving higher-dimensional, for example, twoand three-dimensional, dynamic equations in the nonlinear quantum mechanics.We introduce a method of reduction from higher to lower dimensional equationsproposed by Hirota and Tajiri. As a matter of fact, for the two-dimensional case,the solutions for some similarity variables have been studied by Nakamura et al.They obtained an explosion-decay mode solution by generalizing the similarity typeplane wave solution derived by Redekopp. Tajiri investigated the similarity solutionsof one- and two-dimensional cases using Lie's method.


(I) For the two-dimensional nonlinear Schrodinger equation with r' = 0 in(7.188), i.e.,

i<t>t + p<t>X'x- + q'<f>y'y> + b\<l>\2(t> = 0 , (7.221)

where p, q, b are constants, Tajiri considered an infinitesimal one-parameter (e)Lie's group in the (x',y',t',<f>) space

x = x' + eX(x',y',t',<l>) + O(e2),

y = y' + eY(x', y\ t', 4>) + 0{e2), (7.222)

i = t' + eT(x',y',t',<f>) + O(e2),

4> = (f> + eU(x',y',t',4>) + O(e2).

Then

fc = 4,t + e[Ut] + O(e2),

k* = 4>X'X- + e[Ux>x>] + O(e2), (7.223)

4>vv = <f>v'v' + 4ux>x>] + O ( e 2 ) ,

where

,TT , 8U fdU dT\ 8Y dX dT o dY dX[Ut>] = -w + {di~w)'Pt'- W** - ~dv*x'" W ~ W^'" ~d**tliPx''Similarly

r r r . d2U / 82U d2X\ d2Y d2T[U*'x>] = + (^2^-^ - — j <px, - —vyl - — w

fd2u n d2x \ 2 o d2Y d2r+ {W~ d^>) **'" *!*€?*** ~ 2d*d4>ipx"pt'

d2x 3 d2Y 2 d2r 2 fdu ndx\-Wv" ~W*xl<Pv' ~WVx"Pv + \w~ w)*x'x'

dY dT odX dY

dT dY dT-•QVVVVX'X' - *-QT<Px"Px'y' - 2,—lfx'Vx't',

where

, t . 12m . l2m

Assuming that the 2D-nonlinear Schrodinger equation (7.221) is invariant underthe transformations (7.222) and (7.223), we get by comparing terms in first order


of £,

i[Ur] + p[Ux>x>] + q'[Uylyl} + b{2\<j>\2U + <j>2U*) = 0. (7.224)

The solutions of (7.224) gives the infinitesimal elements (X, Y, T, U) while leav-ing invariant (7.221). For (7.224), Tajiri obtained the following for the infinitesimalelements

X = ax' + kt'x' + 4y# + PJt' + 0i,

Y = ay' + kt'y' - ^x' + pSt' + 03, (7.225)Q

T = 2at' + kt'2 + 03,

U=[iU-a-n(t- {-px'2 - ^y'2) + i7z' + i Sy'} 0,

where a, /3, 7, S, K, 9I, 82, 03, and u> are arbitrary constants. Thus, the similarityvariables and form are given by solving the characteristic equations,

dx' _ dy' _ dt' _ d<j>

T ~ Y ~ Y ~ u • {7-Zb)

The general solution of (7.226) involves three constants, two of them becomenew independent variables and the third constant plays the role of a new depen-dent variable. Note that different similarity variables and form can be obtained byintegration of (7.226) for different choices for values of the constants a, 0, 7, S, K,01,02,03 and win (7.225).

Case (1), a, 7, S, K, 0 I , 02, 03 and w ^ 0 and /? = 0. From the integralsof dx'/X = dt'/T, dy'JY = dy/T and dt'jT = d<j>/U, Tajiri obtained the newindependent and dependent variables as follows

. _ x' (0i«-p7a)t + {61a-pyd3)

V\0W\ («*3-aViGiV _ (02* - q'Sa)t + (02a - q'693)

V\QW\ (K03-a2)y/)Q\ '

and

+ = -JL=e""<*'-»''''>«;(&i7)> Q{t') = Kt'2 + 2at' + 03, (7.228)

(7.227)


where

If 1 \f ibryt' + Oi) + 6(q'6t' + 02) 1+ 2 / T i l [ / o7W\ dt\dt

+± [ (f ^ + £ ^ V dt>+^ [(f tEpiA2 dt.

*pJ \J QV\Q\ J WJ \J Qy/\o\ )

+£[([ n<p±dA *> + %[([ 4«p.#\ dfKt' + afe , V2\\

+ 4 {j + 7)l'and in the above, ax is a step function in Q, i.e.,

f 1 for Q > 0,a i ~ \ - 1 for Q < 0.

Inserting (7.228) into (7.221), we have

pwu + q'wm + b\w\2w - IA ($- + ?L\ + B\ W = 0, (7.229)

with

A=J(K93-OJ) ,

B -" {[»- i s s b i ) ] [ f + f + ^ + " " * -2O<791+Ms)]} •Following the same procedure as in case (1), we can get the similarity variables

and forms for other cases. The results are summarized in the following.Case (2) 9 i / 0 and 82 ^ 0, the similarity variables and forms are £ = x' — 6y',

T — t\ <f> = W(£,T), where 6 — 6i/d2. The differential equation is

iwT + (p + q'02)wx + b\w\2w = 0. (7.230)

Case (3) S ^ 0 (or 7 ^ 0), the similarity variables and forms are £ = x' (ory'), T = t', <j> = expfay2/(4g't')]w(£,T) Or exp[ix'2/(4pt')]w($,T). The differentialequation is

iwT + —w + pwtf + b\w\2w = 0,AT

or

iwT + — w + q'wtf + b\w\2w = 0./iT


Case (2) above is the reduction transformations for essentially one-dimensionalpropagation and symmetry. The reduction of case (3) was first found by Nakamura.As an example of case (1), we give the reductions of the 2D-nonlinear Schrodingerequation to the nonlinear Klein-Gordon equation or the <£4-equation in the following.

If K = 1, w ^ O , a = £ = 7 = (5 = 0 i = 0 2 = 03 = O, £ = x'/t', IJ = y'/t', then

*=^K$+-$- P) ]-«•'>• (7-23i)and

pwtf + q'w^ -LJW + b\w\2w - 0.

But if w = 1, 0i ^ 0, 02 ^ 0, 03 ^ 0, a = /3 = 7 = <5 = K = 0, £ = a:' - (0i/02)*',

f = 2/' - (02/*3)f, then

and

pw^ + 4'wqq — mw + b\w\2w = 0,

where

1 6{ 0 |03 4p0f 4g'0|-

We note that the above equations are the nonlinear Klein-Gordon equation or the04-equation when pq' < 0.

The solutions of the ID-nonlinear Schrodinger equation and nonlinear Klein-Gordon equation can also be transformed into the solutions of the 2D-nonlinearSchrodinger equation through the similarity transformations. Substituting the soli-ton solutions of the ID-nonlinear Schrodinger equation and the nonlinear Klein-Gordon equation into the transformations, we get the well known soliton solutionsextended in the (x,y) plane. The solutions obtained by substituting the solitonsolutions of the nonlinear Klein-Gordon equation into the transformation (7.231)are the self-similar soliton-like solutions,

^4 e x p K^ + ^-^) ] x (7-232)\ ~r sech. / x — - c\ —- — Co , for — > 0, —=• > 0,V b V p + gcj \f t' s u / p + q'c\ p + q'c\

fu / — w (x1 v' \ ui rW T t a n h J c » _ ^ 0 for < 0, 2 > 0,V o y 2 (p + g'cj) \ t' t' / p + q'c{ p + q'cf

which was first obtained by Nakamura.


(II) Tajiri considered further the solutions of the 3D-nonlinear Schrodinger equa-tion (7.188) using the same reduction method. Assuming that the coefficients p, q'and r' are not all the same sign, for example, p, q' > 0 and r' < 0, and using thetransformation,

X = -j=(x'-elt'), Y = -j=(y'-02t), Z=-=(z'-93f), z' = zyj^,

<j> = exp {i \^{x' - Otf) + ^,{y'- W) + ^z'~ W) + 9^\ } Wy'Z)>(7.233)

Tajiri obtained

Uzz - Uxx - UYY + CU- b\U\2U = 0, (7.234)

which is formally the 2D-nonlinear Klein-Gordon equation, where 6\, 82, 63, and 6âre arbitrary constants, and

4p 4g' 4r'

Making transformation once more

a a

^ ax/g7 a^-rq' y^/p ay/q1 ay/-rq' J

U = exp {i [ ± ^ 4 ( - M + \ (a + £) 1} G^'we get

iGT ± f ^/^T^G« ± - ^ L = | G | 2 G = 0, (7.236)

which is formally the ID-nonlinear Schrodinger equation, where a and a are arbi-trary constants. It is well known that the ID-nonlinear Schrodinger equation canbe reduced to the following equation

^ = 2F3 + CF, (7.237)

(7.235)


by the similarity transformation. Using results in (7.233) - (7.235), equation (7.188)with r < 0 can be reduced to (7.237) by the following transformation,

'•(^^•^-(Va)']}-2^'f [ l , 1 , y/l + a2 , (Oi 6>2 \ZlTo7. \ ,]2]

,_( *2_,)1/>(.?a^)I/1^{<f^.ft0+^.fc0VaVl + o:2/ V o / I L2P 29

1/ cw'-g3t' p (x'-erf a(y'-e2t')\

X V y/P a^4 ^ a v ^ 7 )

where /9 is an arbitrary constant. We note here that (7.236) with 6 > 0 is trans-formed into the equation with b < 0 by the substitution r -> —r and £ -> i£.

The 2D-nonlinear Schrodinger equation (7.221) can be easily reduced to theID-nonlinear Schrodinger equation by the similarity transformation

Z = x'-9y', r = t', - | ( p + g'02)i'3] } F(C).

We note here that there are some differences between the similarity variables(7.238) and (7.240). For the wave described by the similarity variable (7.238), thedirection of wave propagation depends on 0, where /? is a parameter for the reduc-tion of the ID-nonlinear Schrodinger equation to (7.237). On the other hand, forthe wave described by the variable (7.240), the direction is independent of (5. Ithas been pointed out that there is a close connection between the Painleve equation

(7.240)

(7.238)


(7.237) and the completely integrable nonlinear evolution equations by Ablowitz etal.. They conjectured that when the ordinary differential equations obtained by thesimilarity reduction from a given partial differential equation are the Painleve-type,the partial differential equation will be integrable. According to this conjecture,it seems that the 2D-nonlinear Schrodinger equation (7.221) has a iV-soliton so-lution, which however is a parallel propagating solution so that the propagationis effectively one-dimensional. On the other hand, the 3D-nonlinear Schrodingerequation has three-dimensional propagating JV-soliton solution which is not par-allel propagating. It is well known that the ID-nonlinear Schrodinger equationhas iV-soliton solutions. Substituting the well-known iV-soliton solutions of theID-nonlinear Schrodinger equation into the similarity transformations (7.236) and(7.239), we can get iV-soliton solutions of the 3D-nonlinear Schrodinger equationand of the 2D-nonlinear Schrodinger equation, respectively. Then, we have three-dimensional propagating iV-soliton solutions of the 3D-nonlinear Schrodinger equa-tion and parallel propagating 7V-soliton solutions of the 2D-nonlinear Schrodingerequation.

Bibliography

Abdullaev, F. (1994). Theory of solitons in inhomogeneous media, Wiley and Sons, NewYork.

Ablowitz, M. J. and Segur, H. (1977). Phys. Rev. Lett. 38 1103.Ablowitz, M. J., Kaup, D. J. Newell, A. C. and Segur, H. (1973). Phys. Rev. Lett. 31 125.Ablowitz, M. J., Ramani, A. and Segur, H. (1980). J. Math. Phys. 21 715.Baeriswyl, D. and Bishop, A. R. (1992). Solitons and polarons in solid state physics, World

Scientific, Singapore.Barut, A. D. (1978). Nonlinear equations in physics and mathematics, D. Reidel Publish-

ing, Dordrecht.Bender, G. M., Milton, K. A., Moshe, M., Pinsky, S and Simmons Jr. L. M. (1987). Phys.

Rev. Lett. 58 2615.Bender, G. M., Milton, K. A., Moshe, M., Pinsky, S and Simmons Jr. L. M. (1988). Phys.

Rev. D 37 1472.Binder, G. M., Milton, K. A., Pinsky, S. and Simmons Jr., L. M. (1989). J. Math. Phys.

30 1447.Bishop, A. R. (1978). Soliton and physical perturbations, in Solitons in actions, eds. by

K. Lonngren and A. C. Scott, Academic Press, New York, pp. 61-87.Bullough, R. K. and Caudrey, P. J. (1980). Solitons, Springer-Verlag, Berlin.Chen, H. H. (1978). Phys. Fluids 21 377.Cooper, F., Lucheroni, C. and Shepard, H. (1992). Phys. Lett. A 170 184.Cooper, F., Shepard, H. K. and Simmons Jr., L. M. (1991). Phys. Lett. A 156 436.Davydov, A. S. (1985). Solitons in molecular systems, D. Reidel Publishing, Dordrecht.Dirac, P. A. M. (1930). Proc. Cambridge Philos. Soc. 26 376.Drazin, P. G. and Johnson, R. S. (1989). Solitons, an introduction, Cambridge Univ. Press,

Cambridge.Duncan, A. and Moshe, M. (1988). Phys. Lett. B 215 352.


Eilenberger, G. (1981). Solitons: mathematical methods for physicists, Springer, Berlin.Fogel, M. B., Trullinger, S. E. and Bishop, A. R. (1976). Phys. Lett. A 59 81.Fogel, M. B., Trullinger, S. E., Bishop, A. R. and Krumhansl, J. A. (1976). Phys. Rev.

Lett. 36 1411.Fogel, M. B., Trullinger, S. E., Bishop, A. R. and Krumhansl, J. A. (1977). Phys. Rev. B

15 1578.Gardner, C. S., Greene, J. M., Kruskal, M. D. and Miure, R. M. (1967). Phys. Rev. Lett.

19 1095.Gorshkov, K. A. and Ostrovsky, L. A. (1981). Physica D 3 428.Grimshaw, R. (1979). Proc. R. Soc. London, Ser. A 368 359 and 377.Guo, Bai-lin and Pang Xiao-feng, (1987). Solitons, Chin. Science Press, Beijing.Hirota, R. (1972). J. Phys. Soc. Japan 33 551; Phys. Rev. Lett. 27 1192.Hirota, R. (1973). J. Math. Phys. 14 805.Hirota, R. (1974). Prog. Theor. Phys. 52 1498.Hirota, R. (1977). Prog. Theor. Phys. 57 797.Hirota, R. (1978). J. Phys. Soc. Japan 45 174.Hirota, R. (1982). J. Phys. Soc. Japan 51 323.Hui, W. H. (1979). J. Appl. Math. Phys. 30 929.Ichikava, Y. (1979). Phys. Scr. 20 296.Jones, H. F. and Moshe, M. (1990). Phys. Lett. B 234 492.Karpman, V. I. (1977). JETP Lett. 25 271.Karpman, V. I. and Maslov, E. M. (1977). Sov. Phys. JETP 46 281.Karpman, V. I. and Maslov, E. M. (1978). Sov. Phys. JETP 48 252.Karpman, V. I. and Maslov, E. M. (1982). Phys. Fluids 25 1686.Kaup, D. J. (1976). SIAM. J. Appl. Math. 31 121.Kaup, D. J. (1977). SIAM. Phys. Rev. A 16 704.Kaup, D. J. and Newell, A. C. (1978). Phys. Rev. B 18 5162; Proc. R. Soc. London, Ser

A 361 413.Kivshar, Yu. S. (1988). J. Phys. Soc. Japan 57 4232.Kivshar, Yu. S. and Malomed, B. A. (1989). Rev. Mod. Phys. 61 763.Kodama, Y. and Hasegawa, A. (1982). Opt. Lett. 7 339.Konno, K. and Wadati, M. (1975). Prog. Theor. Phys. 53 1652.Konno, K. and Wadati, M. (1983). J. Phys. Soc. Japan 52 1.Kundu, A. (1987). Physica D25 399.Lamb, G. L. (1969). Phys. Lett. A 29 507.Lamb, G. L. (1971). Rev. Mod. Phys. 43 99.Lamb, G. L. (1980). Elements of soliton theory, Wiley, New York.Lax, P. D. (1968). Comm. Pure and Appl. Math. 21 467.Liu, S. K. and Liu, S. D. (2000). Nonlinear equations in physics, Beijing Univ. Press,

Beijing.Lonngren, K. E. and Scott, A. C. (1978). Solitons in action, Academic, New York, p. 153.Malomed, B. A. (1984). Phys. Lett. A 102 83.Malomed, B. A. (1985). Physica D 15 374 and 385.Malomed, B. A. (1987). Opt. Commun. 61 192.Malomed, B. A. (1987). Physica D 24 155.Malomed, B. A. (1987). Physica D 27 113.Malomed, B. A. (1987). Phys. Lett. A 120 28.Malomed, B. A. (1987). Phys. Lett. A 123 459 and 494.Malomed, B. A. (1988). J. Phys. C 21 5163.Malomed, B. A. (1988). Physica D 32 393.


Malomed, B. A. (1988). Phys. Rev. B 38 9242.Malomed, B. A. (1988). Phys. Scr. 38 66.Malomed, B. A. (1989). Phys. Lett. A 136 395.Nakamura, A. (1981). J. Phys. Soc. Japan 50 2469.Nakamura, A. (1982). J. Math. Phys. 23 417.Nimmo, J. J. C. (1983). Phys. Lett. A 99 279.Nimmo, J. J. C. (1988). Symmetric functions and the KP hierarchy in linear evolutions,

ed. by J. J. P. Leon, World Scientific, Singapore.Nokajima, K., Onodera, Y. Nakamura, T. and Sato, R. (1974). J. Appl. Phys. 45 4095.Nozaki, K. and Bekki, N. (1983). Phys. Rev. Lett. 50 1226.Nozaki, K. and Bekki, N. (1984). Phys. Lett. A 102 383.Nozaki, K. and Bekki, N. (1985). J. Phys. Soc. Japan 54 2362.Okopinska, A. and Moshe, M. (1987). Phys. Rev. D 35 1835.Ott, E. and Sudan, R. N. (1969). Phys. Fluids 12 2388.Pang, Xiao-feng (1985). J. Low Temp. Phys. 58 334.Pang, Xiao-feng (1994). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing.Pang, Xiao-feng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu.Petviashvili, V. I. and Sergeev, A. M. (1984). Sov. Phys. Dokl. 29 493.Pismen, L. M. (1987). Phys. Rev. A 35 1873.Rao, T. A. and Rangwala, A. A. (1988). Backlund transformation and soliton wave equa-

tion, in Solitons, introduction and application, eds. M. Lakshwanan, Springer, Berlin,p. 176.

Rogers, C. (1985). Theor. Math. Phys. 26 395.Rogers, C. (1986). J. Phys. A 19 L496.Rogers, C. (1986). Phys. Scr. 33 289.Rogers, C. (1989). J. Phys. A 18 L105.Rogers, C. and Shadwick, W. F. (1982). Backlund transformations and their applications,

Academic Press, New York.Rose, H. A. and Weinstien, M. I. (1988). Physica D 30 207.Schochet, S. and Weinstein, M. I. (1986). Commun. Math. Phys. 106 569.Tajiri, M. (1983). J. Phys. Soc. Japan 52 1908 and 2277.Tajiri, M. (1984). J. Phys. Soc. Japan 53 1634.Wai, P. K. A., Chen, H. and Lee, Y. C. (1990). Phys. Rev. A 41 426.Zabusky, N. J. and Kruskal, M. D. (1965). Phys. Rev. Lett. 15 240.Zakharov, V. E. and Shabat, A. B. (1972). Sov. Phys.-JETP 34 62.Zakharov, V. E. and Shabat, A. B. (1973). Sov. Phys.-JETP 37 823.

Chapter 8

Microscopic Particles in DifferentNonlinear Systems

In this chapter, we will discuss the states and properties of microscopic particlesin various nonlinear systems. We will consider motion of microscopic particlesin inhomogeneous media with linearly varying density, in dissipative and randommedia, and in the presence of external electromagnetic fields, traveling and standingacoustic waves, and other types of time-dependent potential fields. Many-bodyeffects and collisions of microscopic particles in perturbed cases will be also discussedin this chapter.

8.1 Charged Microscopic Particles in an Electromagnetic Field

The two-deminsional nonlinear Schrodinger equation of a charged microscopic par-ticle in an electromagnetic field is of the following form, in natural unit system,

i^tfrt) = I -\ [v - \iB{t) x l ] 2 - E(t) -X + b\ <t>{X,t) |2 U(X,t), (8.1)

where X' = (x, y, z) denotes the Cartesian coordinates, V denotes the gradient withrespect to X, and E(t) and B(t) represent a time-dependent uniform electromag-netic field. (Such a field may be realized approximately in the middle of a solenoid.Strictly speaking, the electric field is given by E — (1/2)1? x X). The various con-stants have been absorbed into X, t, B, E and <f>. Thus b can take the values ±1,since the sign of the kinetic energy term can be changed, if <j> is replaced by (j>*.Takagi et al. assumed 6 = — 1 and

B = (0,0, B), E=(E1E2,0), (8.2)

and limited the solution to the following form

<j>{X,t) = cp(r,t)exp (ikz - \ik2t\ ,

397


where f = (x,y) and K is a real parameter. Thus (8.1) becomes the following(l+2)-dimensional equation

{% = [~5A " "(t)D + \u2{t)jfi ~ &{t)'f+ 6 |v |2] *' (8-3)where A is the two-dimensional Laplacian,

D = -i{Xly-yt)>E = {ElE2) and u = B/2.

The electromagnetic field may be eliminated by the following three successivetransformations to noninertial frames of reference. The first step is to transform toa rotating (or Larmor) frame by defining tp via

<p(x,y,t) =ip'[xcos0(t) -ysm6(t),xsin9(t) + ycQsd(t),t], (8.4)

where

0(t) = I dt'w(t').Jo

Thus (8.3) becomes

ijtv'{r,t) = [-Â + \u\ty - E'(t) • r + 6|^(f,i)|2] v(r,t), (8.5)

where

E> = (E[E'2)

with

E[ =E1cos6-E2sin0,

E'2 = Eisin6- E2cos6.

The second step is to introduce a scale factor a(t) which is an arbitrary positivesolution of the equation

ii + uj2a = 0, (8.6)

with initial condition a(0) = 1. We then define the scaled time as

T ( < > = / ' I J F (87)

and transform to the dilating frame by defining <p' via

^'vkp^wr^M^w}' (8'8)

Microscopic Particles in Different Nonlinear Systems 399

here d(= 2) is the dimension of the space. This transformation can eliminate theharmonic potential in (8.5). It can also ensure that the cubic nonlinear term in the(1+2) dimensions is invariant under this transformation. Takagi found that

z j ^ ' ( r » = [-Â - Ei(r) • f + *#'(r,r)|2] V'(r», (8.9)

where

Ei = {a[t(T)}}*E'[t(T)},

and t(r) is the inverse of (8.7).Finally, introducing a time-dependent vector R = (Ri,R2) as a particular solu-

tion of

d2R #

or

R(T)= fT dr' [T dT"E1(T"),Jo Jo

and the velocity W{T): W — {w\,W2) = dR/dr. Takagi further performed a time-dependent translation (or a generalized Galilean transformation) by denning ip via

,/,'(F,r) = %l> [r - R(T),T\ exp \iw{r) • f- %- f dT'w2(r')\ , (8.10)

and found that

i | :V( r» = [-iA + &hKr»|2] T/>(f,r). (8.11)

This is the (l+2)-dimensional nonlinear Schrodinger equation in the absence of theelectromagnetic fields. The solution of (8.11) may be written as

ip(f,r) = g(x,t)exp (ifi2y - -i&A ,\ z /

where fi2 is a known real parameter. We can obtain the single envelope-solitonsolution following approach of Chiao et al. and Karpman in the case of an attractiveself-interaction (b = —1),

g(x,r) = ?/sech[7/(a; - x0 - i;ir)]exp \ivix - -i (v2 - v'2)r\ ,

where rf, xo and V\ are real parameters. Hence one obtains the solution

<P(r, t) = ^ysech j ^ [fit) • P- b'(t)} | exp [iQ(r, t)), (8.12)


where

q{t) = (cosO, -s in0),

b'(t) = [xo + Ri(T)+v1T]a,

Q{f't] = \ ( ^ + fc-f) - ^ ' % ) " | (^ -*?)T ~ lfoTdr'w2(r').

Introducing

v = (vi,v2),

k= (uiCOsO + U2 sin 6,-uisind + u2 cos 6),

(ui,u2) = U + «;(T),

where 6, a, and r are functions of t, defined in (8.4), (8.6) and (8.7), respectively.Equation (8.12) is the solution corresponding to the following initial condition

<p(r, 0) = T]sech[n'(x - ar0)] exp ^ ( O j f 8 + iv • r \ . (8.13)

The direction q(t) of propagation of the envelope \<p\ rotates, and the instanta-neous frequency of the rotation is the Larmor frequency ui(t) or half of the cyclotronfrequency B(t). The center b'(t) (i.e. the radial distance from the origin) performsforced harmonic motion b'(t) + ui2b'(t) = Ei cos 6 - Ei sin#. At the same time thewidth and the peak height are modulated by the scale factor a which itself executesharmonic motion (8.6).

In the case of E = 0, v = 0, and a constant magnetic field and a positive us value,Takagi chose the solution a = coswi of (8.6), so that the solution (8.12) becomes

._ . , sech {77' \(x - x0) - ytan(u)t)]} \i fri'2 «\ . .1

^ t ] = ^ cosM) exp [2 y b ~ " r ) t a n H • (814)

Then under a constant magnetic field, the initial field configuration, <p(r, 0) =7/sech[77'(a; — z0)], will collapse at time TT/2W and rotate by 90° within its life-time. The solution cannot be extended beyond this time. In the case of a pulsedmagnetic field imposed on the system which has a constant value for 0 < t < toand vanishes otherwise, where to is a parameter less than TT/2W, Takagi again chose(8.14) for 0 < t < t0 and a = cos(ojto) — (t - to)w sin((jjt0) for t > to accordingly.Solution (8.12) then corresponds again to the initial condition (8.16) and collapsesat time to + ui~1 cos(u;£o)- However, if we choose

{ cosMto -1)]— , U < t < to

cosMo) , (8.15)cos(wfo)


then solution (8.12) corresponds to the initial condition

<p(r, 0) = rf sech[r){x - x0)] exp I - [ir*u tan(wi0)] } • (8.16)

For t > to, this solution can be expressed as

ip{r,t) = »/cos(a;£o)sech{r7[a;cos(u;£o) - ysm(wt0)] - v'xo}

exp I -if) [t-to + w"1 tan(wio)] | , (8.17)

where fj = r]' cos(cjto)- If the initial phase factor is arranged as in (8.16), then thefield configuration rotates by ujt0 and its scale changes by the factor cos(wio), butthe lifetime is infinite.

The generic soli tons in the case of b = +1 are unstable in the presence of amagnetic field. But, within their lifetime, their motion may be controlled to acertain extent by application of time-dependent electromagnetic fields.

The above model may also be used to describe charged bosons interacting viashort-range force, ip would then represent the quantized field operator, and theabove result would generate a classical solution (i.,e., coherent part of tp) on whichquantum treatment may also be based.

8.2 Microscopic Particles Interacting with the Field of an ExternalTraveling Wave

In the case of a microscopic particle interacting with the field of an external travelingwave, the nonlinear Schrodinger equation can be written as

i^ + B + b^ = £eikX'~°Jt'' ( 8 - 1 8>where t' = t/h, x' = x^/2m/h2, e is the coupling strength which is small, i.e.,e -C 1, u> and k are frequency and wave number of the driving field, respectively.Obviously when e = 0, the solution of (8.18) is given by (4.56). At present, it isgiven by

Mx>,t>) = ^Mi[yX'/2 + (v2 - v>)t/4 - Qo - */2]}Vbcoshl-rjix' -vt' -2XO/T])/2] '

where v, r), 9Q and XQ are free parameters. 77 defines the amplitude and width (I/77)of the microscopic particle (soliton), v is the speed of the microscopic particle, 2xo/r]is the location of its center of mass, and $0 is its initial phase.

According to Faddeev and Takhtajan, the above four parameters of the single-soliton solution form a Hamiltonian dynamic system. r\ becomes the canonicalmomentum conjugate to the coordinate 6 = 60 — {rj* — v2)t'/4, and the velocity v

(8.19)


is the canonical momentum conjugate to x'o = xo — (r)v/2)t'. The correspondingHamiltonian is of the form

*<*>-£(i&r-y)*=K">'-i'') <8-2o>when e ^ O . The perturbed system has the Hamiltonian form, equation (8.18) canbe obtained from the Hamilton equation

.d(j> _ 6Hl8t' ~ 6<f>*'

where the Hamiltonian is given by

H[+] = | ~ dx' | | ^ | 2 - \\^ + 2eK [<AV(**'--'>] | . (8.21)

Cohen employed the adiabatic approximation to study variations of the parame-ters of the microscopic particle (soliton) due to the external field. Assuming a weakcoupling (e <C 1), he neglected the radiation effects and possible formation of othersolitons. Inserting the unperturbed solution (8.19) into (8.21), one can get

H[<j>s] = H(r),v,6,x0,t') (8.22)

= 7 \VV n7? + £ 7 r s i n " — -u? +0\ sech ^—-—— .

4 V 3 J I r\ J I 2rf JThe Hamiltonian (8.22) describes a dynamical system with two and a half degrees offreedom, as it is nontrivially coupled to an explicitly time-dependent driving force.

It can be easily shown that the perturbed one-soliton Hamiltonian (8.22) has anadditional integral of motion

r][v - 2k] = const. (8.23)

This integral follows from the complete and irreduced system, (8.18), possessing anexact conservation law,

/(^) = s r° ( V f r ) dx' =const- (8-24)where rp = </>exp[—i(kx' - uit')}. Equation (8.23) can be obtained immediately byinserting (8.19) for <j> into (8.24).

Cohen chose the integral of motion (8.23) as a new momentum and made thecorresponding canonical transformation in the one-soliton parameters' phase space,and exploited the above mentioned symmetry in the time-dependence of the one-soliton Hamiltonian to reconstructed a new Hamiltonian

flh, P, n. Q) - 1 [1(P + 2*r -$<]-« + 'J^f) • (8'25)


where the newly introduced canonical coordinates are given by

q = v, P = r,(v-2k), Q = ^-.

n = e_'!Ll^xio_ut> + lj (8.26)T] I

The new momentum P is a constant of motion of the system. Therefore, theHamiltonian (8.25) represents a system with one effective degree of freedom and isthus integrable. This fact excludes any possibility of chaotic motion in the reducedone-soliton system. Obviously, there is a possible resonance between the externalwave and the microscopic particle (soliton) in (8.25) which is related to a stable(elliptic) fixed point. To look for the fixed point, we set both dH/dq and dH/dftto zero. This yields two conditions. The first is fio = nir (n = 0,1). The secondcondition is obtained by limiting ourselves to the zero order approximation in e,that is,

q0 = ± [2(ifc2 -u)± v/4(jfc2 - w)2 - P 2 ] V 2 . (8.27)

The nontrivial, multivalued resonance condition for the momentum q is causedby the nonstandard dependence of the Hamiltonian (8.25) on the momentum q.Prom (8.27) we can get two conditions for the existence of a resonance between thedriving field and microscopic particle (soliton), i.e., k? > ui, which is a condition onthe dispersion relation of the external wave alone, and 4(k2 — ui)2 > P2, which is acondition on the parameters of the microscopic particle (soliton) and external wave.The linear stability of the fixed points and (8.27) are determined by the sign of theproduct UR', where R' = d2Ho/dq2 (q = qo) [Ho is obtained from the Hamiltonian(8.25) by setting e to 0], and U = en cosh"1 (irP/2q%). In the first order of e, theresonance condition for q becomes

4 ,, , , , , 47r2Psinh(7rP/2o2)9

4 - 4 ( * 2 - w « 2 + P 3 T e u2, P / O S = 0 ' 8 - 2 8

q0 cosh* (-KPfiql)where the values of go should be taken from the zeroth order approximation (8.27).Prom (8.28), we can see that for a given P, two of the resonant values of q do notexist when

Pg0cosh2[(7r/2)(P/g02)]

47r2Psinh[(7r/2)(P/29o2)] ^ '

which leads to a topological change in the phase plane.The resonant condition in laboratory coordinates can be obtained from Cl — 0,

i.e.,

x d \fv-2k\ ,1


or

*-;MT)-"=0- (829)

However, P is a constant of motion for the reduced system and P — q — 0 atresonance. Therefore (8.29) becomes

Now x'o = qvs/2, and vs is the velocity of the center of mass of the microscopic

particle (soliton). Letting k — v/2 be the soliton internal wave number, and wc = 0

(in the leading order in e), the resonant condition can be written as

uc — kvs = LJ — kvs. (8.30)

This condition is the same as the Doppler shifted resonance between two waves,the external (pumping) wave with k and ui and the carrier wave of the microscopicparticle (soliton) with k and u>c. There are two Doppler shifts at the resonantcondition. The first enters the right side of (8.30), and it is by the center-of-massvelocity of the soliton. The second, entering the left hand side of (8.30), is lessintuitive, and is also by the center-of-mass velocity of the soliton, but it is relatedto wave number of the soliton carrier wave. This shows the particle-like feature ofthe microscopic particle (soliton). When k -> 0, the system reduces to the well-known resonance with a homogeneous ac drive. For UJ -* 0 and P = 0, the resonantvalues of the soliton amplitude are reduced to rj — q = ±\/%j. For u > 0, from(8.27) we can find that there is no resonance for k — 0 due to k2 > u, which meansthat the resonance is unique to coupling to the external traveling field. When thevalues of the wave vectors of the driving wave and the internal soliton wave areclose each other, i.e., K = v/2, this translates to P = 0. From (8.28) we see that inthis limiting case, the reduced system will be at the bifurcation point for all valuesof e. For k = 0 and vs = 0, the resonance (8.30) is reduced to a simple resonance.

8.3 Microscopic Particle in Time-dependent Quadratic Potential

In the following, we consider the states of a microscopic particle placed in a time-dependent quadratic potential,

V(x,t) = di(i)x + d2(t)x2.

The corresponding nonlinear Schrodinger equation for the microscopic particle is

•I 2

i<t>t = -z-txx ~ b\4>\2<t> + V(x, t)<t>, V(x, t) = V dn(t)xn, (8.31)

2 m ^ i


where h is set to 1, di(t) and d,2(t) are arbitrary functions of t except that d,2(t)satisfies certain conditions. Solutions of (8.31) were studied by Nogami and Toyamaby transforming the coordinate from the "laboratory system" to the system which isfixed on moving body in which the center of the microscopic particle (soliton) is atrest at the origin, separating V(x, t) into the center-of-mass motion and the internalstructure of the soliton. The transformed equation determines the structure of themicroscopic particle which turns out to be independent of its motion. This methodis a generalization of Husimi's method for solving the linear Schrodinger equationwith the same V(x, t).

When V{x,t) = 0, the solution of (8.31) is given by (4.53) or (8.19), which cannow be written as

4>{x,t) = B0(x-vt)exp<t \mvx- (eo + -mv2 J t \ \ , (8.32)

where

[kB0(x — vt) = y - sech[fc(a; - vt)],

B0(x) is a bound state solution of the time-independent equation

-^Boxx-bB* = eoBo. (8.33)

For the external potential V(x) = max (a is a constant), Chen and Liu showedthat (8.31) is integrable. Its one-soliton solution behaves like a classical particle ofmass m subject to the potential V(x) = max. The shape of the soliton remains thesame as that of the free soliton given by Bo of (8.33).

For the nonlinear Schrodinger equation with an external potential, Chen and Liufurther showed, by employing a transformation as used by Husimi, that (8.31) withV(x) = mui2x2/2 has an one-soliton solution which again behaves like a classicalparticle. However, the structure of the soliton in this case is different from the freesoliton. Husimi's transformation changes the space variable from x to x' = x — £(£),which is the coordinate relative to the moving origin £(£). Later £(£) will be takenas the center of mass of the soliton. Let the solution <j)(x, t) of (8.31) be of the form

<t>(x,t) = iP(x',t)eimii', (8.34)

where

i _ dm


Inserting Husimi's transformation and (8.34) into (8.31), we get

itpt = -T—^X'X' -b\^\2tp + V2(x',t)ipZm

+ [ml + Vc tf, *)] *V ~ [\me - V(£, t)] i>. (8.35)

The relation

V(x' + t,t) = V2(x',t) + V(t,t) + *'Vi(t,t),

where

v«(t.«) = 2 ^

was used in obtaining (8.35). It can be demonstrated that the position of themicroscopic particle (soliton) £(i) satisfies the Newton's equation,

mt = -Vti£,t). (8.36)

In such a case, we can assume that

<t>(x,t) = g(x',t) exp \imix' + i f L{t')dA , (8.37)

where

L{t)=l-mi2-V{^t),

i9t = -^9i'x'-b\g\2g + V2(x',t)g. (8.38)

Equation (8.38) looks like (8.31). But there are two differences, (i) Equation (8.31)is written in the laboratory system, whereas, (8.38) refers to the moving system,with its origin at x = £(t). (ii) The term V\ is absent in (8.38) and thus the paritywith respect to x' is a good quantum number. The latter allows (8.38) to have abound-state solution such that

/•OO

/ |fl(x',t)|2x'dx' = 0.J -oo

That is, this state is confined in a finite spatial region, and its density \g(x',t)\2

does not have to be t-independent. Equation (8.38) determines the structure of thebound state, which is independent of £(i). The center of mass of the bound state isat x' = 0, i.e., x = £(t).

Nogami and Toyama examined the compatibility between (8.31) and (8.33)based on the Ehrenfest's theorem and obtained that

™£= r(bPx-Vx)pdx, (8.39)J—oo


where

p(x,t) = \<j>(x,t)\2-

The term with px — dp/dx on the right-hand side of (8.39) vanishes because

r PxPdx=I-[PT0O=°-J-oo z

Thus the Ehrenfest's theorem in the usual form can be obtained. For the V(x, t)given in (8.31), the following holds

r vxpdx=vi(t,t).•I — oo

Thus (8.39) reduces to (8.36) which shows the classical feature of the microscopicparticle in the system. If d^{t) is i-independent, then g in (8.38) can be written as

g(x',t) = D(x')e-i*t,

where D{x') is real. We take the solution of the ground-state (8.38) to representthe microscopic particle (soliton). Thus the solution of (8.31) can be

cj}(x,t)=D[x-at)]eiS^t\ (8.40)

where

S(x, t) = mf (z - 0 - et + / L(t')di'.

and D(x) satisfies

~^-Dxx{x) - bD3(x) + V2(x)D(x) = eD(x). (8.41)

In (8.41) we have set £ = 0 so that x' = x without losing generality. The solitonobtained in this way moves in the x-direction like a classical particle of mass m, andno "radiation" takes place. That is, the energy of the 0-field is completely carriedby the soliton. The shape of the soliton, determined by D(x') = D(x — £) of (8.41),remains the same throughout the course of motion. The solution is valid no matterhow rapidly g\ (t) varies with time.

For a linear potential, V(x, t) — V\ (x, t) = ma(t)x, since V2 — 0, we find thatD(x) = D0(x) and e = e0. £(£) is determined by m£ = -a(t). With this £(t),S(x,i) in (8.40) can be worked out, and the results are consistent with those ofChen and Liu, as mentioned in Chapter 4.

For the quadratic potential

V(x) = V2{x) = ±mu2x2,


where w is a constant, equation (8.41) becomes

-^-Dxx-bD* + \mJ2x2D = tD. (8.42)

2m 2If the relative strength of the external potential and the nonlinear self-interaction^TOJ /K ~ 1, where K = mb/2, the contributions of the two interactions to the soli-ton binding are about the same. However, in the case of y/muj/K <C 1, the solutionof (8.42) cannot be found analytically. Nogami and Toyama solved the equationnumerically, using 0.0005 for the strength of the quadratic potential, muj2/2, ory/mu/n ~ 0.36.

Fig. 8.1 The effective potential VeS{x) of (8.43) for the ground state of (8.42). The parameters ofthe model are m = 1, 6 = 1 and mw2/2 = 0.0005. The x, t, and V(x) can be taken as dimensionlessquantities.

Fig. 8.2 The same as for Fig. 8.1, but for the first excited state.

Figures 8.1 and 8.2 show these effective potentials

Veff(x)=V(x)-bD(x)2 (8.43)

for the ground state and the first excited state, respectively. The values of e forthese states are -0.1266 and —0.018, respectively. These can be compared withe0 = -k2/2m = -0.125 and w/2 = 0.032 obtained from mw2/2 = 0.0005 andy/mbj/k = 0.036. Apart from the energy associated with the motion as a classical


particle of mass m in the potential V(£, t), the energy of the system is given by

£ = J°° [ ^ ( ^ ) 2 - \bD* + ima,2*2Z?2] dx. (8.44)

£ is not the same as e. [£ becomes e, only if —bD4/2 is replaced with — bD4 in(8.44).] For the ground state and the first excited state, £ are -0.0401 and 0.0169,respectively. D in (8.44) for the ground state approaches Bo of (8.33), when (8.42)is compared with (8.33), while D for the first excited state is similar to that for thefirst excited state (of odd parity) of the quadratic potential without the nonlinearterm. We take the soliton as the ground state of (8.42). The excited states aremuch more diffuse than the soliton state. The phase S(x, t) in (8.40) can be workedout by using £(£) = £o cos(wi). This 5, combined with the approximation D m Bo,can give the solution of Chen and Liu.

For the inverted quadratic potential,

V(x)=V2(x) = --mu)2x2,2*

Equation (8.41), with £ = 0, becomes

-^-Dxx{x)-bD*-\m<J1x2D = eD. (8.45)2m 2

The nonlinear term — bD3 ensures a bound state solution for (8.45). Consider atrial function D localized around the origin that can produce an effective attractivepotential -bD2. When it is delocalized, D and —bD2 become less attractive andthe energy of the system increases. If D is further delocalized away from the origin,the energy begins to decrease because D is more affected by V(x) which is negative.The degree of localization of D can be measured by a parameter A. Then the energyas a function of A has a local minimum. This implies the existence of a bound state,which is denoted by

D(x,X) = ^sech(\x).

Thus, we can obtain £(D) from (8.44) by replacing mu>2x2/2 with —mu)2x2/2, i.e.,

£ (A) = — A - mbX — —— Iv ' 6m [ 4 V A / J

Setting de(\)/d\ to zero yields two real positive roots if

1 2 / 3 \ 3 1 K4

-mw 2 < - —x— ~0.70,2 \ 4 / 2n2 m

where

K = -mb.2i


The larger of the two roots corresponds to the minimum of £(A).Nogami and Toyama studied this problem numerically and found that a bound

state exists when mw2/2 < 0.0007, or y/muJ/K < 0.39 and mw2/2 < 0.0013.Figure 8.3 shows the effective potential Veft(x) for the bound state obtained formtj2/2 = 0.0005. They found that e = -0.1230 and £ - -0.0434 for this state. Ifmw2/2 exceeds 0.0007, the bound state becomes unstable.

Fig. 8.3 Same as for Fig 8.1, but for V2(x) = -mx2/2.

For the linear-quadratic potential,

V(x,t) = \mJix2 - muj2G(t)x,

where mui2G(t)x is an additional perturbation, Nogami and Toyama found, bysolving the Newton's equation for £(£) in (8.36), that

ft£{t) = OJ G{t) sin[w(i - i)]dt. (8.46)

J—oo

A term in the form of £0 cos(wt) may be added to the right-hand side of (8.46). Theamplitude function D(x — £) remains the same as D in (8.42). The phase S(x,t)can be worked out by substituting (8.46) into (8.40). It is noted that G(t) can bechosen arbitrarily as long as the integral in (8.46) is well denned.

Nogami and Toyama further considered the case of -dependent d2{t) in (8.31).If the variation of d2 (t) with time is sufficiently gentle, then an adiabatic approxi-mation can be used for <j> in (8.40). In this case, e becomes ^-dependent, and et inS of (8.40) has to be replaced by / e{i')di'. On the other hand, if the variationof d2(t) becomes wild, then g(x, t) which was in the ground state of (8.41) can getmixed with higher states and become diffuse. If d2(t) < 0, g(x,t) may escape toinfinity.

Nogami and Toyama carried out a number of numerical experiments with (8.31).They observed that soliton states of a microscopic particle survives any rapid vari-ation of di(t). If d2(t) varies rapidly in the range of -0.0007 < d2(t) < 0.001,[d2(£) = 0.001 corresponds to i/mw/rc = 0.42], it was found that the soliton does


not break even when d,2(t) changes suddenly which was surprising. No significanttrace of "radiation" was found from the soliton.

In general, V(x,t) in (8.31) has a constant curvature as a function of x. Ifthe curvature varies with x, the above derivation does not apply. Hoverer, if thevariation of the curvature is very small over the width of the microscopic particle(soliton) (~ 1/K), the method may still be effective. When the microscopic particleis in a region where the curvature of V(x,t) is negative, i.e., d2V(x,t)/dx2 < 0,the microscopic particle experiences a potential similar to muPx2/2. The solitonstate of the microscopic particle can easily survive. The restriction on the curvaturemay be relaxed if V(x,t) is bounded from below. For example, for the repulsive5-function potential V{x) = G'6(x), equation (8.41), with V2 replaced by G'S(x), is

-J-Dxx - bD3 + G'6{x)D = eD.2m

It was shown by Nogami et al. and Foldy that this equation has a bound-statesolution with

e = ~ £ ' -' = lrn(b-2G'),if 2G' < b. By simulating the <5-function using a Gaussian function with a smallwidth, Nogami and Toyama found that the curvature of this potential at x = 0 hasa very large magnitude.

8.4 2D Time-dependent Parabolic Potential-field

The behaviors of a microscopic particle described by the nonlinear Schrodinger equa-tion (3.2) in higher dimensional systems are of interest. Garcia-Ripoll et al. stud-ied properties of the microscopic particle depicted by the following two-dimensionalnonlinear Schrodinger equation with a time-dependent parabolic potential

* ' i t = ~lA4>+w2(t>+\£{t)(x2+y2)(j>' (8-47)where

dx2 dy2'

h = m = b = 1, and A(<j>) = 0 in (3.2), e is a perturbation parameter. In the case ofa radially symmetric system, the above authors represented the solution of (8.47)by cf)(r,Q,t) = <f)(r,t)eimB, in the polar coordinates, which includes both a typicalradially symmetric problem corresponding to m = 0 and the vortex line solutionwith w / 0 . The simplified equation for cf>(r, t) is

.d<j> 1 9 / d<f>\ \m2 . ,.a e(t) 21 , .


This equation is nonintegrable and has no exact solution even in the case of e(t)being a constant. Its solutions are given by the stationary points of the action

ftl

S = / C(t),

Jt2

where

We can define the following integral quantities for the system

h(t) = J\<l>\2d2x,

h(t) = J \<t>\2r2d2x, (8.50)

'<(<) = 5 / ( W + M 2 + M4) .fx,

where d2x = rdrdO and the integration with respect to 6 yields a factor of 2TTbecause of the symmetry. These quantities are related to the number, width, radialmomentum and energy of the microscopic particles.

It is remarkable that the Ij satisfy the following simple and closed evolutionlaws

£=»• £•'•• £--*<•>*+«. f ->) ' • • <"»The first equation comes from the phase invariance of (8.48) under the global phasetransformations and corresponds to the conservation of particle number. The otherequations can also be obtained in connection with the invariance of the action undersymmetry transformations. The equations (8.51) form a linear non-autonomoussystem for the unknowns, Ij (j — I,--- ,4), that has several positive invariantsunder time evolution, of which the most important is

Q = 2hh - j > 0.

Thus (8.51) can be reduced to a single equation for the parameter I2(t) as follows

(8.49)

(8.52)


If (8.52) can be solved, then the use of (8.51) would allow us to track the evolu-tion of the other 7,'s. By defining R(t) = yfh., which is the width of the microscopicparticle (soliton), and inserting it into (8.52), Garcia-Ripoll et al. obtained the fol-lowing

R + e(t)R=-^. (8.53)

This is a singular (nonlinear) Hill equation. Following approach of Reid et al.,Garcia-Ripoll et al. gave the following general solution of (8.53)

r o l1/2

fl(t)=[^(*) + ^ 2 ( « ) J - (8-54)

where <f>(t) and <p(t) are two solutions of the equation

$ + e(t)$ = 0, (8.55)

satisfying the initial conditions: <l>(t0) — R(t0), 0(io) = R'(to), ip(t0) = 0, andtp'(to) 7 0. W in the above equation is the Wronskian W = cp — constant.Equations (8.53) - (8.55) show that the particulars of the microscopic particle, forexample, its width, are determined by the Hill equations. They are closely related tothe solutions of (8.55) which is explicitly solvable only for particular choices of e(t),but its solutions are well characterized and its properties are known. A practicalapplication of (8.54) is the design of e(t) starting from the desired properties of thewave packet.

Ripoll et al. assumed that e(t) depends on a parameter e(t) = 1 + e(t), wheree(t) is a periodic function of time with a zero mean value and a peak value ofeo. There exists a complete theory which gives the intervals of eo within which allsolutions of (8.55) are bounded (stability intervals) and intervals within which allsolutions are unbounded (instability intervals). Both types of intervals are orderedin a natural way.

In the case of e(t) = 1 + eocos(wt), the regions where exact resonances occurcan be determined by several methods. First, for a fixed eo, there always exists aninfinite ordered series, {wn}, {wj,}, which approaches zero, and if a; belongs to thisseries, equation (8.55) describes a resonance. Second, when ui is fixed, resonancesappear if eo is large enough. Further, a stability diagram can be drawn in the eo-o;plane. The boundaries of these regions are called characteristic curves. If B(e, w) isthe discriminant of the equation, the characteristic curves can be obtained by solvingthe equations B(e,u) = 2 and B(e,u) = —2. In particular, instability regions startat frequencies u> = 2,1,1/2, • • • , 2/n2, • • •. The resonant behavior depends only onthe mutual relation between the parameters but not on the initial data.

The above analysis on problems with the cylindrically symmetry is completelyrigorous up to this point. In the following, we will use various approximations todiscuss other related problems.


Notice that the above derivation is confined to the case of a special symmetry forthe potential and the solution in order to obtain exact results. When we consider ageneral two- or three-dimensional model of the nonlinear Schrodinger equation (3.2)with A((j)) = 0, and these constraints are removed, a corresponding set of coupledHill's equations may still be possible with certain approximations, for example, thescaling laws or the variational ansatz. One such approach requires the wave functionto have a quadratic form in the complex phase,

(f> = \<p\ exp ii âjkXjXk I , (j, k = 1,2,3).

V i* /By introducing this trial function into (8.49), and using some transformations,Garcia-Ripoll et al. obtained the following from the Lagrange's equations for theparameters

* + *(*>* = + 3 r f ^ . (^)

Here Ri, (i = 1,2,3), are the three root mean square radii of the solution for eachspatial direction, i.e., the extensions of the previous R(t) to the three-dimensionalproblem. In this case, ei(t) are not necessarily equal. Equations (8.56) are not inte-grable and form a six-dimensional non-autonomous dynamical system. Numericalstudy on this approximate model (8.56) produces an extended family of resonanceswhich is more or less the Cartesian product of those of (8.54) with minor displace-ments due to the coupling. The numerical simulation of the nonlinear Schrodingerequation (3.2) with A(<f>) = 0 thus confirms the predictions of the simple model(8.56).

Although resonant behavior has been proven in a particular case, it is of interestto find out whether such behavior is also seen in non-symmetric problems. Garcia-Ripoll et al. thus looked for stationary solution (f>(f, t) = xp(r)e~"jjt for a stationarytrap, V = V(f). They wrote the nonlinear eigenvalue problem as

ujip = -Âip + \ip\2ip + V(r)ip. (8.57)

Solutions of the above equation can be used to expand any solution of the non-stationary problem (3.2). By doing so one finds out that the energy absorptionprocess in the time-dependent potential is determined by the separation betweenthe eigenvalues, w; — ujj, of any two different modes of (8.57). If these differencescan be approximated by multiples of a fixed set of frequencies, then an appropriateparametric excitation will induce a sustained process of energy gain (and widthgrowth) such as the one we discussed earlier.

Garcia-Ripoll et al. also studied the spectrum {LJ} in an axial symmetric 3Dpotential V(f) = er(x

2 + y2) + ezz2, using a variant of a pseudospectral scheme


and the harmonic oscillator. The results show that the spectrum exhibits an or-dered structure, with different directions of uniformity that may be excited by theparametric perturbation. Results in ID, 2D, and 3D cases without symmetry as-sumptions were also compared, and are similar despite the fact that the spectrumbecomes more complex as dimensionality increases.

In conclusion, by using the moment technique for the cylindrically symmetricproblem, a singular Hill equation was obtained which is reducible to a linear Hillequation. Existence of resonances can be verified analytically. For a periodic per-turbation, there exist strong extended resonances for relevant parameters of thesolution even when the solution is constrained by conservation laws. It is inferredusing the parabolic ansatz, analysis of nonlinear spectrum, and numerical simula-tions that this behavior is also present in non-radially symmetric problems.

8.5 Microscopic Particle Subject to a Monochromatic AcousticWave

If the potential field V(x,t) given in (3.2) is due to the following monochromaticacoustic wave,

V(x,t) = Vac{x,t) = Acos(fct - kx + 0O)

where k > 0, and 0O is a phase shift of acoustic oscillations in the center of themicroscopic particle (a; = 0). Obviously, V^dît) satisfies the linear variationalequation, Vactt - V&cxx = 0. In such a case, equation (3.2) becomes

i<h + 4>x-x< + 2\<l>\24> = Vacfr ' , * ' ) * . (8-58)

where t/h -> t', and x' -> y/2m/h2 x. This equation describes the coupling betweenthe microscopic particle (soliton) and the field of the linear acoustic wave. If theamplitude of the acoustic wave, A, is sufficiently small, i.e., A <C T;2(1 — v2), whereT) and v are the amplitude and velocity of the soliton described by (4.8) or (4.56),respectively, which is a solution of (8.58) with Vac = 0, then it is natural to employthe inverse scattering method discussed earlier to solve (8.58) in the presence of theinteraction term.

Inserting the perturbation in (8.58) into the general perturbation-induced evolu-tion equation (7.70) for the radiation amplitudes B(\,t), and assuming the solitonto be quiescent, Kivshar et al. obtained

dB in AP |>(fc + 2A)l i { [ f c _ 4 ( A W ) ] + 9 o }

dt> 8 A2+7,2s e c h [ 477 J ' ( }

Using (8.59), Kivshar and Malomed also obtained the emission intensity usingthe same approach as that used in studying emission problems described by theperturbed Sine-Gordon equation.


Here, we integrate (8.59) directly. We can multiply the right-hand side of (8.59)by exp(at'), where a is an infinitesimal small but positive parameter. Physically,this is equivalent to turn on adiabatically a perturbation that was absent at t' =—oo. Then the following can be obtained,

D . , w / , 7T Ak2 ^\ir(k + 2\)] e-i{[k-i(x2W)]t'+e0}B*(X,t') = - - sech M— >-\ j - — , . (8.60

8 \* +r)'i [ Ar) J [k — 4(A^ + rj2)] +ia

Substituting (8.59) and (8.60) into the equation

fw'« = H B " f ) ' (8'61)where

is the spectral density of the microscopic particle number, A = £ + it] is a spectralparameter, v = — 4£, o(A) is denned in (7.63) in connection with

lim _ 1 . = P (^] - iirS(F),*-+oF + ia \FJ

where P represents the principal value, we find

> ) = \(^f -*1 [-^Hp]' (* + n* -1) • (8.62)The total emission rate of the microscopic particle number is given by

where

ffl =-^L8 ech ' f f (v^TVT*)]. (8-64)L«ft'J± 8 ^ - 47J2 [4)7 V ^J

From (8.62) - (8.64) we know that emission takes place provided that k > k0 =4rj2, and it is concentrated at two points of the spectrum, 2A± = i-^/fc — 4r)2.The group velocity of the envelope wave of high-frequency emission is vgr = —4A.That is, A < 0 and A > 0 correspond to waves emitted to the right [with intensity(dN/dt')+) and to the left [with intensity (dN/dt)-], respectively. As can be seenfrom (8.64), the intensity of the waves emitted forward, relative to the sound ve-locity, is greater than that of the waves emitted backwards. Using the conservationof the total microscopic particle number, we can find the decay rate of the soliton,

(8.63)


dr)/dt'. The number of the microscopic particle (soliton) is 7Vsoi = 4T/ (Zakharov etal., 1980), so that

§Mf-According to the above condition, rf < k/4, it is natural to distinguish between

"heavy" (T? « Vk) and "light" (rj -C \/k) solitons. We can estimate the characteristicdecay time T\ for a "heavy" soliton into a "light" soliton using (8.63) and (8.65).The result is given by

From (8.63) and (8.64), we see that the decay rate of the "light" soliton takesthe following form

Integrating this equation results in the following expression for the soliton's decayrate

7] « jVkln(kA2t'). (8.67)

Far from the soliton, the envelope wave of high frequency emission looks like themonochromatic wave </> = a± exp[-i(4Aj.t' — 2A±x')]. By equating the microscopicparticle number flux j± = —4A±aj_ carried by the wave (8.67) to the emissionrate (dN/dt')± [see (8.63) and (8.64)], Kivshar and Malomed obtained a generalexpression for the amplitude of the emitted wave

2 _ 4 \dN]a±-\xZ\[dj\±

Kivshar and Malomed also studied the influence of the random acoustic wavefield on a microscopic particle (soliton). The field is defined by a random initialconditions, Va.c(x',t' = 0) = V0(x'), Vact'(x',t' = 0) = V0{x'), subject to the Gaus-sian correlation, (V0(x)) = (V0(x)) = (V0(x)V(x)) = 0, (V0(x)V0(x)) = V0

2S(x - x),(Vo(x)Vo(x)) = Vo 6(x - x), where Vo

2 < r?, Vg «: rf. The random initial con-ditions give rise to acoustic wave packets with both possible values of the groupvelocities cit2 = ±1.

The potential for the random acoustic wave field in this case can be written as

V&c{x',t') = Yl / dqAjiq)^'*"-"').j=l J-°°

(8.65)

(8.66)


It can be shown that the following correlations hold for the spectral amplitudes

(A1{q)A2(^)) = \ (v2 - ?pj S(q - q'), (8.68)

(A1(q)A1(q')) = {A2(q)A2(q')) = | ^ 02 - ^ \ S(q - q').

The expression for dB/dt corresponding to (8.68) is given by

*= iww) Ldqq sech [ J V ^ J x ^8-69)2A i ( g ) exp{ i [ c i g -4 (A 2 +^) ] f} .

Integrating (8.69) in the same way as it was done earlier, we get

™-^£>~*[*±2«]x (8.ro,2 ^ e-i[ci,-4(Aa+na)]«'

2 - AiM [Cjq _ 4(A2 + ^2)] + ia •

Multiplying (8.69) by (8.70) and averaging the product according to (8.68), we canobtain the average emission rate of spectral density,

(±N,my=i^B.mtm) (,ri)

The emission induced by a random acoustic field is described by the smooth spectraldensity given in (8.71). When rj > 1, the spectral density is smeared over a broadspectral range A2 < r)2. If r) <C 1, equation (8.71) has exponentially sharp maximaat the points Xy' = —2CJT)2 and A- = -Cj/2. The latter lies outside the range ofapplicability of (8.58), while in the vicinity of the former, (8.71) takes the form

( AjV(A)) „ V2 + Urj%2) ± sech2 [ T ( A + ^ } ] . (8.72)

Using (8.72) and the conservation of the total number of microscopic particles,


Kivshar et al. derived the average decay rate of the microscopic particles,

&="i/rdx(^Ni{x))="27r2(^°2+^^ (8-73)That is, at the late stage (when rf1 <C VO/AVQ), the microscopic particle decaysaccording to r)(t') fa rj0 exp(—27r2V0

2£'), which is much faster than that described by(8.67).

The feature that distinguishes (8.73) from (8.65) is the absence of an exponen-tially small factor. This is due to the fact that there exist long-wave componentsin the random wave field with wave numbers q for which the exponential-smallnesscondition rf <C q [see (8.64) ] does not hold. However, we should remember that(8.73) describes soliton decay induced by long-wave component of the random acous-tic wave field [see (8.70) -(8.72)]. The main range of the acoustic wave numbers is\q\ —rf<gl 1]. Thus contribution from short-wave component cannot be covered by(8.58).

The above discussion can be applied to other problems, e.g. emission of acousticwaves by colliding microscopic particles which was observed in numerical simulationby Degtyarev et al. (1974). Within the framework of perturbation theory, explicitevaluations of the total number of emitted particles and its spectral density arepossible if the two colliding microscopic particles (solitons) have the same ampli-tudes, i.e., T]I = T]2 = ri, and their velocities v\ = — V2 = v lying in the intervalm < v2 «; i.

8.6 Effect of Energy Dissipation on Microscopic Particles

As it is known, the energy of a microscopic particle is dissipated, when it is movingin a medium. The dissipation results in damping of its motion or radiation of themicroscopic particle. It was proposed by Malomed et al. that linear dissipationeffect can be adequately accounted for by the following two-term potential

Pdiss = -JO<t> + 7l<j>xx,

where 70 and 71 are dissipation constants, and satisfy 0 < 70, 71 C 77, with 7j beingthe amplitude of the soliton, <j>s in (4.56), of the nonlinear Schrodinger equation withV{x,t) = Pdiss = 0, which may be regarded as a perturbation. The correspondingequation is given by

%<k + 4>x'x' + 2|tf>|24> = Pdiss = -7o<£ + 710*'*'• (8-74)

In the absence of the dissipation, the basic emission equation (7.70) may bewritten in the general form [see (8.59)]

^ £ U e/(A)e«°-"'>*\ (8.75)


where Cl is the frequency of emission, /(A) is an arbitrary function of A.When dissipation occurs, equation (8.75) is modified as the following

^ = -7fl(A) + 6/(A)e«<°-«3>«\ (8.76)

where 7 = 70 + 4r]2ji. Let B{\)e<t = B(\), then (8.76) can be written as

^ ^ = e / ' ( A ) e * ' e - ' < n - 4 A X (8.77)

Multiplying (8.77) by the evident solution of this equation,

B(\) = £-Â) it i(n-i\*)t'

"{A) z(ft-4A2)+7

Kivshar et al. obtained

Changing from B(X) back to B(X), and comparing (8.78) with the non-dissipative equation (8.62), we can infer that the delta function in the expres-sion for the spectral density of the radiation rate are "smeared" as <5(fi - 4A2) -^7r~17/[72 + (fl — 4A2)2]. This phenomenon may be called the Lorenz broadening.Its physical meaning is that, due to dissipative absorption, the amplitude of theemitted wave decreases further away from the soliton. The wave is thus not exactlymonochromatic.

Kodama et al. and Malomed et al. further considered the compensating effectto the dissipative effect by introducing a periodic pumping action. In such a case,the dynamic equation of the microscopic particle can be written as

00

^ ' -0x'x' - 2 | 0 2 | 0 = T0 - l + S W-™) • (8-79)n= — 00

where 7 is the dissipation constant, r is the spatial period of the pumping, whichmay be regarded as a small perturbation i.e., TJ C 1 and 7/fy2 <S 1. The coeffi-cients of the two terms on the right-hand side of (8.79) are chosen such that fullcompensation of the dissipation is achieved for all possible values of the amplitude77 and the velocity v of the microscopic particle (soliton). However, under the actionof the pumping pulses, the microscopic particle generates radiation in the form ofsmall-amplitude waves, which can be considered as "noise". Thus a stationary levelof radiation occurs in the case of one soliton propagating in the medium, which wasstudied by Malomed.

Applying the general inverse scattering transformation evolution equation (7.70)to the radiation amplitudes of the perturbed system, Malomed found an increased

(8.78)


amplitude of the radiation field b(X) generated by the microscopic particle (soliton)under the action of the nth pumping pulse

An6(A) = -7T7Tsech f ^ y ^ ] exP [-4i U + JQV2 + X\v) nT] • (8-80)

Because of dissipation, b(X) evolves according to

b(\,t') = b(\,0)e->tl+iix2t'. (8.81)

Form (8.80) and (8.81), the amplitude b(X, t') can be found for nr < t' < (n+l)r,as the sum of an infinite geometrical progression,

oo

b(X, f) = ^2 An_m6(A) exp[(-7 + 4iX2)(t' - (n - m)r)) (8.82)

rff(A + t//4)] exp[-7 + 4i\2(t' - TIT) - li{r? + t;2/16 + v\/2)nr]-njrsec ^ ^ j exp{-7T + 4i[772 + (A + i;/4)2]T}

Under the condition *yr •C 1, it follows from (8.82) that in an equilibrium statethe occupation numbers of the radiation modes (the spectral density of the numberof quanta) take the stationary values according to (8.61), i.e.,

N'(X) = iMrr^^fl^y ^4 sin {2[JJ^ + (A + V/4)2\T}

provided that

sin I 2r)2 + 2T (A + ^ I > -jr.

If this condition does not hold, i.e.,

2 [ 7 ? 2 + ( A + l ) 2 ] r = 7 r J + 2^ (8>84)

where \0\ <C JT and j is an integer, it follows from (8.82) that

N'(\) - 7r(^r)2 ,rrh2 \W*J-2W]i V W - ( 7 r ) 2 + 0 2 S e c h [ 2r}^ J- (8-85)

In the regime of the soliton's motion with sufficiently low pumping frequency,i.e., with TT]2 » 1, (8.83) and (8.85) describe a spectral wave packet whose centerlies at the point A = —v2/A and whose width is |A + v/4\ « r). AS can be seen from(8.84), inside the packet the spectral density n(A) has a large number ( « TTJ2) ofsharp maxima separated by intervals A A « far)"1; the width of each maximum isSX « 7/77 <C A A due to the underlying condition 7T -C 1.


Adding up the occupation numbers, one can get

N'(X) =l-± |A 6(A)|»e-^™ = |7rsech2 f ^ + ^ l . (8.86)m=0 L ' J

This describes the smoothed spectral structure of the wave packet. In calculationof the total number of quanta scattered in the radiation field,

r+ooNrad = / N'(\)dX,

the smoothed spectral density (8.86) yields the same results as (8.83) - (8.85). Inparticular, iVrad = 27x77. Comparing this to iVsoi — 4rj of the quanta bound for thesoliton, we discover that in the stationary regime iVrad is indeed much smaller thaniVsoi, i.e., iVrad/7Vsol = 7 T / 2 « 1.

Kodama and Hasegawa gave the soliton-like solution of the following nonlinearSchrodinger equation with dissipation and high-order dispersion effects

i<j>t' + 2 fa'*' + l< 210 - -*70 + iHx'x'x' •

The soliton-like solution is

(f>(x',t') = if(x',t') ll + p [2r)H' - 3r?tanh(r?(a;' - 20a))] + \l*'2\ eie

+O(i2J2,1P),

where \t'\ -C O(7"1/2), and

<p(x',tl)=r]Sechlr,(xl-2pcT)}, r, = fae-**', a = ^ ( l - e'^') .

Here </>0 is the initial amplitude of the microscopic particle (soliton). The aboveresults show that the dissipation effect of the medium reduces the amplitude andincreases the width of the microscopic particle (soliton) at the rate of e~27* ande27' , respectively. However, since the area (the amplitude time the width) is con-served, the features of the microscopic particle are maintained even in the presenceof the dissipation loss, but the dispersion distorts the group velocity of the micro-scopic particle by 20a.

Nozaki and Bekki gave the soliton-like solution of the nonlinear Schrodingerequation with dissipation and a driving field

i<t>r + <f>x'x' + 1\4?\(t> = -ij(f> - ieeiwt'.

The solution is of the form

cf>(x',t') = 2risech(2T]x')e-2ia-in/2

- ^ [pe-4i<Tsech2(2r7a;') + p* tanh2(2r?x')] + c{t'),


where

p~t> = iuip + Sir]\c\e2lx + inje2* — jp~,

X = 2a- arg(c), c{t) = ceiut,

p = p[i arg(c)], p = p(£ - 0),

at \x\ —> oo. Ao (< 0) is a constant weight-factor.

8.7 Motion of Microscopic Particles in Disordered Systems

Motion of microscopic particles in disordered media is an interesting problem andwas extensively studied because the competition between disorder and nonlinearitycan result in some new and complex effects. Disorder is the origin of exponential de-cay of the transmission coefficient (Anderson localization). Study shows that weaknonlinearity acting against the disorder changes the exponential length dependenceof the transmission coefficient into a power-law dependence. However, a strongnonlinearity can cause a localization decay for the microscopic particle. Moreover,there exists a threshold transmission value for microscopic particles in the media.This problem was studied by Kivshar et al. and can be described by the followingequation

i<h' + <t>x>X' + 2|</>2|(/) = e(x')<l>, (8.87)

where e(x') = e J3n 6(x' - x'n) describes point impurities with equal intensities e atrandom positions x'n. It can also represent structural disorder of associated systems.If the last term on the left-hand side of (8.87) is neglected, then the equation rep-resents propagation of monochromatic waves in a random inhomogeneous mediumwhich leads to the stochastic dynamic equation,

-<f>x>x> + e(x')<t> = k2(j>,

where k is the wave number, and to the phenomenon of localization of states by ran-dom inhomogeneities due to scattering. Localization means that the transmissioncoefficient T decays exponentially with the system length L. If e(x) is a stationaryergodic random process, then a positive finite number for the localization exists.The localization length X(k) is defined by L~l\nT(k) w -\~l{k). For large L(S> \{k)), a very little transmission is allowed.

Kivshar et al. considered the scattering of microscopic particles (solitons) bya random distribution of point impurities with equal intensities e. Once a solitonis incident on the disordered layer, assuming from the left, it will decompose intoreflected (r) and transmitted (t) parts. After passing through each impurity thewave packet will reorganize itself and become again a soliton plus some waves ofsmall-amplitudes. In such a case, there are two integrals of motion, the energy E


and the "number of particles" N, denned by

/*+oo r+oo

E = dz'[ |«M2+e(z')M2-M4], N= dx'^l2. (8.88)J — oo J—oo

The total energy transmission coefficient is given by T^ = Et/Ei where Et isthe transmitted energy, Ei the incident energy. The transmission coefficient corre-sponding to the "number-of-particles" is given by T ^ = Nt/Ni. Obviously, theconstraints, Et = Et + Er = const., and Ni = Nt + Nr = const, hold. When theimpurity concentration p is low, the average distance between two nearby impuri-ties may be larger than the soliton size. In this limit, the scattering can be treatedas that by many independent impurities. Then T w II,-7), where Tj is the trans-mission coefficient of the jth impurity. This approach is still used in disorderedsystems through the mean transmission coefficients. The transmitted soliton forthe jth impurity is then the incident soliton for the (j + l)th scatterer. Thus wehave (see chapter 4)

Ej+i=EJTJE)(Ei,Nj),

Nj+^NjTJîE^Nj), (8.89)

and

AEj+1 = Ej+1 - Ei= - EjRfHEj,^),

ANj+1 = Nj+1 - Nj = -NjR™ {EJ,Nj), (8.90)

where RÊ) = 1 - TJ£) and R^ = 1- TJiV) are the reflection coefficients of energyand number-of-particles, respectively. These coefficients can be calculated for e d ,by employing the perturbation theory based on the inverse scattering method. Theresults are

Zooh JQ

where

cosh [(7r/4a)(i/2 + a2 - 1)]

and a = N/v. These results, obtained within the Born approximation, are valid ife «; 1 and v2 » \e\a. Using the soliton solution <j>s (4.8) or (4.56) in (8.87) withe{x') = 0, and considering that there are (Ax')p impurities in the interval Ax' andthat energy and number of solitons are function of 2r\ and v, given by N — 4rj and

(8.91)

(8.92)


E = N(v2 - iV2/3)/4, we can derive the following equations from (8.90)

dN 1 f+o°- = - - / Q dyF(y,a), (8.93)

dv 1 f+°° N f+°°

TZ = -2N}0 W - D ^ . o o - ^ j f w<y>°)> («•*)where the distance is measured in units of x'o = 64/npe2, i.e., z — x' /XQ . In thelinear limit, a « 1, equations (8.93) and (8.94) can be solved analytically whichyields v(x) = v(0) — constant. Thus

~ JV(O) ~ E(0) ~ ' t 8 ' 9 5 j

where Ao = v2(0)/pe2 = l/pR\, and Ri is the reflection coefficient of one impu-rity. This indicates that the transmission coefficient decays exponentially, which isthe same as that of the corresponding linear problem, where Ao = \{k0), and fcorepresents the wave number of the carrier packet.

Kivshar et al. studied numerically these quantities by employing the usual

rectangle method for estimating the integrals, and Euler's procedure to integrate

the equations. Results in some cases were verified with a leap-frog scheme. In this

case, they found that the asymptotic change in T^N'BS){z) depends essentially on

the value of the parameter a(0) = N(0)/v(Q) that is related to the nonlinearity of

the incoming wave. The greater the a, the larger the number of quasiparticles in

the soli ton becomes, and the smaller its spatial extension. If a is small, the wave

is similar to a linear wave packet. It can be proved by computing the derivative of

a(z) that the solution ac of the transcendental equation a2 - 2 + G(ac) = 0, with

dy(y2-l)F(y,a)

-W = —7+55 '/ dyF(y,a)Jo

is such that a(0) = ac implies a(z) = ac along the whole disordered layer. Solutionsof (8.83) and (8.94) confirm that a(z) is monotonically increasing (decreasing) ifa(0) > ac (a(0) < ac). By solving the equation approximately and integrating(8.93) and (8.94), Kivshar et al. found that ac « 1.28505. Hence, for the initialconditions a(0) <C ctc, the system evolves, in agreement with the analytical result of(8.95), to a final state in which N approaches zero exponentially while v approachesa constant positive value, satisfying a(oo) = 0 as required (see Fig. 8.4). If 1 « a <ac, the decay consists of an initial, slow transient after which a fast exponentialbehavior appears. When a(0) > ac, it leads to a situation in which both N andv become practically constant and so does a(oo), having some limit value arounda ss 10. The dependence of the transmission coefficients on z is determined not onlyby a but also by the values of iV(0) and v(0). In fact, the smaller the N(0) and


v(0), the smaller the interval needed to reach the asymptotic regime, and the initialslope can be roughly 105 for N(Q),v(0) « 0.1. But both transmission coefficientsapproach their asymptotic constant, nonzero values.

Fig. 8.4 Transmission coefficient T(N) = Nv/N(0) vs. z corresponding to the initial conditionsiV(0) = 0.01, V(0) = 0.5, and a(0) = 0.02 (The solid line represents numerical results while thedashed line shows the analytical results).

The above discussions show that strong nonlinearity can completely inhibit thelocalization effects stipulated by the disorder. This effect appears over a thresholdnonlinearity. Below this threshold, the transmission coefficient approaches zero asthe size of the system increases, either exponentially (Fig. 8.4) or exponentially aftera short transient. Above the threshold value there is an undistorted motion of themicroscopic particle along the disordered system, i.e., the transmission coefficientdoes not decay and localization does not happen in the system anymore, probablybecause of the small soliton width for large values of a.

8.8 Dynamics of Microscopic Particles in Inhomogeneous Systems

Next, we consider the dynamics of microscopic particles in a weakly inhomoge-neous medium without dissipation, from the solutions of the nonlinear Schrodingerequation, (3.2). We assume that the medium has a linearly changing density anda linearly changing transverse component of the refraction index. In such a case,the potential V(x) = E0(t)x is a slowly varying function. It can thus be treatedas a perturbation. The perturbed nonlinear Schrodinger equation then takes thefollowing form,

i<t>v + 4>x<x> + 2 M 2 0 = eP{t')x'(S> (8.96)

where ^2m/h2x -> x', t/H -* t'. Abdullaev proposed the following form for asolitonic solution of this equation

</> = 277sech{27?[*' - i(t')}}eiA^xl+iB^\ (8.97)


with

£(*') = -2e / / p(t'")dt'"dt'",Jo Jo

ZA(t')=e[ /3(t")dt",Jo

B(t') = V « ' - / A2(t")dt".Jo

It shows that the inhomogeneity leads only to the modulation of the soliton velocity.At a constant gradient (/3 < 0), the microscopic particle is decelerated, and thenreverses its direction of motion. If we insert (8.97) into (8.96), we can determinethe relation among £(£'), A and B. A change of variables turns the perturbedequation into an ordinary integrable nonlinear Schrodinger equation. This mayalso be applied to describe the propagation of a soliton in a periodically modulatedmedium. The equation

( t'\r o /

describes behavior of a charged microscopic particle in an ac electric field. Its solitonsolution can be written as

\<t>{x', t')\2 = 4IJ2 sech2 J2rj \x' -x'o- vt1 + 4at'2 sin ( | r ) ] ) • (8.98)

Abdullaev also considered properties of microscopic particles in a non-stationarymedium described by the following nonlinear Schrodinger equation with time-dependent coefficients

i<k- + 2<Px'*< + |4>|2</> = i£l{et')<j>. (8.99)

Its solution is given in (8.97).From the above, we see that the influence of non-stationary perturbed potential

on the microscopic particle in the nonlinear quantum mechanics can lead to aneffective damping or amplification of the amplitude of the microscopic particle. Tothe first order in e, application of perturbation theory yields the following amplitudeof the microscopic particle,

X] = 2r?oe J j{et')dt'

for slowly varying inhomogeneities.For a (5-like impurity inhomogeneity, V(x') = — eS(x' — x'o), we get from (4.20)

- (4.22) that the effective potential of the system is of the form

V(x'o) = 8e773sech2(27j4).


The possibility of transmission and reflection of microscopic particle (soliton),and of oscillating regimes, depend on the initial energy of the microscopic particle.Integration of (4.25) yields the soliton trajectory for e < 0,

X'{t') = sinh-1 l ^ f e i - sin[2\E\(t' - t'0)}\ (8.100)

where

v0 = -4er]3, X' = 2 ^ ,

and E is the energy of the soliton.Phase plane trajectories of (4.25) can be shown when E < 0, which corresponds

to oscillations of microscopic particle around the impurities. E > 0 results in un-bounded motion of the microscopic particles. To show the oscillations of microscopicparticle in the <5-like potential, Abdullaev used an action angle variable (S,0). Theaction is given by

5 = | ipdq = ^2\vQ\ - yfi\E\. (8.101)

The value of the action upon the separatrix to equals 5 = Ss — y/2\E\. Using(8.101), Abdullaev obtained the cyclic frequency of soliton oscillations,

u = ^ = ^2\E\ = (Ss-S).

When S -+ Ss, u(S) -> 0.We now consider motion of microscopic particles in a weakly inhomogeneous

medium. The inhomogeneity can be caused by space-dependent density fluctuationor gradient, Sn(x) = (n(x) — no)/no which is the normalized density deviation fromthe uniform density no- In such a case, the nonlinear Schrodinger equation becomes

* & ' + <j>X'X' + 2\<f>\24> = 8n{x')<j>. (8.102)

For an arbitrary density profile 5n(x), equation (8.102) is difficult to solve ingeneral. However, in the case where the inhomogeneity is sufficiently gentle so thatthe density scale length is large compared with the width of the microscopic particle(soliton) under consideration, 8n(x) can always be approximated by a linear localdensity gradient. On the other hand, the microscopic particle undergoes nonuniformaccelerations in an arbitrary density. To follow the microscopic particle, a time-dependent density gradient, instead of a constant one, can be used. Chen and Liuadopted an adiabatic approximation to study the dynamics of microscopic particlesin such a system. They expanded the density profile Sn(x') around the position of


the microscopic particle xo(t') as

5n{x') = Sn[x' - xo{t') + xo(t')}

~ Sn[xo(t')] + [x' - xo{t')}8n'[xo(t')} + ••• . (8.103)

When 2\5n'/8n"\xi=x » (x' - xo{t')), or 2L » d, where L is the density scalelength and d is the width of the microscopic particle, we can neglect the higherorder terms. Then the arbitrary density profile is approximated by a linear andtime-dependent density profile. If 5n = Sn' — 0, we also need the condition

[x' - xo(t')}25n"(xo(t')) « 4 | C a x .

Combining (8.102) and (8.103), we get the following nonlinear Schrodinger equationfor a linear, time-dependent density gradient

i<k + 4>*x> + [2\<t>\2 ~ B(t') - c{t')]4> = 0, (8.104)

where B{?) = 6n'[xo(t')] and c{?) = Sn[xo(t')] - xo(t')Sn'[x'(t1)]. Applying thetransformations

x' -¥ X = x' - I(t'),

t' ->T = t', (8.105)0 -+ ft = fciXplt')+M(t') >

with

p(t') = f B(z)dz,Jo

/ (O = 2 / p(z)dZ,Jo

rt'9(t')= / \p2(z)+B(z)I(z) + c(z)}dz,

Joequation (8.104) can be reduced to a standard nonlinear Schrodinger equation,

*#- + fe + 2|0'|V = O. (8.106)

Thus the solution of (8.104) can be easily obtained. The direct correspondencebetween the two equations indicates the existence of multi-soliton solutions to(8.104). They are now non-uniformly accelerated. The acceleration is given bya(t) = -d2l(t')/dt12 = -2B(t').

The above equations describe the general propagations of microscopic particlesin an arbitrary density profile in the absence of any source. If some physical processis responsible for the building up of a large amplitude of an initially localized wavepacket, its subsequent time-evolution with the source switched off is then describedby (8.102). Since the initial wave packet is sufficiently localized, we can approximatethe density profile at the position of the initial wave packet by a linear density


gradient. Chen and Liu studied a parabolic density profile, 6n{x') = —a2x'2. Inthis case, (8.102) becomes

i<t>v + 4>x-x- + 2|(?i|20 - aV2<£ = 0. (8.107)

In fact, the one-soliton solutions of this equation can be obtained directly byassuming (f> = AetB. Then, equation (8.107) becomes

(A2)t,+2(A26x,)x,=0,

-(c*V2 + 0v + 0l,) A + Ax,x, + 2A3 = 0. (8.108)

From (4.33), we know that the microscopic particle in this case has a speed givenby xo = 4£cos[2a(i' - t'o)] at position xo = (2£/a) sin[2a(t' - £{,)], where £ is anintegration constant, and is related to the initial soliton speed by 4£ = i 0 \t=t0- I* is

then easily seen that the amplitude A must be a function of Y = ax1 - 2£ sin [2a it' -*{,)] alone. We then have

At- = - 4 f cos[2a(t' - t'0)]Ax-,

6X = 2£ cos[2a(t' - t'o)], (8.109)

9t = -4âsin[2o;(t' - t'0)x' + h(t'),

6 = 2$x' cos[2a(t' - t'o)] + / /i(r)dr + (90. (8.110)Jo

Substituting these equations into (8.108), we get

h(t') = -4£ 2 cos[4a(t' - t0)] + 4TJ2, (8.111)

and

-(Y + 4ri2)A + a2Ax,xl+2A3 = 0. (8.112)

Thus

fa,«**(*£), fory«2,, ( 8 m )

[ 0, for Y » 2TJ.

Combination of (8.108) and (8.113) gives the soliton solution (8.114), i.e.,

(j> = 2?7sech J277 I a;' - — sin[2a(t' - to)]l 1 (8.114)

x exp li \2fr' cos[2a(t' -1 0 ) ] - — sin[4a(t' - t'o)] + Ar]2{t' - t'Q) + 601 j .

This represents a localized object oscillating with frequency 2a and amplitude 2£/ain the harmonic potential well, same as a classical particle. The solution (8.114)


can also be obtained using the adiabatic approximation. Expanding 6n(x') = a2x'2

according to (8.103), we get

5n{x') = [(ax' - ax0) + ax0]2 2 (ox0)

2 + 2(ax' - axo)axo

= 2a2xox' - a2xl = 4a£sin[2a(t' - t'0)]x' - 4£2 sm2[2a(t' - *{,)].

Comparing this with (8.104), we obtain

B{t') = 4ats\n[2a{t' - t'o)],

c(t') = -4esin2[2a(t'-t'o)]. (8.115)

The one-soliton solution given in (8.113) and (8.115) is then obtained from(8.105). It turns out to be identical to (8.114) which is obtained by direct integra-tion. In fact, (8.107) is a special case of (8.31), but the way of solving the equationand their physical meanings are different.

8.9 Dynamic Properties of Microscopic Particles in a Random In-homogeneous Media

It is well known that there always exist fluctuations in physical parameters inany medium. Such fluctuations in external fields and initial conditions result instochastic dynamics of microscopic particles and other nonlinear waves in the sys-tem. Apart from the stochastization of the soliton parameters, processes relatedto soliton-radiated random waves, stochastic soliton and breather decay, as well as"dynamic chaos" of nonlinear wave due to strong instability waves under periodicperturbations, etc. might also exist. In this section, we study the dynamic proper-ties of microscopic particles in such random inhomogeneous media using the meanfield method, the adiabatic approximation and the statistical Born approximation,respectively, as it was done by Abdullaev et al.

8.9.1 Mean field method

We begin with a brief introduction to the mean field method. Consider a nonlinearwave equation

L<t> = e ( z , t)M2, (8.116)

where Q is a deterministic linear operator, L and M are linear integro-differentialoperators, e(x, t) is a random function with prescribed statistics. The wave fieldwill be written as the sum of a mean field ($) and a scattered term, 5<f>, i.e.,

<j) = (4>) + 6(j), (6<f>) = 0 . (8.117)


Substituting (8.117) into (8.116), and averaging over all the realizations of e, wehave

L((t>) = (eM6<fi) + Q(<j>)2 + Q(6<p2),

L6) + eM8<j> - (eM6<j>) + Q(S</)2 - {5<p2)) + 2Q{<f>). (8.118)

In the case of e <C 1, Scj> is found to be

6<J>=êM(ci>), (8.119)

and the equation satisfied by the mean field becomes

L{<j>) = N{<j>) + Q{<j>)2. (8.120)

This equation is closed with respect to {<j>), and can be solved.For a stochastic nonlinear Schrodinger equation, which is of the form

»&- + |0x '* ' + \<!>\2<t> = - e (z ' , t')4>, (8.121)

the mean field satisfies

*(4>)t' + \{4>)x'x' + M2<f>) + {e{x',f)<j>) = 0. (8.122)

According to the mean field method, we can write

<MV> sM<0>2|<0>.Klyatzkin used the Furutzu-Novikov formula and (e(t, x)) — 0 to decouple the mean(e<t>) b y

{e{x',f)<j>{x',t')) = f dt" [°° dx"(e(x",t")e(x',t'))x' ( ^ ' ' 2 ) • (8-123)Jo ./-oo \Se{x",t")/

The causality condition requires that

6<f>v,n _ „ ,Se(x",t") ~ ' <

Integrating (8.122) between zero and t' and calculating again the variational deriva-tive 5(f>/6e(x",t"), we can arrive at

lim ffj/ff} = 2iAe(0)(<t>(x',t'))6(x' - x"), (8.124)

where the correlator (e(x',t')e(x",t")) - 2Ae(x" - x')S(t" - t'). Inserting (8.124)into (8.122) and (8.123), we get

i(4>)v + \(4>)x>x' + \{4)\2(4>) = -iAe(0)(4>). (8.125)


This is the nonlinear Schrodinger equation with damping, and can be solved ana-lytically for Ae (0) -C 1 using the above approach.

8.9.2 Statistical adiabatic approximation

We consider the stochastic nonlinear wave equation

NL<t> = R[e{x',t',<j>,<l>tl,<t>x,,•••)], (8.126)

where NL is a nonlinear operator for the nonlinear Schrodinger equation,

NL = idf + dx.x, + b\<f>\2,

R is a perturbation operator, e(x, t) is a random function which satisfies the Gaus-sian distribution with zero mean (e) = 0, and (e(x',t')e(y', t')) = BicTe (x'—y1, t'—r),where le is a correlation length, re is a correlation time. For lE,re —> 0,B(x' —y',t' — T) « 2crf <5(a:' — y')5(t' — r) which corresponds to the (^-correlation ran-dom process. e(x',t') is a weak random perturbation, i.e., e < l . We can thereforeinfer that the parameters of the microscopic particle (amplitude, velocity, phase,etc.) change smoothly with time, while their mutual correlations remain the sameas in the case of a free particle.

For the one-particle stochastic nonlinear Schrodinger equation with damping 7and under a random field e(t)

i<t>v + \<t>*'x' + |</>|2</> = -*70 + e(t'), (8-127)

applying the perturbation theory described in Sections 7.4 and 8.6, its solution canbe obtained as

<PS = 2f,(f) sech{2r,[x' - £(*')]} exp [ * ^ y V ~0+ # (f)] , (8-128)

where the soliton parameters satisfy the following equations which are obtainedusing the inverse scattering method, based on the adiabatic approximation,

^L = -2 _ ^(t')sme(t')dt' ~ 1T] 2cosh[Trv(t')/2r](t')}'

dS _ nr *2<t') cosfl(Q sinhMf )/2iKf)]

dt' Si,* cosh2^! , ] 'dv TT£vsm0(t')

dt' ~ At] cosh[7ro/2?7]'

— = 2(v2 + n2Nl - 7 r £ c o s h 6 > ( t ' )dt' v -1- / ; 7rcOsh[7n;/2»?]'

(8.129)


Abdullaev et al. expanded these parameters in the forms of

v = vo + vi H ,

0 = dQ + 0! + • • • ,

V = Vo + Vi H

£ = & + 6 + • • •,

where î, 7?i, î, î « e, and

vo = constant,

Vo - Vie~2lt'',

*H = *to(*' = 0), (8.130)

eo = 2 ^ ' + e i + ^L(e-27t '_1) i

In the case of 7 = 0, we have

d7? 7r£(t')sing0(t')dt' 2cosh[7ro0/2»j0]"

K '

Prom this, Abdullaev et al. obtained the fluctuation in the soliton's amplitude (for7 = 0) which is given by

,n2v _ *& U sm[4K2 + yl)t'\ \{Th) ~ coBh[,nW2v,] I " 4(«g + ijg)f / " (8"132)

Obviously, re 7 0 which leads to small-parameter corrections Te/t's -C 1. Thestochastic growth in the energy of the microscopic particle is a result of interactionwith the fluctuating field. The linear amplitude increases with t' is accompanied byoscillations of frequency 4(v$ + T]Q). For &{VQ 4- Vo)t' » 1, we obtain the diffusioncoefficient

D = lim M l = TO'

" t'->oo f 4cosh2(7rv0/277o)

The mean values (v2), (^2), and (ff2) could be also obtained in a similar way.We now study the parametric interaction between a microscopic particle and a

random field. The nonlinear Schrodinger equation, describing propagation of themicroscopic particle in a medium with fluctuating parameters, has the followingform in this case

i<h + \<Px'X' + \<t>2\ = -u2(t')x'2<f>. (8.133)


For u — LJQ, the microscopic particle (soliton) performs periodic oscillations aboutx' = 0. In the adiabatic approximation the parameters can be denoted by

§-"• £•«• %-¥«* wThe coordinate of the center of mass of the microscopic particle (soliton) satisfies

0 + w2 ( O £ = O. (8.135)

Let ui2(t') = CJQ[1 + e(t')], where e(i') is a random function. Abdullaev et al.expressed the mean values (£), and (y) (y = £) by

whose solution corresponding to the initial conditions £(0) = 0 and y(0) = 1 is

(0 = — sin(w0Oi (y) = cos(w0i')-

The second moments are given by

(^2) = 2(i/),

(iy) = (y?-"Ue),

with

(e) = (£y) = o, <2/2) = i, *' = o.

The solution of this set of equations for <72Wo is

{e{t']) = 2 ^ K ^ ' ~ e~a^%t'12 [ c o s ^ ^ + ôsin(a,ot')] } .When t' « I/0"?' the microscopic particle is accelerated stochastically due to fluc-tuations in the system parameters. The time of acceleration is t' 2> ts, wheret, w 1/2^,, t' = 1/afwg.

To further illustrate the stochastic parametric soliton resonance, Abdullaev etal. studied the following nonlinear Schrodinger equation

i<t>v + ^<t>x'x' + \<!>\2<t> + ii<t> ~ ie{x',t')e-2'1ixl4> = 0, (8.136)

where 71, and 7 are increment parameters related to the damping effect of thesystems, e is a random function defined by the field of partially coherent pumpingand medium properties. In the following, e is assumed to be a function of thecoordinate t' only. Then for 7 = e = 0, equation (8.136) can be converted into anonlinear Schrodinger equation with one-soliton solution (8.128).


Now we assume that e{t') = eo+e(t'), where e(t') is a Gaussian random function.Utilizing the adiabatic approximation in the perturbation theory, we obtain thefollowing for the amplitude of the microscopic particle (soliton)

~ = [eoe~W - 7 + ie-M] . (8.137)

Given the initial condition r\ = rjo, equation (8.137) can be solved to give

h0 = — (l - e-2^*') - 27 l*\ (8.138)

/i i=2 r e(t')e-27lt'cft'.J-oo

Averaging (8.138) over all possible random values of the mean soliton amplitudeyields

fo(f)> = ijb exp (ho + i(/if)^ . (8.139)

For a 5-correlated random process e, we have

»7(f) = % exp [2(2a2 + e0 - t') - 27 l(e o + 4a2)t'2] .

Expression (8.139) indicates that (??(*')) grows as the particle propagates for arange of parameters because energy is pumped into the soliton from the randomfield, which leads to the stochastic parametric resonance. When £o = 7, the dissipa-tive energy loss is exactly compensated by the interaction with the regular part ofthe pumping wave. Ignoring the dissipation of the pumping wave, we have 71 = 0in this case. From (8.139), Abdnllaev et al. showed that (7/) = 7joexp(4cr2t'). Thisclearly shows that the growth of the mean amplitude is due to fluctuation of e. Insuch a case, the probability density function of the microscopic particles (solitons),in terms of their amplitudes, is given by

{hl) = 0, (hi) = ^ [l - e-4^] = 8<72t'(l - 27lt')-

Therefore, the amplitude of the microscopic particle has a log-normal distribution.

8.9.3 Inverse-scattering transformation based statistical perturba-tion theory

Radiation of microscopic particle in a random inhomogeneous medium was studiedby Abdullaev et al. using the statistical perturbation method. The following form

(8.140)


was chosen for the nonlinear Schrodinger equation

ith + 4>x'x' + 2\<j>\2<j> = ef(x',t')R(<j>), (8.141)

where R((f>) -4 0 when \<f>\ -> oo.When e = 0, the solution of (8.141) is given by (8.19). Here we will examine

the statistical features of the radiation field for some special types of perturbation.The correlation functions for the random function f(x',t') have the form

(f(x,t)f(x',t')) = <%Bx{x - x')Dx{x - x');

(f(x,i)r(x't')) = G\B2{X - x')D2(x - x1).

The mean spectral density of the energy of the wave radiated by the microscopicparticle per unit time can be expressed as

{P{x))=8^^)^ny (8.142)

The Jost coefficient in the inverse scattering method satisfies the equation

Algebraic calculation then leads to the following

A{\,t,ri) = I {e-*^-4)*/" [(A - £ - iijtanhz)2^,]^*'**]

-\-^-R*[(f>sy*^}}dz.Lcosh z J J

The change in the number of quanta M is given by

dN s^dNk 2 f°° , . _ . . ,

where

/(A) = »^ ( A > * j - g j r - ^ = 8 r ? 2 [ ( A _ e ) + v2]2 / _ -

x [al\e\2r]i\A1\%(K)d2{J-)) + <72

2|e|2|^2(«)|%(«)d2(^+)) (8.144)

+ <T2|£|2|A2(«)|262(«)d1[4-)] - 2r,2crme2b1(K)d1U

+))A1(-K)A*2(K)]} ,

with

(4±))^±4[(X-02+V2}-^xl,

Ax= f dze-Wo-^-^^Rseclfz,J —OO

/•OO

A2= dzJ^+^-^^WiX - it] tanhzf.J — OO

(8.143)


In the case of a (5-correlated random process, and R = (j>, Abdullaev et al.obtained

(p(\)) = ^ V 2 A 2 P - a 2 + '?2]22«£* cosh2[7r(A2 - £2 + T / 2 ) / 4 ^ ] "

There are two maxima in the above spectral density, pi « £2<r2f and p2 «£2°WK2, located at A = ±Am, respectively, where Am = i/(f2 ~ V2) « 6 Theirwidth is of the order of 77. The total mean power emitted by the microscopic particle(soliton) can be obtained and is given by

/•OO

P'= p(X)d\ = 4eV?^, (77 « 0,

or

F ' = £ C T ? y ? e x p ( " i ) ' forr7>>eNozaki and Bekki investigated the stochastic behaviors of microscopic particles

described by the following nonlinear Schrodinger equation, with radom phases inboth time and space in the presence of a small external oscillatory field

00

i<h + <t>x'x> + 2M2<?i = «(ei - e2\<fi\2)<t> + iez<{>x,x, - ^ ^ einuot>,n=—oo

where Ej (i = 0,1,2,3) are small positive constants. The results show emission ofsmall-amplitude plane waves with random phases. Statistical properties of randomphases give the energy spectrum of the microscopic particle (soliton) and planewaves in the systems.

8.10 Microscopic Particles in Interacting Many-particle Systems

In interacting many-particle systems, such as quantum liquids, solids, molecular andmagnet as well as nuclear hydrodynamics, the states and properties of microscopicparticles described by the nonlinear Schrodinger equation will be changed due to theinteraction among them. Barashenkov and Makhankov gave the dynamic equationof the microscopic particle in this case as follows

ifo +A<f>- aKJ> + b\(j)\2<l> - a 6 | 0 | 4 0 = 0, (8.145)

where t' = t/H, x\ — Xi^2m/h2, (i = 1,2,3), and a5\(f>\4(fr a5 > 0 is a nonlinearinteraction arising from the many-particle interaction. This equation was usedin Chapter 4 to discuss multi-particle collision in nonlinear systems, but its solitonsolutions were not given. The work of Barashenkov and Makhankov on this equationis introduced in the following.


In one dimension and for #5 = b — 1, we can make the following transformation,

^vmrw <*<[fr+*»r]- * - » ^ + « -with p0 > 0, 2p0 + A > 0, and

Po = o ( 1 + V l + 4 a i ) , —=- = - 2 + - — ± — - — | — .2 po 4ai 4|ai |

Then, equation (8.145) becomes

i<Pr + <fx'x> ~ Po(Po + 2A)<p + 2(2p0 + A)\<p\2ip - 3|<^|V = 0. (8.146)

Equation (8.146) is more convenient for studying condensate excitations. By intro-ducing two parameters, A and p0, in (8.146), the coupling can be eliminated. Herepo is arbitrary, even though ^4/po is required to take certain value. Thus, equation(8.145) can be called the "drop" form, and (8.146) is the "condensate" form. Thedrop form can be reduced to be condensate form if at\ > —1/4. The term ai(x',t') = eiait'4>(x',t').

For the drop boundary conditions of

<j>(x',t') ->0, a t | z ' | - > c c , (8.147)

the energy and particle number integrals are given by

^ ' { | < M 2 - 2 M 4 + Î 6 }> N = j_ d*W> (8-148)respectively. Equation (8.145), with the boundary condition (8.147) and 4>t> — 0,can be integrated twice to give

r A V2

4>dr = eie° I l \ . (8.149)\ 1 + y/1 + 16a1/3cosh[2v/^T(a;' - x'o)} j

Thus a moving soliton can be obtained via the Gallilei conversion

0o->lx'-^t' + eo,cosh[2V-ai(z' - x'o)] -> cosh[2v

/=ai"(a;' - vt' - x'o)). (8.150)

From (8.149), the soliton-like solution is known to depend on three parametersx'o, 60, and v and the solution exists when — 3/16 < a.\ < 0. In such a case,Barashenkov and Makhankov gave N and E as

N(ai) = \/3arch ->• 4v/=S7|ai_+0,[1/I + I6ax/3J

/ 3 \ 3E{a{) = I v2 - - - a i l N(ax) + ^v^n-


When ai -» -3/16, N grows without limit with the width of the soliton, and itsamplitude approaches <f>o = V3/2. Such behaviors of the solution of (8.149) isextremely proper, and implies that the particle density in the soliton increases toits peak value due to an attractive force, but the three-body repulsion begins to actat small distances. This compensates for the attractive force and the increase inparticle density so that further increase in the number of paxticles N gives rise toa growth in soliton size, and the binding energy of the particle remains constant.This bound state of a large number of particles is naturally regarded as a drop oftheir condensed state ("fluid") as shown in Fig. 8.5.

Fig. 8.5 The bound state of the nonlinear Schrodinger equation in the 04-</>6 field for cx\ G(-3/16, 0), and particle numbers AT; = N(ai), Ns > N4, > N3 > N2 > Ni.

If a\ = —3/16, the solution (8.139) gives the condensate <j)c — e%e°\f3/2, i.e.,N — 00, which corresponds to a transition from the drop state to the condensatestate. Such a behavior of the model, along with the saturation effect, is one of themost remarkable features of this system.

Barashenkov and Makhankov studied the solutions of (8.146) under the nontriv-ial boundary condition |(,o|2 —> po- Integrals of motion of (8.148) are divergent inthis case. They were thus rewritten as follows

E = r dx' [|YV|2 + (M2 - Po)2(M2 - A)] ,J—oo

rOO

N= dx' (M2 - po), (8-151)J—oo

where A is a real constant. Barashenkov and Makhankov considered the dispersionof small oscillations of this condensate which can be expressed as

<p(x',t') = ^+X(x',t'),


Thus 771 and 772 satisfy

WJ71 - K V - 2/90(po - A)rfx - 2ipl(p0 - A)fj2 = 0,

-W772 - K2r?2 - 2po(po - A)r)2 - 2ipl(pQ - A)i)i = 0.

Setting the determinant of the coefficients of the above system to zero, one can get

u,2 = K2[K2 + 4po(po -A)}.

This is the Bogolubov dispersion, ignoring the change in the interaction due tovariation of the sound velocity in such a condensate

i;s0 = lim — = 2\fpo(p$ - A).

For small amplitude nonlinear waves, \f)ei6{-x'•*'). (8.152)

Introducing y = 6X>, we get from (8.146) the following dynamics equations

pv = -2{yp)x,,

yv = 1 (^L - \ ^ j - 2yyx, + [(Po - p)(3p - Po - 2A)]X,. (8.153)

Choosing new coordinates, r = -\/ei', £ = \/l{x' -vst'), and by means of "reduc-tive perturbation theory", we expand p and y in powers of e about the condensate,P = po, y = 0,

p = po + epi + £2P2 H ,

y = eyi+ £22/2 -I •

Substituting these into (8.153) and letting the coefficients of e and e2 vanish, we get

Vsop'i - 2poy[ =<%,

vsoy[ =2(po-A)p1,

Vsop'2 =Pi+ 2(i/ipi)' + 2poy'2 = dT, (8.154)

vsoy2 = Pi + 2j/iJ/l - I 1 - + 2(p0 - 4)p2 + 3(p2)',•'Po

where the "/ " and "•" denotes differentiations with respect to £ and t, respectively.The first (or third) of (8.154) gives y[ = vsop'1/2p0, while the second and the fourthreduce to the KdV equation

2VPO(PO-A)PI - |pi" + 3(2p0 - A){p\y = 0,

or

dTpi ~ dfn. + 3(p2)f = 0, (8.155)


by the scale transformation

t = ty/\2{2po-A)\, r = f Vs°2y/(2p0 - A)3

It can be shown that (8.155) has a one-soliton solution which is given by

Piilf) = -^sech2 ~ (f + If - ft) , (8.156)

or

p(a ; ' , t ')=|po-2^37s e c h 2[^7(a ; ' -ôt ' + — t'-ôjj} . (8-157)

where b' = b/2(2p0 - A) > 0. Expressions (8.156) or (8.157) describes a localizedrarefraction wave moving in the condensate at a velocity which is close, but lessthan the sound velocity. The quasisolitons and the background arise in an initialperturbation decay. However, the decay of a compression perturbation occurs in thesolitonless part to give a dispersive packet of Bogolubov waves (some sound waveswith small wave numbers). The rarefraction soliton (8.157) is accompanied by theemission of "linear" (Bogolubov type) waves moving in the same direction as thesoliton, but faster, with the group velocity vg — 2(k2 + v2o/2)/y/k2 + v2

0. Thesewaves are usually called "foregoers".

Barashenkov and Makhankov obtained localized solutions of (8.146). Here, weuse (8.152) and assume <p(x',t) = <p(£'), with £' = x' - vt'. Equation (8.146) thenbecomes

PP" ~ \p'2 + v6'p2 - (6')2p2 - P2(p - Po)(3p -po- 2A) = 0, (8.158)

p0' + y(pg-p) = O.

Inserting the second equation into the first of (8.158), and integrating again, we get

r f v2\~1/2

2tf' -Q = J d{p - p0)-1 {p2 -Ap-jj

= (^-s/q2 + {2p0-A)q + c). (8.159)

J Q

Here

1 A V2 1 / 2 2\q = p-p0, c = pZ-Apo- — = ^(vio-v'!).

There exists a soliton-like solution of (8.159) at c > 0 or v2 < v20. It shows that the

microscopic particle is localized in these systems in this case. Inverting the integral


(8.159), one can get

q±(n = , — . (8.160)H±K^' s/A2 + v2 coshpVcK' - &)] ± (2Po - A) K '

The second of the above solutions, q~(£'), is singular since 2po > A. The first isregular, and takes a different form in each of the following four regions, A > p0,p0 > A > 0, -po/2 < A < 0, and A < -po /2.

In the region p0 > A > 0, the soliton is at rest and

w = e"- y i c o s h K o Q E ' - ^ ]( l + ^ / A j ) s i n h 2 [ ^ s o ( ^ ' - 4 ) / 2 ]

(fb is an even function of (z' — XQ). It describes a bubble in the condensate, whosedepth depends on A, the smaller the A, the greater the rarefaction in the bubble.

In the region -po/2 < A < 0 and A < —po/2, we have

Vk = e«- V\A\Sinh[vso(x>-x>o)m

(1 + \A\lpo) coshVv.o(a:' ~ x'o)l2\

Here ^ is an odd function of {x1 — x'o) and has a kink form.Phases of both solutions were found via (8.159) and (8.152) to be

A V2V9" COSh(l/ - tu)Pfc,6 = , - , (8.163)

y/(2po - A)/y/A2 + v2 + cosh(2i/)

where v - {\l2)yjvlo-v2{x' - vt' - x'o), cos(2u) = (Ap0 + v2/2)/(p0\/A

2 +v2),and the sign of A determines whether the solution is a kink or a bubble. For A = 0,or ai = —3/16, Barashenkov and Makhankov gave a solution with mixed boundaryconditions ip(x',t') -t 0 for x' -¥ —oo, and ip(x',t') -> y/poe16" for x' -> +oo. Inthe case of finite energy, the solution is

^ ( X } = ^l + 2poexp[±2po(x'-x'o)}' ( 8 ' 1 6 4 )

It connects two stable vacuum states, the condensate \(p\2 = p0 and the trivial statel^l2 = 0. When8v = constant, (8.164) describes a wave at rest. UQV = (v/2)x'-uit',the wave given in (8.164) moves together with the condensate.

Obviously, integrals of the hole number and energy for solutions (8.163) are

Nk,b = - arch f ,WA2 +v2\

Ek,b - sTc (j + A.) + [APO + \{v2 - A2)] Nk,b.

(8.161)

(8.162)


For small amplitudes c (hence velocity v close to the sound velocity vso), themodule of (8.163) is

r 2c 1 1 / 2

P » ( ^ O = [ p , - ^ + o 2 r o s h | 2 v^ ,_ s ) ) + 2 A i _ A j • (8.165)

This coincides with the approximate solution (8.157) since

(2po-A)2-(A2+v2)=4c,

or

s/A2 + v2 ~2Po-A- 2°2/90 - A

at c <C \2p0 — A\. y/c2 + v2 in (8.165) is then replaced by 2po — A, and we can get

f 2c/(2po-A) Y'\.[ c/(2Po-A) ^Pk'b \Po l + cosh[2v'c(?-ft)]J " I * cosh2[2VE(e-^0)]i '

with v = vk - 2c/vs0, b = 2c/(2p0 - A).Besides the above solutions, the <j>6 model also has other localized solutions, for

example, the rational solution. However, in the region of A > p0, there are nolocalized solutions.

Barashenkov and Makhankov demonstrated the stabilities of the above hole-like excitation and the drop-like solitons by the spectral analysis and variationalmethod. In the first case (particle-like solutions), the drop is stable. There are twotypes of localized excitations of the condensate: the bubbles (8.161) with —1/4 <ai < -3/16, and the kinks (8.162) with c*i > 3/16. They are unstable. Thesesolutions were also obtained by Cowan et al. through numerical simulation.

8.11 Effects of High-order Dispersion on Microscopic Particles

The effects of various high-order dispersions on the properties of microscopic parti-cles were studied by Karpman, Kundu, and Pathria and Morris. Karpman obtainedthe solutions of the following nonlinear Schrodinger equation

i<t>t' + ^<f>x'x' + \4>\2<t> = R - on<j)xix<x'x', (8.166)

where the operator R is denned as

n . d3 d4


with 0:3 and on being real and small coefficients. Karpman expressed the solutionof (8.166) as

4> = aF ft', x' - f vdA exp \U f a2dt' + iO(x', t')} . (8.167)

where a and v are the amplitude and velocity of the microscopic particle, respec-tively. Variations of a and v with time are ignored. Inserting (8.167) into (8.166),Karpman obtained

6 = -Clt' + kx',

n=±k2-a3k3-a4k

4, (8.168)

v = k — 3a3&2 - 4«4fc3,

and

\FV + \a2Fa + ia3F^ + aiF^m + a2 (\F\2 - i J F = 0, (8.169)

with

a2 = l - 6 a 3 A ; - 12a4A;2,

0 3 = 0 : 3 + 4a4fc,

04 = 0:4.

For small 03 and 04, the solution of (8.169) can be written as

F(f,0 = sech (^j + /(£, t'), (8.170)

with O2(i') > 0. The term f(£,t'), at sufficiently large £ and £', describes theradiation field. Karpman gave an asymptotic expression for f(£,,t') which is

/(f,f) = î:Cêxp(-^|M)x

eê^Jdtvj -|£l)©M- (8-171)

where j (= 1,2) is the number of the radiation modes, kj are their wave numbers,given by fci,2 ss (-a3 ± y/a^ + 2o2a4)/(2a4), Wj are the corresponding group veloc-ities in the reference frame specified by £ and t', Vj = a2kj - 3a3k

2 — 4a4fc?, Cj areconstants (of order 10), and Q(Z) is the step function

ri (z>o),O (^- \0 (Z<0).


The 9-function in (8.171) states that the fronts of the modes with fc = fcj,2 prop-agate with v = vifl in the directions specified by the signs of 1,2- The wholeexpression (8.171) is exponentially small, because Ifci^la"1 » 1.

When af » 2a2|o4|, kx « a2/(2a3), and |&i| < \k2\, (8.171) is approximatelyreduced to

/ « , * ) » ^ f e i e x p ( - ^ | f c i | ) e*«6 ( | | * t * | - Ifl) @(Vl0, (8.172)

where C\=C and i>i « — O2/(4a3). Under such a condition the radiation field doesnot depend on 0:4 in (8.166). Therefore, we essentially have a third-order nonlinearSchrodinger equation. When a\ <C 2a2|a4|, we have

^ R i - ^ R j J ^ - , vi w -v 2 w -\hr-, (8.173)y zO4 y 2a4

with 04 > 0. In such a case, the third derivative term in (8.166) is insignificant andwe arrive at the fourth-order nonlinear Schrodinger equation. The soliton radiationis in the form given above, with 04 > 0, C\ = — C^ = C. For a4 < 0, the solitondoes not radiate. Karpman showed that (8.172) may be valid even at |fci| < \k2\-However, if |fc2| < |fci| is sufficiently small, one should use the full expressions(8.171).

Based on conservations of particle number and momentum, i.e.,

and the above solution, Karpman also obtained the variations of the amplitude aand the velocity v of the microscopic particle with time, which are caused by theradiation of the microscopic particle (soliton). They are

da{t') y/2\C\2 ( TT \

dt> * 4^4 6XP V v/2a4 J '

L ao J

where

a(t')=\ V \ôj


and

Karpman further obtained solutions of (8.166) with R = ia^d3 jdxz anda^d^/dx4, respectively, and showed that the radiation of the microscopic particle(soliton) leads to a decrease in the amplitude of the microscopic particle. He alsostudied acceleration of the microscopic particles which arises due to conservation ofmomentum.

Karpman also studied resonant radiation effect of microscopic particles due tohigh order dispersion, described by the following nonlinear Schrodinger equation

l(j>f + -zfyx'x' + & M 2 0 = - ( « * l M 2 < / V Jri0L2(i>\(j>^)xl + W*3<£z'z'z' + OH<t>x'x'x'x',

with both linear and nonlinear (cubic) dispersion terms, third and fourth deriva-tives. He obtained the amplitudes and asymptotic form of the resonant radiations.According to him, the soliton-like solution of the above equation can be written as

<j>{x',t') = exp{i[/iaf/2 + g(s')]}

yjAoAi[Al sinh2(/za;') + A\ cosh2(/xz')]'

where

. , ai +2a 2 \A0 ,]

^ ^ = 2C— a n \Ai t&nh^x >\ '

C2 = ^[4(O l + 2a2)ai - (o! + 2a2)%

2 _ y/2C*vP+q*+qAl~ 2C*~ '

2 _ y/2CV+^-gA°~ 2^ '

at 0.3 = 0 4 .

De Oliveira et al. obtained soliton-like solution of the nonlinear Schrodingerequation,

i(f>v + -^<t>x'x- + \<l>\2<t> = + i a i |<? !> | 2 0 x ' - ie\4>\2<j>,

which is given by

t(x\t') = A(t>)Sech[^-êie(*'>t'\

0(f) =90- \alt' + In (l + fs4?t') ,


where

Mt) = ~-3==, /i' = 1 + 00(01-202), cj(t') = u>osJl + 8eA'o2t>/3,

ao and 02 are constants.The influences of the nonlinear dispersion on microscopic particles described by

the <^6-nonlinear Schrodinger equation (Section 8.9) were investigated by Pathriaand Morris. The generalized nonlinear Schrodinger equation is

i<fa +<l>x'x' +b\<l>\2<t> + ai\\i<t> = -ia-2{\<t>\2)x<4>-ia3\<l>\2<l>xl. (8.174)

Applying the following gauge transformation,

</>(x',t')=ip(x',t')ei^''t'\


i<Pv+<Px'x'+b\<p\2<P + Pi\<p\*<P + ifo\<Px'\2<P + ilh\<p\2<Px' = 0 , (8.175)

where fi\ — ai + 26012 — Sa3 + AS2, fo — "2 + AS, /?3 = a3, and 6 is an arbitraryconstant.

Solutions of (8.174) which satisfy the integrability condition are constructedfrom the known solutions of the mixed nonlinear Schrodinger equation. For (8.174)with 4ai = a\ — 0.201$, if a3 = 0 (so that 4c*i = a\) and b > 0, the solitary wavesolution corresponding to the one-hump solution of the cubic nonlinear Schrodingerequation is given by

4>{x',t') = W ^ s e c h ^ z ' - vt') + z'oK^''*'}, (8.176)

6(x',t') = \{x'~ vt') - ^ p tanh[0?(z' - vt) + x'o] + lu

where rj = (v/2)(v/2 - c') > 0, c', v, XQ and l\ are arbitrary constants. Obviously,the solution (8.173) differs from the above soliton of cubic nonlinear Schrodingerequation only in its peculiar phase. The solitary waves given in (8.176) can exhibitalso the same clean interactions as the solitons (4.53) of cubic nonlinear Schrodingerequation (4.37).

Choosing 6 = - (2a 2 + a3)/8 and setting the forces 2/?2 + 03 = 0 in (8.175), weget an equation for tp

iff + Vx'x> + b\<p\2<p + 0i\tp\*<p + i/32\<p\l.<P + i & M V ' = 0, (8.177)

where

Pi = ati- 77(2(22 + a3)(2a2 - 3a3), /?2 = - - o 2 , P3 = "3-ID 2


Since 2/32 +/?3 = 0, equation (8.177) can be written in the following equivalent form

iff + fx'x- + F(<p)(p = 0, (8.178)

where F((f) is a real function and is given by

Pathria et al. obtained solutions of (8.174) using the transformed equation(8.178). Writing ip(x',t) as

ip(x',t') = f{x' - vt')eih{x'-cy), (8.179)

where / and h are real functions with h(y) = (v/2)y 4-l\, v, c and l\ are arbitraryconstants. 6(x',t') can then be obtained and is given by

6(x',t') = 25 j \<p\2dx = 2(5 f / V - vt')d(x' - vt)

- i (2a2 + a3) / f(x' - vt')d(x' - vt').

Inserting (8.179) into the transformed equation, (8.178), we get a cnoidal wavetype equation for g = f2(x' - vt'),

(91)2 = - ^ V - (2b - vfo)g3 - (2vc - v2)g2 - AC'g

a(g-gx){g-g2)g2 = 0, (8.180)

where C is an arbitrary constant, g' = dg/dx', g\ and gi are determined by thesolution of g, a — —Afii/Z. Solution (8.179) may be represented in terms of theelliptic functions. If C — 0, these solutions can be expressed in terms of elementaryfunctions, including oscillatory, singular, phase jump, and solitary wave solutions.These are four-parameter families of solutions, with arbitrary constants v, c, l\ andx'o. The parameters v and c representing, respectively, the speeds of the carrier andenvelope waves of ip, partly determine the form of the solution. When Pi < 0, andgi and g2 are real and #1 > g2 > 0, the solitary wave solution of (8.174) is

M( 9l92. , a , o J e^ '> , (8.181)

gi + (gi -52)sin/i2(n')J

B(x', t') = _^_t«3 yijtanh"1 [ y f tanh(O')] + \& ~ d>) + tlt

where

n' = ]f^f^(x'-vt') + »'.


When fix > 0, a in (8.170) is replaced by —a. If g\ and g?, are real, and<7i > 0 > gi, a solitary wave solution of (8.174) can be obtained and is given by

[ n 1/2

fl2 + (52-<?i)sinh2(fi')J

If gi and ,92 are real, and gi > g 0, then g is oscillatory and

0( '*)~lft + te-fli)co8>(n')J

9{x',t')^-2-^±^JJt^ L/I tanh(n')l + ^-cO+Zt.4 V PI LV ffi J 2

Therefore, equation (8.174) has solitary wave solutions for both positive andnegative values of $\. Note that (8.174) is invariant under the Galilean transforma-tion

x* ~A2(x' + 2A2Bt'),

t* = AH', (8.182)

4,* = -Mx',t')eiA2B^+A2m'\

where A is an arbitrary nonzero real constant. Then, since

i # . -<l>*x.x. -b\<j>*\2p -aÛ* +i<X2\P\l.f +ia3\P\2<t>*x* - 0 ,

we can construct new solutions for (8.174) using the above approach and (8.182).The conservation laws of the system described by (8.174) were studied by Pathria

et al.. If <p(x',t') is a rapidly decreasing function of x' and P(x')dx>tp(x',t') ->• 0 asx' -> ±oo, for any polynomial P(x'), then the following are the conserving quantitiesfor (8.175)

N0 = J\v\2dx',

J—oo

E2=r [i^'i2-^4-^1^6]^'-Pathria et al. numerically solved the following equations using a pseudo-spectral

split-step discretization,

i<h + 4>x-x> + \<P2\<t> + I 0 I V - 2i\<P\2<P = 0, (8-183)


for the initial condition

^ ^ = 71 v—2—J e x p r —2— + t a n h (—4—) {• +1 . /V-35 \ f.f a;'-35 , /V-35\"n

2sech (-2VTJ x exp I1 [ — i - + tanh ( ^ V T J j j ' (8184)

which corresponds to two initially well-spaced solitons of the form (8.176). Theone initially on the right moves to left with speed of unity, and the one initially onthe left moves to right with half the speed. The two solitary waves emerge fromtheir encounter with their shape and velocities unchanged, although they may bedisplaced from the position they would have occupied had the collision not occurred.The elastic collision of the waves, which is shown in Fig. 8.6, demonstrates thestability of the microscopic particles (solitons).

Solution of the following nonlinear Schrodinger equation

i<h + <t>x'* - \\<l>\2<t> - ^ | V - i\4>\l-4> ~ 2 # | 2 0 , » = 0 (8.185)

can be found and it is given by

r A iV2K ' |.4 + 3 s i n h V - 2 * ' - 1 5 ) J

9(x',t') = 2 tanh"1 | \ tanh(i' - It' - 15)1 + x' - 15.

L 2 JFor the initial condition <po(x') given in (8.184), results of numerical simulationagree with theoretical prediction that a bell-shaped solitary wave propagates fromleft to right at a speed of 2, as shown in Fig. 8.7.

Fig. 8.6 Collision between microscopic particles described by (8.183).


Fig. 8.7 Solitary wave solution of (8.185).

The properties of microscopic particles described by

i<t>f + 2 ^ W + l ^ = 7<t> ~ a{\<t>\2<t>)x< (8.186)

were investigated by Shchesnovish and Doktorov using an adiabatic approximationof perturbation-induced evolution of the soliton parameters based on the Riemann-Hilbert problem. 7 and a in (8.186) are constants. The results show that the shapeof the microscopic particle was distorted and the emission of linear waves occurs insuch a case.

Crespe et al. studied the stability of soliton-like solution of the following non-linear Schrodinger equation,

i<t>f + 2<Ax'x' + \4>2\4> = i$<t> + ie\(t>\2x>x> + ifJ-'\4>\2(t> - " ' | < / > | V

The soliton-like solution was found to be

4>{x',t) = A{x'yv^-iut'\

where A(x') and 0(x') are some unknown functions which are determined by theoriginal equation. It was found that in general the microscopic particles in such acase are unstable except in a few special cases and that there are some regions inthe parameter space where soliton solutions are stable.

Various profiles of microscopic particles described by the following nonlinearSchrodinger equation,

i<t>r + \x<x< + | 0 | 2 0 = -ic4> - i04>x'x' + *£|<A|V - « * M V (8-187)

were studied numerically by Moussa et al.. They found that soliton-like solution ofthis equation is of the form

<t>{x',t') = 7/sechfrV - A)]e'<"'V/3+«')>


where

rf = ^ [ 5 ( 2 £ - P) ± \/25(2e-/02-480cd)],

A and 0' are arbitrary constants. The results reveal various shapes of the solitarysolutions in such a case.

8.12 Interaction of Microscopic Particles and Its Radiation Effectin Perturbed Systems with Different Dispersions

We first consider collision between microscopic particles in a perturbed system whichwas investigated by Malomed. The microscopic particles are described by the fol-lowing nonlinear Schrodinger equation with a nonlinear damping

i f c + <l>x'z> + 2\4>\2<j> = £<j>R{4>), (8.188)

where R{4>) is a perturbation, e is real small number. Malomed considered

R(4>) = (\<t>\2)X', (8-189)

and

m = iP r midx, (8.190)•K J-oo X' -X

Here P stands for the principle value of the integral. R{(j>) given in (8.189) specifiesthe dispersion effect, and that in (8.190) describes a nonlinear Landau dampingeffect. Obviously, when e = 0, the soliton solution of (8.188) is given by (8.19).When e ^ 0 the microscopic particle experiences an acceleration induced by thenonlinear damping. In the lowest order approximation, the equations of motion forthe amplitude 77 and velocity v of the microscopic particle are r/t = 0, and vt = S'er]n,with 5' = 128/13 and n = 4 in the case of (8.189), or 6' «a 7.443 and n = 3 in thecase of (8.190). The accelerated motion of the microscopic particle can generateradiation and self-damping. However, these effects are very weak, and depend onthe amplitude of the microscopic particle.

Consider the collision between two microscopic particles and collision-inducedchanges in their amplitudes. Such changes are useful in understanding of energytransfer from the slower microscopic particle to the faster one. As it is known, thesum of amplitudes is conserved because the number of quanta bound in the solitonis Nso\ = 4r) which is conserved for the soliton (8.19). We now examine the changesin the amplitude and velocity of the microscopic particle in the system describedby (8.188), using the perturbation theory.

Assuming that the relative velocity v of the colliding microscopic particles (soli-tons) is large compared to their amplitude, Malomed considered the lowest order


approximation and expressed the full wave field during the collision as a linearsuperposition of the unperturbed solutions (8.19),

# M ) = (&oi)i + (<M2. (8.191)

Inserting (8.191) into the right-hand side of (8.188), both (8.189) and (8.190)become polynomials in {<j>so\)i and (</>soi)2- The number of quanta is conserved inthe system described by (8.188), i.e.,

rOO

N = \<j>\2dx' = constant.J-00

It can be shown that

d d r+o° r+00-(TVso,)! = -T. / |(<?W)i|2cte = -i / dx!4>itf)Pi(x') + c.c, (8.192)a i " ' J-00 J-00

where

Pi =e(&oi)i(&oi)2[(&,i)2]*' (8.193)

Further analysis shows that evolution of the first soliton is dominated by the termof the polynomial given in (8.193) in the case of (8.189). Indeed, inserting thepolynomial (8.193) into (8.192), one notes that the dominating term must containone power of (0soi)2 and one power of (<>gOi)2 (or their derivatives), lest the integralon the right-hand side of (8.192) will be exponentially small (the integrand willcontain a rapidly oscillating exponent), and that either (0Soi)2 or (< >*ol)2 must beexpressed by its derivative, which gives an additional large multiplier w v. Usingr) = TVsoi/4 and (8.19), (8.193) and (8.192) in the following

-I /-+OO J

J ( 1 ) ? ? i = i L ^ ( i V s o i ) i 'we get

<5(1)f?i,2 = 4«7i??2Sgn(i>1,2 - v2,i). (8.194)

The superscript (1) in (8.194) indicates result obtained in the first order of e. Ev-idently, (8.194) satisfies the conservation law S^rji + 5^r)2 - 0. For (8.190), theresult is similar

SWVl,2 = 80JiffeK2 - W2.1)"1' (8-195)

Malomed considered an ensemble of solitons with different initial amplitudesand velocities. Solitons with the largest initial amplitude will acquire the largestvelocity according to vt< = 8'erf1, and from (8.194) and (8.195), their amplitudes willbe further increased due to collisions with the slower solitons. Malomed estimatedthat if the mean distance between the microscopic particles (solitons) is L and acharacteristic value of the initial amplitudes is 770, then the time T during which the


microscopic particles (solitons) will separate into "large" and "small" ones is aboutT « y/Lhl/t for (8.189) and T » L/(en0) for (8.190).

Using perturbed theory and numerical method, Malomed also investigated col-lision of microscopic particles described by (8.188). The collision-induced changesin the amplitudes of both microscopic particles and their radiation losses in thecollision process were obtained for the case of relative velocity of the microscopicparticles being much larger than their amplitudes. The collision between two mi-croscopic particles of amplitudes 771 and rj2 and the relative velocity v(v2 ^ vh V2)results in a radiation loss which depends on the changes S^7]n (n = 1,2) in theamplitudes of the colliding microscopic particles, i.e.,

f - 4 ^ 1 , 2 ^ , 1 , for (8.189),S{2)Vi,2={ 16s2

2 . , f i 1 Q n . (8.196)I 2~I?l,2l?2 1. f ° r ( 8 - 1 9 0 ) -

However, the changes in the amplitudes of the microscopic particles produced bythe adiabatic (non-radiative) exchange should also be considered, which are firstorder effects in e and are given by (8.194) and (8.195). Therefore, total changesin the amplitudes of the microscopic particles should be the sum of the above twoeffects.

Contribution of the collision to the change in the velocities of the solitons can becalculated. However, this effect, unlike the energy exchange and the radiative losses,is not of principal importance since the velocities of the solitons change continuouslybetween collisions.

Malomed also investigated radiative losses in collision processes of microscopicparticles obeying the following nonlinear Schrodinger equation, with high-order(third order) dispersions

i<k + <t>x'x' + 2\<j>\24> = iei4>z'x-z' + ie2\4>\2<l>x> + ie3<P4>x', (8.197)

where e\, £2 and £3 are coefficients of the third-order linear and nonlinear disper-sions, respectively, which are some real and small perturbation effects. Malomedcalculated the radiative losses using perturbation theory based on the inverse scat-tering method. In his treatment of the collision process, he also assumed that therelative velocity of the two microscopic particles is much larger than their ampli-tudes, i.e., v = \v\ — i>2| ^$> T}i,ri2. Hence, what we discussed in the above is stillapplicable here.

In the lowest order approximation, Malomed expressed the solution of (8.197)in the form of

<t>(x',t') = J2 [tfiSiV.f) + <ty(nV>*')] , (8.198)71—1

where the correction 8<pn\x' ,t') axe determined by the following linear equation

^ 1 ) + M ^ + 4 | ^ > | a ^ 1 ) + 2 [ ^ ] a ^ W = -4 | ^ |V ( 1 ) +2[^ ) ]V ( 1 ) ' . (8.199)


It is easily seen that the lowest-order approximate solution to (8.199) is givenby keeping only the first term on either side of the equation, and the fully modifiedsolitonic wave form can be expressed as

*% =4$+64$ =*%*», (8.200)

where v — vi — v2 is the relative velocity of the colliding microscopic particles, and

8 =-*^-ta,nh[2r)2(x' - vt')].

The collision results in the following change in the phase of the first microscopicparticle

A0i = 6{x' - vt' = -co) - 0(x' - vt' = +co) = 16—. (8.201)

As discussed in Chapter 4, the expression of the collision-induced phase shift inthe unperturbed nonlinear Schrodinger equation is

Thus (8.201) is nothing but the lowest-order term in the expansion of (8.202) inpowers of IJ"1 . Furthermore, the collision between the microscopic particles givesrise to a shift in the center of mass of the first soliton, and a corresponding expansionwhich starts from the term v~2.

According to the inverse scattering method, the radiation part of the wave fieldis described by the spectral amplitudes (reflection coefficient) B(X) and the spectralparameter 2A (radiation wave number). The basic ingredient of the perturbationtheory is the general perturbation-induced evolution equation for B(X), which isgiven by

^ = -a(A)e-4i*V f~dx' {[^>{X,x')fP*(x')+

[^2>{\,x')fP(x')), (8.203)

where P(z') denotes an emission-generating perturbing term on the right-hand sideof (8.197), and l i2)(A, x') are components of the Jost function, which are relatedto (j){x') by the Zakharov-Shabat equations (4.42) or (3.22). The quiescent (v - 0)one-soliton solution is given by (8.19). The transmission coefficient in (8.203) isa(A) = (A — ir))/(\ + ir)), and

^ ( A , * ' ) = ^ - [ A + ^tanh(2^ ' ) ] ,

^ ( A , x') = -^-eiXx'+4^t'sech(2r1x'). (8.204)

(8.202)


Since it is assumed that \v\ — D2I ^ ^1,^2 in the lowest order approximation, acorrection corresponding to the modified soli ton wave form given by (8.200) can befound. Substituting (8.200) into the Zakharov-Shabat equation (4.42), the correc-tion to ip(A, x1) can be found to be

8i> = -id I ?2) j . (8.205)

Inserting (8.200), (8.204) and (8.205) into (8.203) with the perturbing termiei<t>x'x'x', and integrating (8.203) over t', we obtain, within the lowest order ap-proximation,

o /,N f°° JtdB(X) .167^772A 17T?2 + 3A2 ,_ / nX \Bfin(A) = L ^M = l 15, ^+A» S6Ch (M) • ( 8 -2°6 )

Equation (8.197) (with ei = €3 = 0) conserves the number of quanta, the mo-mentum and the energy,

r+00

N = / <feW)|2,J—00

r+00

p=i <&'#:,,j—00

Thus the spectral densities of the radiation parts of these three conserved quantitiesare

N'(X) = ^|Bfin(A)|2,

P'(A) = - ^ | B f i n ( A ) | 2 ,

4A2

e'M = -—\BRn(\)\2,

so that the net number of quanta, momentum, and energy carried by the radiationare given by

/"+00

iVrad = / dAJV'(A),J—00

r+00

Prad = / d\P'{\), (8.207)

/>+oo

EraA = / dXE'(X).J—oo


The same quantities for the unperturbed soliton are

NSOI=4T], Psol=2r}v, Esoi = -—ri3+r)v2.

Substituting (8.206) into (8.207) and integrating the spectral density, the totalnumber of quanta emitted by the first microscopic particle (soliton) can be obtainedand is given by

„(!) _ fienrhmeiV , , .JVrad - 1 ^ I »?W) (H.ZUii)

where

T f+o° , rz(17 + 3^2)l2 ^firzs

Using the conservation of the net number of quanta and (8.208), the collision-induced change in the amplitude of the microscopic particle (soliton) can be ob-tained as

8m = -\N&. (8-209)

For the second soliton, changes in the amplitudes of the microscopic particlesare given by (8.208) and (8.209) with 771 and 772 replaced by their transposes. Usingthe momentum and energy conservations we can find the collision-induced changesin the velocities of the microscopic particles through the balance equations for themomentum and energy in the center-of-mass reference frame defined by rjivi +Tj2V2 = 0. In fact, from (8.206) and the assumption of \v\ — t;2| > 171,772, o n e c a n

conclude that the spectral wave packets emitted by the two microscopic particlesare centered at Ai = ui/4 and A2 = t^/4, respectively, and the widths of thepackets are given by SXn « 77,, (n = 1,2). According to (8.207), this implies that,in the first-order approximation, the emitted radiation carries zero momentum, andST]I/ST]2 = 771/772 in such a case. Based on these, Malomed obtained changes in thevelocities of the particles 6v\ and Sv2 as

771^1+772^2 = 0. (8.210)

Substituting (8.207), (8.210) and 771 1 = -772K2 into the energy-balance equation,Malomed determined Svi,

viSvi = — (771 + 772 - 771772) 771.

For the effects of the other two perturbing terms involving €2 and 63 in (8.197),on the emission of radiation and on the amplitudes and velocities of the solitons, atedious calculation cannot be avoided. As mentioned above, the particular cases ofei = 0, €2 = 2t3 and £3 = 0, £2 = 6ei can be integrated exactly. The soliton-solitoncollisions are elastic in these cases. Thus one may infer that the contributions to


B(X) due to the second and third terms, respectively, in (8.197), or the first [see(8.206)] and second, may cancel each other. If this is true, all the above results areapplicable to the general case, (8.197), if the parameter ex in (8.206) and (8.208) isreplaced by

1 1Ceff = Cl - 7e2 + o€3'

D 6

Besley et al. used multiscale perturbation theory in conjunction with the inversescattering method to study the interaction of many microscopic particles (solitons)obeying the following nonlinear Schrodinger equation which has a small correctionto the nonlinear potential,

i<t>f + 2 <f>x-x< + \<f>\2, </>*] = 0 ,

subjecting to the initial condition <j>(x',0) = 4>o{x') for certain initial field cf>o(x').In the limit of the perturbation term F[<p, <j>*] being small, it can be assumed thatthe microscopic particles are all moving with the same velocity at the initial in-stant. This maximizes the effect each microscopic particle has on the others as aconsequence of the perturbation. Over a long time, the amplitudes of the micro-scopic particles remain the same, while their centers of mass move according to theNewton's equation, with a force for which the above authors presented an integralformula.

Interaction of microscopic particles through a quintic perturbation was also stud-ied by the above authors. They give more details since symmetries, which are relatedto the form of the perturbation and to the small number of particles involved, allowthe problem to be reduced to a one-dimensional problem with a single parameter,the effective mass. They calculated the binding energy and oscillation frequency ofnearby solitons in the stable case, when the perturbation is an attractive correlationto the potential. Numerical results verified accuracy of the perturbative calculationand revealed its range of validity.

8.13 Microscopic Particles in Three and Two Dimensional Nonlin-ear Media with Impurities

Properties of microscopic particles described by the nonlinear Schrodinger equationin two and three dimensional systems were studied by many scientists, for example,Gaididei et al., Desyatnikov et al., Infeld and Rowlands, Pokrovsky and Talopov,Germanschewski et al., Konopelchenko, and so on. Here the work of Desyatnikovet al. and Gaididei et al. will be briefly discussed.

Gaididei et al. investigated dynamics of nonlinear excitation in two-dimensionalsystems with impurities. In their study, the following assumptions were made, thedensity of excitations is low, the impurity is located at site 0 of a lattice consisting ofsimilar hosts, and the impurity substitution for the host does not distort the lattice


significantly. The dynamic equation of the microscopic particle in the system iswritten in the form of the following nonlinear Schrodinger equation

where

V2 = £ j + | ^ , and Jdr\<f>(f,t)\2 = l. (8.212)

E(r) in (8.211) is the continuum limit of the on-site excitation energy E^. It deter-mines an energetic profile for an excitation in the vicinity of the impurity molecule.Gaididei et al. considered only some general properties of soliton dynamics in thevicinity of an impurity. E(r) is given as an axially symmetric Gaussian function

E(r) = Ee-Wrrf, (8.213)

where E is the strength of the impurity and ro is its radius. Letting

_ f ht limbrl 1 imErl

equations (8.211) and (8.213) are replaced by

iifr + V V + M V = V(p)<p, V{p) = ee-'2, (8.214)

where e characterizes the strength of the impurity and

jdp\ip\2 = N. (8.215)

In general, the two-dimensional nonlinear Schrodinger equation possesses brightvortex soliton solutions, i.e., localized solutions with an internal velocity (spin), andsome unstable solutions that may either disperse or collapse. These two types ofsolutions are separated by the so-called ground state solution whose width does notchange in time. The ground state solution to (8.214) with V = 0 is approximatelygiven by

^s = Bssech(-^\eiT, (8.216)

if the particle is initially at rest and centered at p = 0. The ground state amplitudeBs and width As are given by

/ 12 In 2 , /21n2 + l / o n i ^

*- = V4E2=T' ^ V S I ^ T ' (8-217)

respectively. Substituting (8.216) into (8.215) yields N = Ns = 11.7. For the initialconditions, (p = (p(p, 0) with N being larger (smaller) than A s, the solution, <p =

(8.211)


<p(p,r), of the two-dimensional nonlinear Schrodinger equation collapses (disperses)in finite time.

To study the motion of a microscopic particle (soliton) in the neighborhood ofan impurity, Gaididei et al. assumed that the radius TQ of the impurity is largecompared with the width of the microscopic particle. In this case we can expandthe impurity potential V{p) in powers of p. Keeping only terms of the second order,i.e., V(p) ss e(l — p2), the problem is reduced to that of a microscopic particlemoving in a two-dimensional parabolic potential, given by

V(p,r) = -e(T)p2, (8.218)

where e(r) is an arbitrary function of time. Transforming to the noninertial frameof reference in which the center of mass of the excitation is at rest, we have

cp(p, T) = *(p, T)exv^k-p-%-j*df [k{ff + e(f)R2(f)] j , (8.219)

where

i? = l |dr . r>(r ,T) | 2

is the position of the center of mass and p ' = p — R(r) is the coordinate in the new

frame of reference, R = dR/dr. It can be shown that

R - Ae{r)R = 0, (8.220)

and

i*T + V2,* + |* | 2$ + e(r)p'2* = 0. (8.221)

Equations (8.220) and (8.221) show that in the parabolic potential, the external(position of center of mass, R) and the internal [^(P,T)} degrees of freedom ofthe microscopic particle are separated. This is a unique property of the harmonicpotential. Anharmonic terms in V(p) would lead to coupling between the externaland internal degrees of freedom of the microscopic particles. The center of massof the microscopic particle behaves like a time-dependent oscillator. To considerthe internal motion of the microscopic particle in the impurity field, we introducea change of variables which is referred to as lens transformation

*(p,r) = —$(£,T)exp [t (T + £p'2)] , (8.222)

where a(r) is the width of the microscopic particle, and

£ = ~i \"> •* = / O/~\ and a, = ——a(r) Jo a2(r) dt


are new space and time variables, respectively. Substituting (8.222) into (8.221)yields

i $ T + [ I F + i ~ k ] $ + 1 $ | 2 $ ~ A^2$ ~$=0> (8-223)

where

A = io 3 a -e ( r ) a 4 . (8.224)

The same problem was also investigated by Karlsson et al. using the collectivecoordinate method. We are interested in localized solutions ($ -> 0 for £ -» oo) of(8.223). If A is a positive constant, we can set $ T = 0 and (8.223) becomes

( | ^ + ~ ) * - A£2$ + ||2$ - * = 0. (8-225)

In the case of A > 0, equation (8.225) has localized solutions, and in this case,(8.222) describes a non-collapsing soliton. We solve (8.225) now.

If £ -> C/A1/4, $ ->• A1/4^, Gaididei et al. obtained from (8.225)

{w+m)*+m2-*2)* = 7K*- (8-226)The corresponding energy is given by

E = Jdtz(jt2- \\*\4+em2 + ^ I $ I 2 ) • (8-227)

The following form can be assumed for a trial function,

*=\/^sechG9' (8-228)where /? is a variational parameter. Inserting (8.228) into (8.227) and minimizingE with respect to /?, we obtain

where £ denotes the Riemann zetafunction and Ns = 47rln2(21n2+l)/(41n2—1) ~11.7 is the number of excitations in the ground state. For N < Ns, returning to theoriginal variable, the soliton solution of (8.221) was found to be

*= i r ) / 3 ^ s e d * e x p [* ( T + i p l 2 ) \ • (8-229)The soliton dynamics in the parabolic potential (8.218) is governed by (8.220)

for the center of mass and (8.224) for the width of the soliton. Equations (8.220)


and (8.224) belong to the class of Ermakov-Pinney equations, the solution of whichwas obtained by Pinney and it is

r 4A i 1 / 2

a(t)=[u2(t) + |^t;2(i)] . (8.230)

where W = iiv — uv is the Wronskian and (u,v) is the fundamental set of solutionsof the respective linear equation y — 4e(t)y = 0, which coincides with (8.220) as faras the motion of the center of mass is concerned. If

e(r) = J S , _ ? ! 2 i M = c o n s t . (8.23I)

then the impurity is an acceptor of excitations. Inserting (8.231) into (8.220) and(8.230), Gaididei et al. obtained

R(t) = Ro cos[cj(t + t0)],fe2 A

a\t) = A* cos2M* + *i)] + 2m\Elr2A2 ^ 2 M * + **)]•

Here Ro, A, t0 and t\ are arbitrary constants, w = yf2\E\/m/r0 is the frequencyof oscillation of the center of mass. The width a(t) of the soliton oscillates at afrequency of 2u. Here, w depends on the depth |JE| of the impurity, the radius ro,and the mass of the microscopic particle. The above approach is valid if the radiusof the impurity ro is larger than the width of the soliton (i.e., a < 1).

IfQ2

e(r) = —^[1 + Acos(fir)], (8.232)

then the acceptor molecule oscillates around its equilibrium position with the fre-quency fi. The acceptor can perform either translational or vibrational motion. Theparameter A is proportional to the amplitude of the impurity oscillations. Inserting(8.232) into (8.220), we get

R + nl[l + X cos(fi-r)].R = 0, (8.233)

which is the Mathieu's equation. The (fioi A) plane can be divided into a stabilityregion [R(T) remains finite as r approaches to infinity] and an instability region(parametric resonance). For the first resonant range of |S7 — 2Slo| < fioA/2, Gaididei,et al. obtained an approximate solution of (8.233) which is given by

R(T) = Ro{e<T cos[ft(r + n)] + e~7T sin[n(r + To)]}, (8.234)

where RQ and ro are constants that depend on the initial position of the microscopicparticle and its initial velocity. 7 is given by

7=i[(^)2-(fi-2fio)2] • (8.235)


We can see from (8.234) - (8.235) that when an acceptor oscillates about its equi-librium position with a frequency in the range given, it cannot trap the microscopicparticle. The amplitude of the oscillations of the microscopic particle increasesexponentially with the amplification coefficient 7.

Gaididei et al. also obtained numerical solutions of (8.223) using the split-stepFourier method. The numerical results are consistent with the analytical resultsgiven above.

Desyatnikov et al. investigated solutions of high-dimensional <^6-nonlinearSchrodinger equation

; < ^ - + V2<?!> + I<£|V - |0|4</> = 0, (8.236)

where

v ' - ^ l + ^L + ldx'2 dy'2 + dz12 '

and t' = t/h, x' = x\f2m/h, y' = y\/2m/h, z' = zy/2m/h. In the case of two-dimension, Quiroga-Teixeiro and Michinel found that there exists a 2D vortex soli-ton of (8.236) which was shown to be stable by means of numerical simulation.However, Kruglov et al. found that the helical vortex soliton of this model, withan amplitude periodically modulated along the propagation direction, is unstable.Meanwhile, this 2D vortex soliton with the quadratic nonlinearity, although ex-ists as a stationary solution, is subject to a strong azimuthal instability. This wasdemonstrated numerically by Firth and Skryabin, and Petrov et al., and observedexperimentally in optical fiber by Petrov et al. A similar strong azimuthal insta-bility of the 2D spinning soliton has been predicted by numerical simulation withsaturable nonlinearity by Petrov et al.. However, in the 3D case, numerical simu-lation showed that there is also a 3D soliton solution to (8.236). Desyatnikov et al.first obtained for radially symmetric soliton solution without the spin. Their workwill be introduced in the following.

Let r = \Jx'2 + y'2 + z'2 and assume the solution of (8.236) has the followingform

<j>(x',y',z',t)=eikt'i>(f).

Substituting the above into (8.236), we get

i M O + ^r(r=) - ty{?) + V-3W - V>5(r> = 0, (8.237)

The boundary condition is defined by means of the asymptotic expression,

%l)[f) « a(fc)(l + cr2), at r -)• 0;

rP(r) » W.e-y/krt a t r _> oo. (8.238)r

where c-{ai -a2 + fc)/6, a(k) and B(k) are functions of k.


Desyatnikov et al. obtained numerically the solution of (8.237) for different kby the shooting method as shown in Fig. 8.8. Prom this figure we can see that theeffective size of the soliton increases with k, which results in a natter distribution ofthe field at the center of the soliton. This corresponds to a decrease in the curvatureparameter c in (8.238). When c = 0, the amplitude of the soliton a(k) at r = 0approaches to two limiting values: ai,2(&) = ± y l + >/l — 4fc/2.

Fig. 8.8 The zero spin 3D soliton solutions of (8.237). The values of the propagation constant kare indicated near the curves.

In any dimension, there is a similar upper boundary for the values of k at whichsolitons exist. In ID, one has, instead of (8.237), an equation

x{>" - kip + V>3 - tp5 = 0 ,

which has the well-known exact soliton solution

<d,2(x') = 4fc

K ' 1 + >/l-(16/3)fccosh[2>/ib(s/ - x'o)}'

which is the same as (8.149) except that —c*i is now replaced by k. Obviously, thissolution exists if k < &max = 3/16. In 2D, it was found, using the same shootingmethod, that fcmax & 0.18. If fcmax < feax < fcmax , the upper boundary for asoliton solution of (8.237) to exist decreases as the space dimension increases. Themost important physical characteristic of the 3D soliton is its energy,

/•OO

E(k)=4n ip2{f]k)^dr.Jo

In 2D, the soliton is interpreted as a spatial cylindrical beam. Its energy reachesa finite value Eîn' w 11.75 at k = 0. In ID, the energy of the soliton vanishes ask -> 0. The divergence of the energy of the 3D soliton at k -»• 0 is a consequenceof a minimum energy required the 3D zero-spin solitons, whose numerical value is-Emin/47r ss 15. Therefore, there is a U-shaped dependence of E(k) on k in 3D.This means that there are two soliton solutions, with different values of k, at eachenergy value E > Emin. This is a distinctive feature of the 3D case, which is alsoknown in the saturable model.


The spherical spatiotemporal coordinates can be used to construct solutions to(8.237) in the form of 3D solitons with an integer spin m ^ 0. et al. searched forsolutions of the form

tf>(z',2/',2',t') = eikt>+im*Mr',6), (8.239)

where cos# = z/r, and (p is the usual angular coordinate in the transverse plane{x,y). Then, equation (8.237) is transformed into

I T r a + o • oaE sm0T£ ~ , • \ n ~ ty + ip3 - ip5 = 0. (8.240r2 or \ dr ) r2 sm6 86 \ 06 J r2 sin2 6

A variational approximation was developed to describe the 3D spinning solitonsand the following trial function was used

V>(r,0) = tf(r)sin0. (8.241)

Inserting (8.241) into (8.240) and integrating the equation with respect to 6, butkeeping an arbitrary dependence of \P(r) on r, the following can be obtained.

^ + 2 £ _ 2 * _ w 4 * , _ 2 4 t t 6 = 0 _dr2 r dr r2 5 35

Solutions to this equation corresponding to different values of k are displayedin Fig. 8.9. This is a spinning soliton. These solutions were found numerically bymeans of the shooting method adjusted to the obvious boundary conditions that\P(r) must vanish linearly at r -> 0 and exponentially at r —> oo. Similar to that inthe ID case obtained by Teixeiro and Michinel, it was found that the slope of thefunction \P(r) at r = 0 increases with k up to a maximum value at k = 0.09, andthen decreases after this (in the 2D case, a maximum was reached at k = 0.145).The energy of the spinning soliton (8.239) is given by

/•OO piT

E = 2n r2dr ip2 (r,6) sin 6d6,Jo Jo

or

E(*) = ^ / *2{r,k)r2dr.«* Jo

The energy given by the latter expression is displayed as a function of the prop-agation constant k. The minimum of the soliton energy is Emm/4-K « 62.6, which islocated at A; = kcr « 0.033. This suggests that the spinning soliton with m = 1 canbe stable at k > kcr. It was not possible to have soliton solutions to (8.242) whenk exceeds some maximum value, which was shown to coincide with fcmax « 0.15,the upper bound of the region where the zero-spin solitons exists. Thus, the upperbound for multidimensional solitons does not depend strongly on the value of thespin, but on the spatial dimension. Desyatnikov et al. verified that the upper boundfor 2D solitons with m = 0 almost coincides with that of m = 1 (fcmax « 0.18 for

(8.242)


Fig. 8.9 3D spinning solitons with m = 1. The solid and dashed curves show, respectively, thefunctions <Sf(r) and $f(p) [see (8.241) and (8.243)] with the same values of k, as indicated.

both m = 0 and m = 1). The ratio of the minimum energy of the m = 0 andm = 1 soliton in the 2D case (which are E^n

= ^0 for the soliton with m = 1 and- mfn = 1 1 > 7 5 f o r t n e zero-spin soliton) is 50/11.75 ~ 4.26. Comparing this to thesame ratio in the 3D case, 62.6/15 ~ 4.17, we find that, in any dimension, formationof a spinning soliton requires energy which is roughly four times that necessary forthe formation of a spinless soliton. Thus, experimental generation of the spinningsoliton is expected to be harder than that of zero-spin, but not impossible.

Desyatnikov et al. also gave the solution of (8.237) in the form of (8.239) incylindrical coordinate system (p, y>,r). tjj(r',9) in (8.239) is replaced by

^(p,T) = *(p)sech(/it'). (8-243)

The form of \&(p) is shown in Fig. 8.9. The soliton solution found in such a casehas same features as that in the spherical coordinates, as discussed above.

Bibliography

Abdullaev, F. (1982). Lebedel Institute Reports on Physics 10 3.Abdullaev, F. (1982). Theor. Math. Phys. 51 454 (in Russian).Abdullaev, F. (1994). Theory of solitons in inhomogeneous media, Wiley and Sons, New

York.Abdullaev, F., Abrarev, R. M. and Darmanyan, S. A. (1989). Opt. Lett. 14 131.Abdullaev, F., Darmanyan, S. and Umarev, B. (1985). Phys. Lett. A 108 7.Abdullaev, F., Darmanyan, S. and Khabibullaev, P. (1990). Optical solitons, Springer,

Berlin.Abdullaev, F.,et al. (1985). Contributions of III Inter, symposium on some problems of

statistical mechanics, Dubna, JINR, 3-7.Ablowitz, M. J. and Segur, H. (1981). Solitons and the inverse scattering transform, SIAM,

Philadelphia.


Barashenkov, I. V. and Makhankov, V. G. (1988). Phys. Lett. A 128 52.Benjanian, T. B. and Feir, J. E. (1967). J. Fluid. Mech. A 21 1439.Besley, T. A., Miller, P. D. and Akhmedier, N. N. (2000). Phys. Rev. E 61 7121.Chen, H. H. (1978). Phys. Fluids 21 377.Chen, H. H. and Liu, C. S. (1976). Phys. Rev. Lett. 37 693.Chiao, R. Y., Garmire, E. and Townes, C. H. (1964). Phys. Rev. Lett. 13 479.Chirikov, B. V. (1979). Phys. Rep. 52 263.Cohen, G. (2000). Phys. Rev. E 61 874.Cowan, S., Enn, R. H., Rangnekar, S. S. and Sanghera, S. S. (1986). Can. J. Phys. 64 311.Crespe, J. M. S., Akhmediev, N. N. and Afanasjev, V. V. (1996). J. Opt. Soc. Am. B 13

1409.Degtyarev, L. M., Makhankov, V. G. and Rudakov, L. I. (1974). Sov. Phys. JETP 40 264.de Oliveira, J. M., Cavalcanti, S. B., Cerda, S. C. and Hickmann, J. M. (1998). Phys. Lett.

A 247 294.Desyatnikov, A., Maimistov, A., Malomed, B. (2000). Phys. Rev. E 61 3107.Faddeev, L. D. and Takhtajan, L. A. (1987). Hamiltonian methods in the theory of solitons,

Springer, Berlin.Firth, W. J. and Skryabin, D. V. (1997). Phys. Rev. Lett. 79 245.Foldy, L. L. (1976). Am. J. Phys. 44 1196.Gaididei, Yu. B., Rasimussen, K. <j>. and Christiansen, P. L. (1995). Phys. Rev. E 52 2951.Garcia-Ripoll, J. J. and Perez-Garcia, V. M. (1999). Phys. Rev. A 60 4864.Garcia-Ripoll, J. J. and Perez-Garcia, V. M. (2004). E-print,

http://xxx.lanl.gov/abs/Patt-sol/9904006.Garcia-Ripoll, J. J., Perez-Garcia, V. M. and Torres, P. (1999). Phys. Rev. Lett. 83 1715.Germanschewski, K., Grauer, R., Berge, L., Mezentsev, V. K. and Rasmussen, J. J. (2001).

Physica D151 175.Husimi, K. (1953). Prog. Theor. Phys. 9 381.Infeld, E. and Rowlands, G. (1990). Nonlinear waves, solitons and chaos, Cambridge Univ.

Press. Cambridge.Karlsson, M., Anderson, D. and Desaix, M. (1992). J. Opt. Soc. Am. 17 22.Karpman, V. I. (1967). JETP Lett. 6 277.Karpman, V. I. (1991). Phys. Lett. A 160 531.Karpman, V. I. (1993). Phys. Lett. A 181 211; Phys. Rev. E 47 2073.Karpman, V. I. (1994). Phys. Lett. A 193 355.Karpman, V. I. (1998). Phys. Lett. A 244 394 and 397.Karpman, V. I. (2000). Phys. Rev. E 62 5678.Karpman, V. I. and Shagalov, A. G. (1991). Phys. Lett. A 160 538.Karpman, V. I. and Shagalov, A. G. (1999). Phys. Lett. A 254 319.Kerner, E. H. (1958). Can. J. Phys. 36 371.Kivshar, Yu. S. and Gredeskul, S. A. (1990). Phys. Rev. Lett. 64 1693.Kivshar, Yu. S. and Malomed, B. A. (1989). Rev. Mod. Phys. 61 763.Klyatzkin, V. I. (1980). Stochastic equations and waves in randomly inhomogeneous media,

Nauka, Moscow (in Russian).Kodama, Y. (1985). J. Stat. Phys. 39 597.Kodama, Y. (1985). Physica D 16 14.Kodama, Y. (1985). Phys. Lett. A 107 245.Kodama, Y. (1985). Phys. Lett. A 112 193.Kodama, Y. and Hasegawa, A. (1982). Opt. Lett. 7 339.Konopelchenko, B. G. (1995). Solitons in multidimensions, World Scientific, Singapore.Kruglov, V. I., Logvin, Yu. A. and Volkov, V. M. (1992). J. Mod. Opt. 39 2277.


Kundu, A. (1987). Physica D25 399.Makhankov, V. G. (1990). Soliton phenomenology, Kluwer, Amsterdam.Malomed, B. A. (1987). Sov. J. Plasma Phys. 13 360.Malomed, B. A. (1988). Phys. Scr. 38 66.Malomed, B. A. (1990). Phys. Rev. A 41 4538.Malomed, B. A. (1991). Phys. Rev. A 43 3114.Malomed, B. A. (1991). Phys. Rev. A 44 1413.Menyuk, C. R. (1986). Phys. Rev. A 33 4367.Moussa, R., Goumri-said, S. and Aourag, H. (2000). Phys. Lett. A 266 173.Nogami, Y. (1991). Am. J. Phys. 59 64.Nogami, Y. and Toyama, F. M. (1994). Phys. Lett. A 184 245.Nogami, Y. and Toyama, F. M. (1994). Phys. Rev. E 49 4497.Nogami, Y., Vallieres, M. and Van Dijk, W. (1976). Am. J. Phys. 44 886.Novikov, S. P., Manakov, S. V., Pitaevsky, L. P. and Zakharov, V. E. (1984). Theory of

solitons, the inverse scatteaing method, Consultants Bureau, New York.Nozaki, K. and Bekki, N. (1983). Phys. Rev. Lett. 50 1226.Nozaki, K. and Bekki, N. (1986). Physica D 21 381.Pathria, D. and Morris, J. L. (1989). Phys. Scr. 39 673.Petrov, D. V. and Torner, L. (1997). Opt. Quantum Electron 29 1037.Petrov, D. V., Torner, L., Martorell, J., Valaseca, R., Torres, J. P. and Cojocaru, C.

(1998). Opt. Lett. 23 1444.Pinney, E. (1950). Proc. Am. Math. Soc. 1 681.Pokrovsky, V. L., Talopov, A. L. (1986). Thermodynamics of two dimensional soliton

systems in soliton, ed. Trallingen, S. Zakharov, V. E. and Pokrovsky, V. L., North-Holland, Amsterdam, p. 73.

Quiroga-Teixeiro, M. and Michinel, H. (1997). J. Opt. Soc. Am. B14 2004.Reid, J. L. and Ray, J. R. (1984). Z. Angew. Math. Mech. 64 365.Shchesnovish, V. S. and Doktorov, E. V. (1999). Physica D 129 115.Stevenson, P. M. and Roditi, I. (1986). Phys. Rev. D 33 2305.Takagi, S. (1991). Phys. Lett. A 160 251.Zakharov, V. E., Manako, S. V., Noviko, S. P. and Pitayersky, I. P. (1984). Theory of

solitons, Plenum, New York.

Chapter 9

Nonlinear Quantum-MechanicalProperties of Excitons and Phonons

In the remaining two chapters, we will continue to discuss applications of the non-linear quantum mechanics. We will discuss nonlinear properties of exciton, phonon,proton, polaron and magnon. Through these discussions, and what we already knowabout electrons and helium atom from Chapter 2, we can get further understandingon properties and motion of microscopic particles in these nonlinear systems. Thesediscussions will also establish the nonlinear quantum mechanics theory presented inChapter 3 as the correct theory for describing properties and motion of microscopicparticles in nonlinear systems. We will begin with motion of exciton and phonon ina molecular crystal.

9.1 Excitons in Molecular Crystals

Let's first consider a particular molecular crystal, the acetanilide (CH3COHNC6Hs)^ or ACN. Two close chains of hydrogen-bonded amide-I groups which consistsof atoms of carbon, oxygen, nitrogen and hydrogen (CONH) run through the ac-etanilide crystal. Its crystal structure has been determined and a unit cell of ACNis shown in Fig. 9.1. The space group is D\\ (Pbca) and the unit cell or factorgroup is D2h for this crystal. The average lattice constants are a = 1.9640 nm,b — 0.9483 nm, and c = 0.7979 nm. There are eight molecules in an unit cell and atthe amide-I frequency, each of these has one degree of freedom (d.f.). Thus, thereare three infrared-active modes {B\u, B2u, and B3u), four Raman-active modes (Ag,Big, Big-, and B3g), and one inactive mode (Au). However, at low frequency (< 200cm"1), each molecule exhibits 6 d.f. (three translations and three rotations. Thisgives 48 low-frequency modes: 24 Raman active modes (6Ag+6Bi9 +6B2g+6B3g ),18 infrared-active modes (6Biu + 6B2U+6B3u) and six (Au) modes corresponding tothe acoustic modes of translation and rotation). All of these active modes are seenin infrared absorption and Raman experiments. ACN is an interesting system be-cause the nearly planar amide-I groups have bond lengths which are close to thosefound in polypeptide (see Fig.9.2). Since physical properties of such a hydrogenbonded amide-I system are very sensitive to bond lengths, study of ACN revealed

471


some new phenomena. For example, in the experiments of infrared absorption andRaman scattering, a new amide-I band red-shifted from the main peak at 1666cm""1 by about 16 cm""1 when the crystalline acetanilide is cooled from 320 to 10K. No other major changes occur from 4000 to 800 cm""1. The intensity of thisnew band increases steadily from room temperature till 70 K. The band at 1650cm"1 is not present in amorphous materials or ACN methylated at the positionwhere hydrogen-bonded distances occur, but it is recovered after annealing. Simi-lar phenomena can be observed in Raman scattering experiments (Alexander 1985,Alexander and Krumbansl 1986, Benkui and John 1998, Careri et al. 1983, 1984,1985, 1998, Davydov 1968, 1973, 1975, 1977, 1979, 1982, 1985, Eilbeck et al. 1984,Kenkre et al. 1994). Pope and Swenberg 1982, Scott 1990, 1992, 1998, Scott et al.1985, 1989, Silinsh and Capek 1994, Tekec et al. 1998, Xiao 1998).

Fig. 9.1 Various views of the unit cell of ACN, with cell parameters a = 19.640 A, b = 9.483 A,and c = 7.979 A.

Fig. 9.2 Comparison between peptide groups of ACN and a protein molecule.

As it is known, the characteristic feature of the amide-I group, CONH, inpolypeptides is the amide-I mode which mainly involves stretching of the C=O

Nonlinear Quantum-Mechanical Properties of Excitons and Phonons 473

bond. This mode is observed as an infrared absorption peak at 1666 cm"1 in ACNand near this value in a wide variety of materials, including the amide-I groups.The corresponding spectroscopic evidence of the new band at 1650 cm"1 has beenmentioned earlier, but detailed measurements of the crystal structure and specificheat as a function of temperature preclude assignment of the new band to (1) aconventional amide-I mode, (2) crystal defect states, (3) Fermi resonance or (4)frozen kinetics between two different subsystems. The correct assignment is theself-trapping of the amide-I vibrational energy. This is based on the following ex-perimental facts: (1) the 15N substitution induces a small shift to the amide-I at1666 cm"1, and the new band is also shifted by the same amount; (2) deuteriumsubstitution at the NH position strongly affects both the amide-I and the new bandin a complicated way; (3) upon cooling a decrease in the integrated absorption ofthe normal amide-I and a corresponding increase in the integrated absorption of the1650 cm"1 band are observed; (4) the 1650 cm"1 band and the amide-I band showthe same dichroism over the temperature range integrated; (5) the measurementsof specific heat, the dielectric constant and the volume expansion as a function oftemperature rule out the occurrence of rotational isomerism or of a polymorphictransition which would affect some other infrared and Raman absorption bands,but not the new band. The self-trapping mechanism of the amide-I vibrational en-ergy used by Scott and Eilbeck et al. comes from the Davydov model of vibrationenergy transport in alpha-helix protein which was described in Section 5.6. Theyhave given a good account of the properties of first excited state in this model.Scott and co-workers, Alexander and Krumbansl have also obtained an exponentialdependence of the absorption intensity on temperature, exp(—/?T2), and explainedthe experimentally observed intensity change of the new band with decreasing tem-perature in terms of a complementary polaron or soliton of the self-trapping stateof exciton on the basis of the Davydov model (Davydov referred to the intramolec-ular excitation occurred in the systems as exciton in such a case). However, thered shift of a few cm"1 given by this model is much smaller than the experimentalvalue of 16 cm"1. This indicates that these models need further improvement anddevelopment.

Pang and co-workers also proposed that the new band of amide-I is caused bythe self-trapping of amide-I vibrational quanta (exciton) in their soliton model (thevibrational quantum is referred to as exciton, in accordance with the exciton the-ory by the Davydov). But this model is different from Davydov's soliton modelor Alexander's complementary polaron model. Here it is the nonlinear interactionbetween the localized amide-I vibrational-quantum and low frequency vibrationsof the lattice (phonons) that results in the self-trapping of exciton as a soliton.The mechanism of this soliton model can be described as follows. Through theirintrinsic nonlinear interaction, an amide-I vibrational quantum acts as a sourceof low-frequency phonons and causes shifts in the average positions of the latticemolecules in the ground state. These shifts, or lattice distortions, in turn react,


through the same nonlinear interaction, as a potential well to trap the amide-Ivibrational quantum and prevent the energy dispersion of the amide-I vibrationalquantum via the dipole-dipole interaction that occurs in neighboring peptides withcertain electric moments, resulting in the soliton. Evidently, the soliton is formedby self-trapping of excitons interacting with low frequency lattice phonons. Thisis a dynamic self-sustaining entity that propagates together with the lattice defor-mation along the molecular chains. The main properties of the soliton is that itcan move over macroscopic distances with velocity v and retain its wave form, en-ergy, momentum, and other quasi-particle properties. Using this model, Pang andcoworkers also gave an exponential dependence of the infrared-absorption intensityon temperature which is consistent with the experimental result (Pang 1987,1989a,1990, 1992a-c, 1993a-k, 1994b-c, 1996, 1997, 1999, 2000a, 2001a-g, 2004a, Pang andChen 2001, 2002a).

Prom these studies, we know that collective excitations can result from localizedfluctuation of the amide-I vibrational quanta and structural deformation of themolecular chains due to photoexcitation in the molecular crystals. The Hamiltonianof the systems given by Pang is as follows

H = Hex + Hph + -Hint

n n n

+ 2ltf E P n + f £ (#n - Rn-lf + mXl ^{Rn+1 " Rn-l)^n n n

+mx2 £(i?n+i - Rn)rnrn+i - m%2 £ ( f t n - i - Rn)rnrn-i-n n

This Hamiltonian includes vibrational excitation of amide-I caused by localized fluc-tuation, vibration of lattice molecules caused by the structural deformation of themolecular chains, and the interaction between the two modes of motion in the crys-tals, respectively. Here m is the mass of the amide-I vibrational quantum (exciton),Wo and u>i are the diagonal and off-diagonal elements of the dynamic matrix of thevibrational quantum, U)Q is also the Einstein vibrational frequency of the exciton,rruxi\rnrn+i/2 the interaction between the nearest neighboring excitons caused bythe dipole-dipole interaction in the molecular chains, rn andpn = m{rn)t the normalcoordinates of the ith excitons and its canonical conjugate momentum, respectively,M the mass of a molecule or peptide group in the unit cell, 2î = dw^/dRn and2^2 = dJi/dRn the change of energy of exciton and of coupling interaction be-tween the excitons for an unit extension of molecular chain, respectively, Rn andPn = M(Rn)t the canonically conjugate operators of displacement and the mo-mentum of the molecule and /3 the elastic constant of the molecular chains. Theterm Hex in H is the Hamiltonian of harmonic vibration of amide-I including theoff-diagonal factor, Hph is the Hamiltonian of harmonic vibration of the molecular

(9.1)


chain and H\nt is the interaction Hamiltonian between the two modes of motion.This Hamiltonian is significantly different from that of the Davydov model used byScott and co-workers. As far as the vibration of amide-I is concerned, we adopt aharmonic oscillator model with optical vibration that includes an off-diagonal fac-tor, which comes from the interaction between the excitons, and interaction withthe lattice phonon. Thus, the vibrational frequencies of amide-I are related to thedisplacements of lattice molecules, which shows the occurrence of the interactionbetween the amide-I vibrations and displacement of lattice molecules. Their rela-tionship is described by

du2

(J2(Rn) KU2 + -^-{Rn - Rn-l) ="20+ Xltfn - Rn-l),OHn

U>2(Rn) » U2 + X2{Rn - -Rn-l).

Inserting the above into the vibrational Hamiltonian of amide-I and taking intoaccount the effect of the neighboring lattice molecules on both sides of the excitons,(9.1) can be obtained. Therefore, the Hamiltonian given above has high symmetryand a one-to-one correspondence. On the other hand, the Hamiltonian in the Davy-dov model does not have such a one-to-one correspondence. In other words, there isno interaction between the amide-I vibrational quantum and displacement of the lat-tice molecule resulting from the interaction between the neighboring excitons in theDavydov model (see Section 5.6). Thus, the Davydov model has encountered manydifficulties in studying dynamics of energy transport in the systems. The Hamil-tonian given above includes not only the optical vibration of amide-I, but also theresonant interaction caused by the dipole-dipole interaction between neighboringexcitons; and it also takes into account both the change of the relative displacementof the neighboring lattice molecules resulting from the vibration of amide-I and thecorrelation interaction between the neighboring excitons. Therefore, the Hamilto-nian given above has higher symmetry and one-to-one correspondence in Hp andHint. It can give a more complete description on the dynamics of the systemscompared to the Davydov model and other models.

Since the amide-I vibration and the vibration of molecular chains are all quan-tized, we introduce the following canonical second quantization transformation,

r- = v S ( 6 - + ^ Pn = - ' V ^ 6 - - (9-2)R —\î———(n -4- n+ \P

iro9" p — : V i 9 (n+ -n Wro?n In o\Kn~ 2 ^ \/ 2MNu ( 9 q) ' " - ^ V 2M [ q q) ' ^ '

1 ' q q

where i = y/—\, w, = 2A//3/M sin(rog/2) is the frequency of the phonon with wavevector q, N the number of unit cells in the molecular chain, ro the distance betweenthe molecules, b+(bn) and a+(aq) are the creation (annihilation) operators of theexciton and phonon, respectively.


Using (9.2) and (9.3), (9.1) becomes

H = £ £ ° (bnbn + I) ~ J$>+6n+1 + bnb++i) + £ ^ ? (<a, + \)n ^ ' n 9

+ £ [9(q)(btbn + bnbi) + 9l(q)(b+bn+1 + bnb++1) (9.4)n,q

-gziqMb^ + MiU)] (a, + ol^c*""",

where e0 = ^ o , ^ = hcjl/4(jj0, and

Due to the fact that the collective excitations generated by the localized fluctua-tion of the excitons and structural deformation of the molecular chains are coherent,Pang proposed the following form for the trial wave function of the systems

|$) = \<fi(t))U(t)\0)ph = 1 f 1 + 5 > n ( t ) 6 + J |0>ex • U(t)\O)ph, (9.5)

with

U{t) = exp J ^ ^[«n(t)Pn - TTnîin] 1 , (9.6)

or

U(t) = exp J 5> , ( t )< - a;(t)o,] 1 ,

where |0)ei and |0)ph are the excitonic and phononic vacuum-states, respectively,A' the normalization factor. We assume hereafter that A' = 1 for convenience ofcalculation unless it is explicitly mentioned. ipn(t), un{t) = (<J?|i?n|$) and 7rn(t) =($|Pn|) are three sets of unknown functions. Here,

is not the Davydov wave function (neither \D\) in Section 9.5 nor \D2) in (5.72))which is given by \ipo) = ^2n fnit)b^\Q)ex. It is not an excitation state of a single


particle, but rather a coherent state, or more accurately, a quasi-coherent state. Itcan be approximately written as

\y) ~ 1 exp W <pn(t)b+ |0>« = i exp fa M * ) # ~ ¥>»(*)&»] j |0)e»-

The final representation is a standard coherent state. The above derivation is math-ematically justified in the case of small <pn{t) («-e., <pn(t) •€. 1). Therefore, the wavefunction can be viewed as an effective truncation of a standard coherent state,which is referred to as a quasi-coherent state. However, it is not an eigenstate ofthe number operator N = £n6+&n s i n c e N\<P) = J2n'Pri(t)b+\0)ex = \<p) - |0)es.Thus, \<p) indeed represents a coherent superposition of the one-quantum amide-Ivibrational state and the ground state of the exciton. But the number of quantumare determinate in this state and it contains only one exciton because

N = (<p\N\<p) = J > | & + 6 » = J>»MI2 = I-n n

Therefore, \tp) given in (9.5) not only exhibits coherent feature of collective excita-tions of the excitons, caused by the nonlinear interaction generated by the exciton-phonon interaction, which makes the wave functions of the states of the exciton andphonon (the wave function of the phonon is also standard coherent sate) symmetri-cal, but it can also make the number of the excitons conserved in the Hamiltonian(9.4) as well. Meanwhile, the wave function given above has another advantage.The equation of motion of the soliton can be obtained from the following Heisen-berg equations for the creation and annihilation operators of the exciton and phononby using (9.4) - (9.6)

ifi^($|6n|*) = (*|6ni ff|*), i f i j^KI*) = ($K, H|#>. (9.7)

The equations of motion for ipn(t), aq(t) and a*_q(t) are

iftipn = hujQipn - J(<pn+1 + <Pn-l) + ^ {29(<l){aq + <X-q)<PnQ

+(9ii.q) - 92{q)) [(aq + alg)<pn+1 + (aq + a*_q)ipn^]} eiro"n, (9.8)

iMq = hwqaq + ] T {2g(q)\<pn{t)\2

n+[9i(q) ~ S2(9)](v;v>n+i + V»;Vn-0} e^ 0 9 " , (9.9)

iha*_q = -nuqa; - Y, {ig{q)\<pn{t)\2

n

+ [9l(q) ~ 02(g)](Vn¥>n+l + <P*nVn-l)} e'*™". (9.10)


From (9.9) and (9.10), we can get

n *

Prom (9.3), we get the Fourier transformation of the variable un = ($|.Rn|$),

where x = nro and

Thus we get in the long-wave length approximation

where uq as t;og, o = o \/P/M. Now multiplying the above equation byexp(iroqri)/\/N, summing over the wave number, q, and making the continuumapproximation, we get

d2u{*,t) 2d2u(x,t) _ 2hro(Xi + X2) d\<p(x,t)\2

dt2 V° dx2 ~~ MCJQ dx ' K '

At the same time, (9.7) becomes

i hd ^ = ^0-2JMX,t)-Jr29-^

+ 2hr0(Xl+X2) y ( g | t ) g « M . (9.12)Wo OX

Inserting (9.11) into (9.12), we can get the nonlinear Schrodinger equation for mo-tion of the exciton

where

2m~Jr°-Here m is the effective mass of the exciton. This shows clearly that the excitationand motion of the excitons in the ACN is a nonlinear problem, and the excitons in

(9.13)


the laws of nonlinear quantum-mechanical systems satisfy or obey truly the non-linear quantum mechanics, its motion can be exactly described by the nonlinearSchrodinger equation. An exciton in such a case has the properties of microscopicparticles as described in Chapters 3 -6 . Obviously, the nonlinear interaction energy(G) is provided by the coupling between the excitons and phonons, and the excitonself-trapes as a soliton by this nonlinear interaction. The soliton-solution of (9.13)is of the form

¥>(*, t) = ^sech[M'(:c - x0 - vt)] exp j *- [ ^ ~ * o ) - Esolt] } , (9.14)

<x,t) = - n ( g l + a a ) tanh^Ca - x0 - vt)], (9.15)2u;op(l - sJ)

where

G 4 / i 2 r g ( x i + X 2 ) 2 , _ £M ~ 4J' Mvlwlil - S2) ' ^ ~ r o '

us = —, e = fouio — 2J, e0 = &«;o) (9.16)

Using

/Afw, . / 1

and the results given above, we found that aq(t) in (9.3) is given by

~ 2 M ^ W o ( l - S2)ÂTW(, S m h \2fi ) 6 -a"e • ( 9 J 7 )

The energy of the soliton can be found by using (9.14) and (9.15) and is given by

E - T Hdt-r 2 7 + ^ fJ F 4- M ^ 2

where

F - P 01 fi4(Xl+X2)4

is rest energy of the soliton. The mass of the soliton is given by

M rr, i ^ 4 ( x i + y 2 ) 4 ( l - 3 g2 / 2 - g y 2 )

Therefore, the energy and the rest energy of the soliton is about h4(xi +X2)4/3/32JCJ^(1 - s2) and tfixi + XiYI^J^t lower than those of the exciton,


E' = £0 - 2 J + \mv2 and E'o = e0 - 2 J, respectively, but the mass of the soliton isgreater than that of the exciton, m. The exciton-soliton is thus stable.

Pang et al. also studied the influence of anharmonic vibration of the moleculeson the soliton. The phonon's Hamiltonian in (9.1) is replaced by

H'vh = ^M ^ P " + 2 £ ( j R n " jRn-l)2 + lXl{Rn ~ Rn~l)3- (9-18)

n nIt was found that only terms such as (ftu0 - 2J)ip(x, t) in (9.13) are changed. Thusonly the phase, velocity, energy, amplitude and width of the soliton are affected.

9.2 Raman Scattering from Nonlinear Motion of Excitons

In the following three sections, we will discuss physical phenomena arising fromnonlinear excitations and motions of the excitons, i.e., Raman scattering, Mossbauereffect and infrared absorption, which were studied by Pang.

In order to study the Raman scattering, we first diagonalize the soliton Hamil-tonian (9.4) following procedures of Eremko et al.. Then we transform it to theintrinsic reference frame which moves with the soliton at velocity v. In this case,(9.4) is replaced by

H = H-^2 Mv(b+bk + a+ak), (9.19)k

where

Since the soliton excitation is always connected with the deformation of molec-ular chain or intramolecular distances, it is necessary take this deformation intoconsideration in (9.19). In the light of Eremko et al., such a transition is real-ized by the replacement (Ri,Pi) -» {a+,aq) -> (A+,Aq) according to the relations:A+ = a+-a*/</N, Aq = aq-aq/\/~N, where A+ (Aq) is the creation (annihilation)operator of the new phonon. The coherent phonon state (lattice distortion) thenbecomes the vacuum state of the new phonon, \0)ph = exp[a9(t)a+ — a*(t)aq]\0)ph,where Aq\0)ph = 0. We can now carry out the canonical transformation for thepartial diagonalization of the soliton Hamiltonian by the following transformation

n

£vA(™)VA(n') = <W, ^2^>l(n)ipx'(n) = 6X\<.A n

The so-called partial diagonalization of the Hamiltonian means diagonalizationof the part of the Hamiltonian which does not contain the creation and annihilation


operators of the new phonons. In the continuum approximation, the conditionimposed onto the function ip\{ri) to realize such a diagonalization is equivalent tothe following problem on the eigenfunction >p{n/\) and the eigenvalues E\,

h & - * & - * '*«*vo+* -«] * {:)=*» {:) •It has a unique bound state solution,

fn\ HI ,, . (ihvn\^ ( S J = V 2 S e C h ( / X n ) e X P f e J '

Es=e0-2J-^-fj.2J, (9.20)

and an unbound state solution given by

fn\ u, • tanh(un) - ikr0 I ihvn ., \f , 1 = - — F — ^ exP 7T? !• ikron ,

U ; V2n(n - ikr0) y\2Jr0 J'fc2 2

Ek = e0-2J--^ + (kr0)2J. (9.21)

Es is less than that of lowest unbound state E\ = £o — 2J by n2 J in such a case.Thus, the partially diagonalized Hamiltonian in (9.19) becomes

G2

H = EsBfBs + J2 EkB+Bk + h{uq - vq)A+Aq + —k q

+ 7 ^ E h ^ - v^)(Atai+<«;)(! - fis+B«) (9-22)

+ ^ J2P(k>MB°B-k + BtBs)(Atq + Aq)qk

where

xsechj2^-^],

F ( M ) = V 2 i \ ^ [x i (e i""0 " ^ ^ + X2{eiqr0 ~ e~i9r0)]

x / i ^qro 1\ (G - 4ikr0J)[G + Ai(l + q)r0J] J


are the coupling constants of the bound state and the unbound state of the newphonons, respectively. Obviously, the distribution amplitude of the amide-I vi-brational excitations, (9.20) and (9.21), can be obtained if an exact deformationpotential is given a priori. This treatment allows us to see that the Hamiltonian(9.22) or (9.19) describing the molecular chain in the reference frame moving withvelocity v has a soliton state given by (9.20), with energy

and a delocalized state corresponding to the exciton in the undeformed chain withenergy Ek given in (9.21). The former is a localized coherent structure with a sizeof the order of 2nro/(J. that propagates with velocity v and can transfer energyEso\ < £Q. Unlike a bare exciton that can be scattered by interaction with phonons,the soliton state describes a quasi-particle consisting of the exciton plus the latticedeformation and hence it already includes the interaction with the acoustic phonons.Therefore, the soliton is not scattered and affected by this interaction, and it canmaintain its form, energy, momentum and other quasi-particle properties over amacroscopic distance in the absence of external fields and/or impurities.

This result shows that the soliton energy is lower than that of the lowest excitonstate (fc = 0) by G2/48J. Thus, an energy gap between the soliton state and theexciton state occurs which corresponds to the binding energy of the soliton. This isdue to the fact that the soliton, unlike an exciton, is localized, and is a dynamicallyself-sustaining entity as a result of self-trapping of the exciton through interactionwith the deformation of the molecular chain. Because of the self-trapping, theenergy of the exciton drops by about G2/24J which is the same as the deformationenergy of the lattice,

i

The energy of the soliton formed by this mechanism is about G2/48J less than thatof the exciton. Thus, the energy gap between them is G2/48J. The energy gap canresult in a red shift from the main peak at 1666 cm"1 in the infrared absorption andRaman spectra. This red shift of about 16 cm"1 has been observed experimentallyin ACN, as mentioned earlier.

Using the following values for the physical parameters of ACN, J = (3 — 4)cm~1,X = hxi/2uo = (56 - 62)PiV, 0 = (4.8 - 13)N/m, M = 2.25 x 10"25 kg, X' =%2/2w0 = (6 - 8)PN, ro = {2- 4.5) A, w0 = (2 - 4) x 1014 s, wx = (3 - 6) x 1013 s,Pang obtained the binding energy of the soliton or the energy gap to be 18.1 - 33cm"1 which is consistent with the experimental value of 16 cm"1. In contrast, thebinding energy of the soliton in the Davydov model is about 3 - 7 cm"1, whichis too small compared to the experimental value. Obviously, the soliton or thelocalization of the exciton is formed due to the nonlinear interaction energy G =


(1.5-3.2) x 10~21 J which strongly suppresses its dispersive energy J = 0.795x 10~22

J. This indicates clearly that the observed red shift is caused by the formation of thesoliton. Thus it is experimentally confirmed that the exciton is actually localizedin ACN.

Identification of the corresponding Stokes component in the Raman scatteringspectrum can be one of the ways of experimentally determining the energy gapbetween the soliton and exciton, and hence the solitons themselves. It is, therefore,necessary to calculate the differential cross-section of the Raman scattering due tothe solitons. We assume here that the Raman scattering process is activated bysome intermediate states of the molecular chains associated with the soliton, e.g.,electronic excitation whose Hamiltonian is

He = Yên(k)DtnDkn,fcn

where D~£n (Dkn) is the creation (annihilation) operator of the electronic excitationof the nth band with wave vector k and energy en(k). At the same time, let theincident light wave be quantized in the volume V = NroS' and be denoted by theHamiltonian

HQ = ^huJQC+(rCQr,

where Cqa (CQ^) is the creation (annihilation) operator of the photon with wavevector Q, energy HUJQ, and unit polarization vector ^(Q). In such a case, theinteraction Hamiltonian among the soliton, electronic excitation and light waveleading to the Raman scattering can be denoted by

fcfc' q

^J2Y,X^(q)DtnDkn,B+_qBk(B+Bkl - B+B)+ (9.23)

fcfc' q

v^V £ Un<T(Q)(C+, + C-Q,)(D+n - D_Qn),Qan

where Xnni and Xnni are the interaction coefficients among the excitons and solitonsand electronic excitations with different wave vectors, respectively, Una(Q) is thecoupling constant between the electronic excitations and the light wave, i.e.,

I 2TTUna{Q) = -ienWJ-fr-rrjeAQ) • dn,

where dn is the dipole moment of the transition from the ground state of electroninto the state of the nth band, the z-axis is directed along the molecular chain.


Pang calculated the transition probability per unit time of the electron-light-soliton (exciton) system from the initial state,

H = c^0Jo)QBt\o)ex n i«)"'|ouio>e,q Vn«-

to the final state,

IA) = C+j0)QB+\0)eX]l-}=(at)n<\0U\0)e,q Vnq-

caused by the perturbation potential, (9.23). Here the initial state consists of thephoton with wave vector Qo and polarization e,ro(Qo), the soliton moving withthe velocity v, the new phonon and the electronic excitation, while the final stateconsists of the photon with the wave vector Q and polarization e^Q), the excitonand a certain number of ordinary phonons and electronic excitations. In the above,\0)Q and |0)e are the vacuum states of the photons and the electronic excitationstates, respectively. According to the quantum mechanical perturbation theory, theprobability of transition from the initial state to the final state in the present casecan be determined by

|* (* . . . -<J. . ) - | [£EP<->KJ

f dh f * dt2 12 dh{fk\Hm,{h)Hmt{t2)HXnt{t3)\m) | , (9.24)

J—oo J—oo J—oo 1

where

Hint(t) = ei"°</fitfint(*)e-i"°t/'\

Ho = H + HQ + He - H ^ kvD+nDkn - h ^(Q • ^)C+aCQcr.kn Qo

However, we are more interested in the long-time behavior of dW/dt. By straight-


forward calculation, the final transition probability can be obtained approximately,

.. dW TTV2 jVl ( 7 \90

AS, -* w i r t ^TTA^ ( 2 ^ ) (9-25)

x {2[(Afcs(0) - W)2 + (%)2][v/(Afcs(0) - W*) + (h-yy - V(A*.(0) - W2)2]}~V2

Y^ (Qoz ~ Qz)Uaono(Qo)[Xnno(Qoz - Qz) + Xnno(QOz - Qz)]U*n{Q)â [SnoiQoz) ~ fiuQn(en(-Qz) + huQ)]

,(n n m (n , Xnn0 {Qz -Qoz)+ Xnno (Qz - QOz)U;n(Q) 2

HQz - QozWîQo) {£no{_Qoz) + tu,Q<7)i£n{-Qz) + fu,Q)

xL + tanh2 t ^ y S - (QOz - Qz)r0\\

+ tanh2 U (J*f + (QOz - Qz)r0) \ ,

where

The cross-section of Raman scattering is closely related to the transition prob-ability, (9.25). This transition probability has a sharp peak when the frequencyof the scattered light is in the vicinity of the frequency of the incident light, UJQ0 .Naturally, we are of interest only in the cross-section of Raman scattering in sucha case which is

}_dN_ _ (fid _ V'2WQ0 dW _ ijujjoufyUjy/J (_J__\9°N ds duQdfl ~ 4TT2 dt * 16TT3/j,3Jh2C6 \2A2uQ)

x {2[(At.(o)-w)2 + (hj)2} [ v ^ l T W + W - V(Afcs(o)-wy]}"1/2

X V - V^ (QOz - Qz)U.0n0(Qo){Xnn0(Qoz ~ Qz) + Xnno(QOz - Qz)]U*an{Q)

k ™o l£n° ( g ° ^ " f^Qn^n(-Qz) ~ fUQ)}

, (Qz - Q0z)U,0n0(Qo)[Xnn0(Qz ~ Qoz) + Xnno(Qz - Q O z )^ n (Q) ] '[(eno(-Qoz) + hwQlT)(€n(-Qz) + tvujQ)}

x sech2 < -— sin 00 — (cos 60 cos 6 + sin On sin 8 cos ip) >

I 2/xc I uQo JJx cos 60 — (sin 80 sin 0 cos <p + cos 60 sin 6) , (9.26)

L w Q o Jwhere dCl = dOdip sin 6 is the element of solid angle in the direction of propagationof the scattered light, #0 is the angle between the incident light Qn and the 2-axis,9 is the angle between Qo and the scattered light Q, tp is the angle between the


projection of the vector Q on to the zy-plane and the X-axis. Obviously, the cross-section of the Raman scattering is directly related to the size of the energy gap,the direction and intensity of incident light, the dipole moment and its orientationcorresponding to the transition from the ground state to the resonance electroniclevel. The distribution and intensity of the scattered light are different for differentfrequency of the incident light. If the frequency of the incident light is in thefollowing region,

0<huQo <fj,2J-en{0),

or

fi2J + en(0) <huQo <2en(0),

then (9.26) represents a resonant Raman scattering. The inequalities ensure notonly the smoothness of the cross-section as a function of CJQ0 but also the appli-cability of the above perturbation theory to the calculation of the cross-section.However, when the frequency of the incident light satisfies /j? J <gi HLJ <g. en(0) forall n, then (9.26) represents a nonresonant Raman scattering. In such a case, partof the energy of the incident light is converted into the excitation energy of theexciton-soliton. Thus, the angular-dependent part in (9.26) may be approximatelytreated as a constant.

Detailed examination of (9.26) show that the Raman scattering has the followingproperties. (1) When the frequencies of the scattered light is close to that of theincident light, the scattering cross-section is the largest. (2) For a fixed direction ofthe scattered light, the cross-section and the shape of the spectral line are mainlydetermined by the factor,

{2[(A*,(o) - wy + (h7y}W(Aks(o)-wy + (h7y - (At,(o) - w)\yi*'

regardless it is a resonant or a nonresonant Raman scattering. (3) In general, thescattering indicatrices (angular dependence of the scattering cross-section) are dif-ferent for the resonant scattering and nonresonant scattering. For the former thescattering indicatrix depends strongly on the orientation of the dipole moment ofthe transition from the ground state to the resonance electronic levels. For thelatter, when the frequencies of the scattered light UIQ are close to that of the Ra-man peak, its scattering indicatrix is independent of the coupling strength, and hasa simple form. (4) The intensity of the scattered light reaches a maximum whenthe frequency LJQ approaches U>Q0 + 2 J - G2/48J — e0. This maximum scatteringintensity is associated with the transition of the system from the soliton state whichis accompanied by lattice deformation, to the delocalized exciton state without lat-tice deformation. Therefore, the soliton becomes unstable as the incident light getsscattered. These characteristics of Raman scattering and the asymmetry exhibitedin the spectral line shape (sharper on the shor wave side and smoother on the long


wave side) which is connected to the corresponding energy dependence of the den-sity of the delocalized states, allow us to confirm the existence of soliton state ofexcitons in the organic molecular crystal ACN.

9.3 Infrared Absorption of Exciton-Solitons in Molecular Crystals

In the light of Hamilton theory in the nonlinear quantum mechanics described inChapter 3, Pang expressed the effective Hamiltonian corresponding to (9.13) as

HeH = hI G J r ° K | 2 + \ e ^ - i G M 4 ) d x - <9-27)The corresponding descrete form is

tfeff = \ $>»¥>;+! + <pn-i<P*n) + £ (\e\<pn\2 - \G\vnA • (9.28)

In earlier discussion, we considered only properties of exciton-soliton which isformed through coupling of amide-I oscillators (excitons) with longitudinal acousticphonons. To explain the temperature dependence of intensity of the infrared ab-sorption of the exciton-soliton, one has to take the interaction between the solitonand optical phonons in ACN into account. It is believed that this phenomenon isrelated to the zero-phonon line spectrum arising from random thermal modulationsof the exciton-solitons by the optical phonons. In this case, the Hamiltonian of thissystem can be written as

H = HeS + Hop + H' (9.29)

where Hes is the Hamiltonian of the effective amide-I oscillator system given by(9.28), which satisfies Hefc\m) — huj(m)\m). Here the eigenstate and the energyeigenvalue are collectively denoted by \m) and hui(m), respectively, in terms of aquantum number m. Hop is the Hamiltonian for the optical-phonon system whichis taken to be a set of harmonic oscillators,

#op = £MQn) = E [ - ^ ( | ^ ) + \M*UIQ^ , (9.30)

with

hn(Qn)$Kn (Qn) = fan U n + \ J $«„ (Qn) = ^n(«n)$ K n (Qn), K« = 0, 1, 2, • • • ,

W(JV) = 5 > n ( « n ) ,n

where hn, Qn and u>n are the Hamiltonian, the normal coordinate and the eigen-frequency of the nth harmonic oscillator, respectively, in which M* is the effectivemass, and $Kn(Qn) is an eigenfunction of hn characterized by the integer Kn. H'


in (9.29) describes the interaction between the amide-I oscillator system and theoptical phonon system which we take to be linear in the Q'ns, i.e.,

#' = £M¥v)Qn<- (9.31)n'

We can find that

Vn,(¥>„,) = ~ VXnll<p2n,e(k,itfe**-' sin(£ • un,,), (9.32)

where e(k, 77) is a polarization vector associated with the wave vector k and branchr) (r) = 1,2,3, two transverse and one longitudinal), An» is an interaction constantin which the index n" indicates the range of the oscillator-phonon interaction.

In order to study the properties of the entire system given by (9.29), we usean adiabatic approximation, i.e., the phonon state depends adiabatically on theamide-I oscillator state. This can be justified on the ground that the characteristicfrequency of about 1650 cm"1 of the amide-I oscillator system is much higher thanthe maximum frequency of the phonon system which is about 200 cm"1. In orderto obtain the adiabatic potential Um associated with the state \m) of the amide-Ioscillator system, we neglect the change in wave function of the amide-I oscillatorsystem caused by the perturbation H'. Thus the effective Hamiltonian #eff of thissystem for the state \m) can be written in terms of a set of the Hamiltonians of thedisplaced harmonic oscillators of the phonon system

HeS(m) = M"0 + £ \M*ul,Ql, - £ ^-^~ + 5>n'(m)Qn,

h2 d2

= Um - ! £ U^dQ2! = J2hn'&"' - Gon'(m)] + const., (9.33)

where

qn,{m) = {m\Vn,(m)\m), QOn>(m) = ~ | ^ r -

Using this Hamiltonian, we can find the infrared absorption coefficient arisingfrom the soliton motion. In accordance with the first order Born approximationin quantum radiation and theory of absorption, the absorption coefficient per unitvolume of the amide-I oscillator system described by (9.33) can be expressed as

B/n'M = ^ r / d f e " i U " (Y,(mn,\P(t)\mf)(mf\P\mn.)

I]£r(/Sn lX1inB)m/)Je(lwW-u(' . ')M , (9.34)

n' k'n I T


where

pn\ _ eiHetlt/hpe-iHcttt/h

r(«n ' , n'n,, mn, m/) = / dC,n'\, [<«' - Con' W l * ^ , [<n' - Con'(mf)],J — oo

in which

_ /M*ayCn' — Qn'\j r ,

In the above equations, mn/ («;„<) and mf{K'n,) with u>(Nn) = ^n/ W r i ' (K«') an(û(Nf) = X)n'

wn'(«n') a r e the quantum numbers denoting the initial and the finalstates of the amide-I oscillator system (the nth harmonic oscillator), respectively.The quantities, c, P, e' are the speed of light, the effective dipole moment of theamide-I oscillator and the refractive index, respectively, Vo is the volume of the totalsystem, (• • • )T denotes the average over all initial states in the canonical equilibrium.Following procedure of Takeno, we get

1 / 1\1 - o \Kn' + o ôn' (mn, mf), for K'n, = Kn>,L V L)

T(K > K! m mA-\ V^~o—Con'(mn,m/), for n'n, = «„/ + 1,

L \iini,Kni,mn,mf) — \ V 2

-W-^-Con'(mn,m/), for «'„, = «„/ - 1,

.0, for Kn, ^ Kn>,Kn: ± 1 ,

and

nEr^"''<"m«'m/)e i K <' )"w ( K n' ) i t - i i[ i + z"'(*'m"'m/)]'»' «^, ra'

where

Zn,{t,mn,mf) = -Con'(mn,m/) [(«„, + l je-^- '* + Kn-e-ia;»'* - 2«;n. + l] ,

C0n'(TOn,"l/) = Con'(mn) - Con'(m/) < 1.

In accordance with the above approximation procedure, the thermal average inthe integrand of (9.34) can be expressed separately for the amide-I oscillator stateand the phonon state, which are denoted by the angular bracket with subscript sT


and aT, respectively. Taking first the average over the phonon state, we get

lY[[l + Znl(t,mnl,mf)}\ =(expA\'£iZn,(t,mn,mf)\ (9.35)\ n' / aT \ In' \ I aT

= expl (expA y^Zn,(t,mn,ms) - l \ \ ^ exp / ^ Z ; ( i , m n , m / ) )[\ Ln' i /aT) L\n> laT.

where the symbol exp^(^n , Zn>) is a leveled exponential function that levels offthe product ZniS by neglecting terms which contain higher orders than unity inany of the Zni s, and the superscript (D) attached to the bracket symbol indicatesa cumulative average, as proposed by Kabo. Using (9.34), equation (9.35) can bewritten as

B, ,, s - 4™ f°° dte-i"t [Sp -Zo(T,mn>mf) -Zi(T,t',rnn,m,)\

x(mn\P(t)\mf)(mf\P\mn)sT>

Z0(T,mn, mf) = ^2((%n,(mn, mf) (Kn> + - ) ) ,„/ \ \ ZS I aT

Z1(T,t,mum}) ^ ( ^ K . m , ) [fa +l)eiu'lt + W™*1*])^.n'

The function exp[Zi (T,t,mn,m,f)] can be expanded as

exp[Zl{T,t,mn,mf)] = l + Z1{T,t,mn,mf) + -Zf(T,t,mn,mf) + --- .

The first, the second and the third terms on the right-hand side are contributions tothe zero, one and two-phonon processes. In the general case, one and multi-phononprocesses give weak side-band effects. Therefore, we may pay attention exclusivelyto the zero-phonon process and the main absorption line. Then we get

xlê-z°(T>m«>m>Hmn\P{t)\mf)(m}\P\mn)\ . (9.36)

In order to determine Zo(T, mn, mf), we have to find out Con(win> TUf). Evaluat-ing the matrix element of H' in (9.31) with respect to \m) for low frequency phononsand in the adiabatic approximation, £on' is separated into two parts, phonon andsoliton,

Con'(mn,mf) S ^nki2)(mn,mf), (9.37)


where

CHmn,mf) = - ^ E dmn\QiQ*i + Q*iQi\mni) - (mf\QiQ* + Q*Qi\mf)).

Here

Fn, = V v A eCn') (^-) - jL=, un. S iwfc,

uOn' is the speed of the nth phonon mode. Since the n'-dependence of e(n') andvOni comes mainly from ojn>, we can neglect the n'-dependence of Fn> and rewrite itas F. Inserting the above formula into (9.36), and using the Debye's approximationfor the frequency spectrum of phonons, Pang finally obtained

Z0(T, mn, mf) = Z0(T) = [<(2>]' -L- ^ , ^ + | ) ) T

« [c i 2 ) ] 2 ^ 2 E^; (^ ' )^ + - (9-38)

where C and A, of which the explicit expressions are omitted here, are constants,and 6 is the Debye temperature of the ACN. If T «; 6, we have Z0(T) = C + XT2.

Until now, no specification has been given for the quantum number m. It maybe divided into two parts, one being associated with the bound states or the solitonstates of the microscopic particle, the other with continuum or scattering statesof excitons, denoted by ms and me, respectively. Correspondingly, the absorptioncoefficient can be divided into the soliton absorption line J3S/(W) and the excitonabsorption line Bex(u), appearing at about 1650 cm"1 and 1665 cm"1, respectively,i.e.,

Bn'f(w) - Bgf[y,) +BeaE(w).

Prom (9.37) and (9.38), the following relation can be derived

(ms\QnlQ*n, +Q*n,Qn-\ms) » {me\Qn.Q*n, +Q*n,Qn.\me).

Thus we have,

Cm(msn,ms/) » Con(men,mef),

or

Z0(T,msn,msf) » Zo(T,men,mef).

Since the mean square displacement of the exciton system associated with thestate |me) is a sum of a large number of oscillatory functions with respect to the


space variable n', the temperature dependence of Bei(u) is very weak as comparedto that ofBsf(wy When the soliton density is so small that their mutual interactionsare negligible, we can consider only a single excited state corresponding to the firstexcited state of single-soliton states. Pang then approximately obtained

B«M = -££- f°° dte~^(Px{t)Pi)m,Tl (9.39)c en VQ j _ o o

t cnvo J—oo

where

* (o) M = ^ w*Ls V ' •n ( W 2 - C J 2 ) 2 + w*w2'

and Ns and Pi are the soliton number and an effective dipole moment associatedwith a single soliton state, respectively, u>3 is the eigenfrequency of the soliton, Ws

and A are, respectively, a natural width of the single-soliton absorption line and aconstant. Since the soliton energy is much higher than that of phonons, Bs{u)does not depend on temperature in the above approximation.

From (9.39), we see that the soliton absorption line spectrum Bsf(u) is a productof the phonon-free soliton absorption line spectrum Bs (w) and the Debye-Wallerfactor exp[—(c + aT2)]. This indicates that the intensity of infrared absorption dueto the nonlinear excitation of the excitons in ACN depends on the temperatureexponentially, which is consistent with results obtained by Alexander and Krum-bansl, i.e., exp(—/JT2), using a polaron model. This temperature dependence ofthe infrared absorption line at about 1650 cm"1 is analogous to that of localized orresonant harmonic phonon modes due to impurities in alkali halide crystals, wherethe impurity-induced infrared absorption spectra have a characteristic temperaturedependence in the form of exp(—aT2) (a = constant) at low temperatures. Al-though the physical origins of the localized modes are different in these two cases,the Debye-Waller factor results from the same origin, i.e., random thermal modula-tions of low frequency optical phonons. Furthermore, the temperature dependenceof the exciton absorption line, if any, is quite different from that of the soliton inthe absence of the Debye-Waller factor in Bex(uj). If we identify Bsf(w) and Bex^)with the absorption spectra of the ACN at about 1650 cm"1 and 1665 cm"1, re-spectively, which were observed experimentally, the experimental results in ACNcan be explained. Therefore, it is firmly believed that soliton can exist in ACN andthe above theory gives a correct description of it.

Pang et al. used again the exciton-soliton model and other methods for theinteraction of the amide-I vibrational quanta (exciton) with optical phonons tostudy infrared absorption at finite temperature. They used an improved theory forcolor centers in solid state and the nonlinear quantum perturbation method. Theirnumerical result for the exponential temperature dependence of the intensity of the


anomalous band at 1650 cm"1 is basically consistent with the experimental data.

9.4 Finite Temperature Excitonic Mossbauer Effect

From the above discussion, we can see that there can be subsonic soliton {v < vo)in the system if linear harmonic vibration of the molecular chains is taken intoconsideration. The subsonic soliton is formed by self-trapping of excitons throughinteraction with localized deformation of lattice. In other words, the lattice mustbe distorted for the soliton to occur. This means that positions of atoms in thelattice will shift. Thus, the states of atoms will be changed, and the nuclei at thelattice sites will be excited or activated in such a case. 7-quantum emission of theactive nuclei would then follow. If the 7-quantum is exactly equal to the molecularexcitation energy, i.e., when the molecular chain does not change its state in thecourse of emission, an observable Mossbauer effect will occur in the system. In fact,when an active nucleus emits a 7-ray, the transition energy, AEnm, in principle,may be distributed between the 7-photon, the nucleus that emitted the photon,and the chain as whole, and eventually between vibrations of the chains (in the casebeing considered, instead of pure phonon vibrations, we have localized self-trappedsolitonic state which is a result of the exciton-phonon interaction). The energyneeded for a nucleus to leave its site in the chain is at least 10 eV. But the recoilenergy which a molecular chain can receive as a whole is small (N > 1) and doesnot exceed several tenths of an electronvolt. As a result, an atom whose nucleus hasemitted a 7-ray cannot change its position in the lattice, so that it may be neglected.Thus the transition energy can only be distributed between the 7-quantum and thesolitons. A Mossbauer transition occurs if the solitonic state of the chain is notchanged, and the 7-photon receives the entire energy of the transition. Therefore,solitonic excitation and motion in acetanilide could cause changes in Mossbauereffect, compared to the case in which pure phonon modes spread over the wholechain. In the latter, the lattice remains unchanged after the 7-quantum emissionof an active nucleus. We can thus prove the existence of soliton by experimentallyobserving the changes in Mossbauer effect in ACN.

The purpose of this section is to discuss the properties and changes of Mossbauereffect arising from this mechanism and to calculate the corresponding 7-radiativeMossbauer transition probability by means of the above theory, (9.1) - (9.6). Thisdiscussion is also useful in understanding and clarifying the natures of the solitonin such systems and in facilitating development of soliton experiments. Since thedomain of the lattice where the soliton is present is subject to a deformation whichtravels along the molecular chain with a velocity v < vo- In the context of ouranalysis, the molecular crystal is in equilibrium with a thermostat having T ^ 0,so that we can use a model that couples the soliton with thermal phonons.

For the system with a finite temperature, the collective excitations and motionsof the excitons and phonons will be changed due to the thermal effect. A straight-


forward result is the thermal excitation of the phonons. Therefore, t/(t)|O)p/i in(9.5) must be replaced by the following

K) = U+\u) = exp { 5>n,(i)a+ - <(t)a,]l IJ -^(4)<\Q)ph, (9.40)I qn ) q V ?'

where

, vvform a complete phononic set which represents the elementary excitation of singlephonon due to finite temperature T ^ O K . Using (9.4) - (9.7) and (9.40), we canobtain the equations of motion for the exciton and phonon in the system. However,because the molecular crystal being studied is in contact with a thermostat at finitetemperature T ^ O K , similar to the calculation of the expectation value of (9.7),we can calculate the thermal mean values of these quantities using

Y = (Y) = Tr{Pvv)phY = *£(v\p\v)ph($\Y\$), (9-41)

V

where the density matrix, Y^V(V\P\V) IS giv e n by

, I M ( x (v\exp(-Hph/KBT)\v)

= W\exV[-Zq^q(atao)/KBT]W)EM^M-Eq^q(ataq)/KBT]\uy { "

Here the diagonal matrix elements of the Hamiltonian are

<*|ff|*> = {*\Hea + Hmtl*) + (av\Hph\av).

Making use of the method discussed earlier and the following relationships

(av\{aq + a+)\av) = ~[anq(t) + a*n_q(t)]; (av\a+aq\av) = (vq + \an-q\2);

Y,exp ( - - ~ ; ) H e x p « , a g ) e x p ( - a n , a + ) H = {vq + l)exp[-|an?|2(i/, + 1)];

vq = exp ( —*= - 1) ; exp(-Wn,n±1) = (ang|Q;n±ia); (9.43)

Wnn±1 = exp i ^2 \a*n±qanq - - ( | a n + i 9 | 2 + |an?|2) \

I q L ^ )

f" (-lTK+^pî


we get, after some tedious calculations, the following equations of motion for theexciton and phonon in the molecular crystals at finite temperature

&u ,a2u_ftro(xi + x2)aM2 , „ , .dt2 °dx* - 2Mu0 dx ' [ '

ih9-^ = [so - 2Jfl(T,,)MM) - JrlB(T,q)d-^>

+ ^+X2)rodu(x,t)

u>o ox

Thus the motion of the exciton in the nonlinear quantum systems at finite temper-ature still satisfies the following nonlinear Schrodinger equation

*&§& = [so - 2JB(T,q)Mx,t) - JrlB(T,q)9-^

+G\<p(x,t)\2<p(x,t). (9.46)

Its soliton solution is

tpfat) = A/^Psech \^l(x - a* - vt)] e*(»-"«."«0V 2 [ ro J

= A{T) sech \^-{x -xo- vt)] e<(S«-""»i«). (9.47)L ro J

It can then be obtained that

"<*'" = 2 S r ^ l a " h [ ^ ( l - * ° - 4 (9'48)and

r=tfrl(Xl+X2)2

ulMvlil-s^y

MT) = , m(T)=,nexp[^], k= * 2 > ,

a4j =JBiT^ « b ) H "s2)" h^¥^F^]x

^ [ ^ t + ^ ^ ^ 4 22 2 (9-49)

eo(T) = £0 - ~eW" =e0- 2JB(T, q), husol = eo(T) + ff[* " / i ' ( T ) ]>

TTirQ Z771

, 2 ^ ^ forfcBr»^? !

I 2 + "fef ' f°r fcBT « ^"(9.47) represents a subsonic soliton (v < v0, s = v/vo < 1). It can be shown thatthis soliton is stable because its rest energy is lower than the bottom of the exciton

(9.45)


band at about l/3J/x2(T). The presence of the energy gap in the spectrum of theexcited state of the molecular chain is one of the reasons for the better stability ofthe soliton. This soliton state describes a quasiparticle consisting of the exciton andlattice deformation. It thus already includes interactions with the acoustic phonons.Therefore, destruction of the soliton requires the removal of the lattice deformation.That is, in order to split the soliton into an exciton and an undeformed lattice it isnecessary to expend sufficient energy to allow a transition from the soliton state toa free exciton state. The transition probability to a lattice state with no distortionis proportional to the Prank-Condon factor which is negligibly small for a longmolecular chains. Otherwise, since solitons always move with velocity less than thevelocity of the longitudinal sound in the chain, they do not emit phonons. In otherwords, their kinetic energy cannot be transformed into energy of thermal motion.The soliton is thus stable.

Prom the above discussion, we see that when the subsonic soliton is formed byself-trapping of the exciton through interaction with the acoustic phonon in themolecular crystals at finite temperature T ^ OK, the lattice molecules or atomswill have a displacement given by (9.48), i.e., the part of the crystal lattice wherethe soliton is present is subjected to a deformation. The states and positions ofatoms and nuclei in this region of the molecular chains will be changed. The activenuclei may emit 7-photons in such a case and the Mossbauer effect could occurin the crystals, as mentioned above. In the following, we evaluate the probabilityof the 7-radiative Mossbauer transition resulting from the active nuclei which isassumed to be located at node no in the molecular chain.

Following the approach of Gashi et al. and Ivic et al., the interaction potentialdue to the emission mentioned above can be expressed as

V;nt = A'Tr(xL,PL,aL)T(P,uno), (9.50)

where ~K{XI,,PL,GL) is an operator which describes the internal state of the activenucleus located at node no and depends on its coordinate XL, momentum PL andspin 07,, F(P,uno) is an operator related to the emitted 7-photon, P is the mo-mentum of the emitted 7-quantum, una is the position vector of the center of massof lattice node. Obviously, um = noro + Rno, in the emitting process, where Rno

denotes the recoil displacement of the emitted nucleus from its node. In general,T(P, uno) can be represented in term of a periodical potential or a plane wave, i.e.,T{P,uno) = dexp{iP • uno/h). Thus (9.50) becomes

Vint = Av{xL, PL, <jL)eiPnafa>heiPR»°lh, (9.51)

where d and A' are constants related to the characteristics of the molecular chainand nucleus, and we can set A = dA'. In accordance with the quantum theory ofradiation, the transition matrix element due to the change of the state in such a


case can be written as

Tn^m = (m9m\ViBt\n$n) = i($™|etf-*.o/VAnof t/*|$n)-

(m\ir(xL,PL,aL)\n). (9.52)

Since TT{XL,PL,(TL) depends only on the fixed numbers of internal degrees of free-dom of the nucleus, it is unnecessary to know its explicit form if we are onlyinterested in the emission of the molecular crystals. Thus the matrix element,{m\n(xL,PL,crL)\n), can be treated as a constant value when the change of innerstate of the nucleus is small, and can be absorbed into A. Then the relative transi-tion probability can be determined when the same soliton is found in the molecularcrystals before and after the emission, and is given by

rn_>m = i(*|e"M»o/»|*)e"J-»o'V*. (9.53)

|$) in (9.5) and (9.40) represents the amplitude of the soliton. Following the pro-cedures of Ivic et al. and Gashi et al., Pang inserted (9.5), (9.40), and

R — V^ . r (n 4- n+ \Pinr°1

K"o - 2 , y 2NMcoqeq{aq + ~«)e

into (9.53), and obtained

Tn^m = i 4 $ > + \<pn\2)TuV{n)ein°F°-plh,

n

where

*•„(»,=n«p U<~, - v . + m t <-'>-iy;w>', (,54)with

^ = i]f^q(p-êinoroq and ^ = - -

Since the molecular crystal being considered is in contact with a thermostat attemperature T ^ 0 K, we should take the thermal average over the phonon statesfor the matrix element (9.53) using the density matrix (9.42). Equation (9.53) thenbecomes

(Tn-+m)ph = ^ E ((! + \Vn?)Tvv{n)ein^p/h) hn

= I J ( 1 + Wn\2)(Tvv{n))phein°f°-pl\n

{Tvv{n))ph = Y,P"uTVu{n) = (Tsol)phexp ( - £ {^£- Lq + | U ,v \ q 9 \ / J


where {Tsoi)vh contains the matrix element for the solitonic part. Inserting anq

and a*_?, which are related to the soliton solution (9.47), obtained from previousequations of motion into (9.54), the soliton part of the matrix element, (Tsoi)ph,can be written as

(T l) ., - CXP ( 1 V ( X l + X2){P • e"")gr°^"|2 ciarn(nn-n) }{lsol)ph~exp<^ N ^ W o M t ; 2 ( 1 _ s 2 ) M 2 e j -

After tedious calculations, Pang finally obtained

where

2Mv20(l - s*)NiJo ^ q)-

The Mossbauer transition probability is given by

/ = <W»-™.> = KT^)*? = i 2 a p | - ^ ^ ^ ( - . + | ) |

where

From (9.55), we can obtain the following properties of the Mossbauer effect.(1) The transition probability is a product of the subsonic soliton part related to

the temperature and the phononic part. Generally speaking, if the molecular chainis populated exclusively by solitons, the Mossbauer effect is different from that inthe case where pure phonon modes are spread over the whole chain. The latterleads to a probability with a factor

f ^ (P • eq)2 . ( huq \ \

(9.55)


in (9.55). In the former the soliton transition probability is manifested by thefactor in the curly brackets in (9.55), 3?{- • •}, which is to be multiplied by thefactor corresponding to the pure phonon states.

(2) The transition probability given above depends on temperature T. Temper-ature influences the probability through the following factors, the amplitude of thesoliton, A(T), /x(T) and

_ 1Vq ~ exv(hwq/KBT) - 1

in the case of pure phonon states. Knowledge of these parameters allows us toestimate the value of the transition probability. In general, although an exactcalculation for the probability is very difficult, an approximate numerical estimationof (9.55) is possible by using generally accepted values for the parameters givenearlier. In Fig. 9.3, the average of the relative transition probability f/A2 is shownas a function of temperature T a t u = 0 o r s = 0. Here A = 2 .25xlO~ n mis usedfor the average wave-length of 7-photon.

Fig. 9.3 The average relative transition probability f/A2 is shown as a function of temperatureT for s = 0 and A = 2.25 X lO"11 m.

From this figure, we see that the transition probability overall decreases expo-nentially as the temperature increases, except at very low temperature. The effectof temperature on the probability comes from the phononic part. It can be shownthat f/A2 is 3-3.5% at T = 300 K and s = 0. Therefore, the transition probabilityis very small at 300 K.

The average value of the soliton part of the transition probability in (9.55),

A _ a [1 + 1 ^ csch (i*£°\ einvt]2

A2-R[1+2»(T)CSChW(T))e J 'can be evaluated at high and low temperature limits using the parameters givenin Section 9.2, and the results are fs/A

2 = 1.1738 at T = 300 K and fs/A2 =

1.1857 at T = 10 K, respectively. As far as the influence of temperature on the


transition probability is concerned, the value of the soliton part at low temperatureis larger than that at high temperature, but the difference between them is verysmall in the long-wave length approximation. This demonstrates that the effect ofthe temperature on the soliton part of the transition probability of the Mossbauereffect is also very small. Furthermore, it can be shown that the contribution of thesoliton state to the transition probability is larger than that of the pure phononmodes in this model, particularly at high temperature.

(3) The transition probability depends closely on the strength of the couplingcoefficient, (xi + X2) and /i(0). From (9.55), we see that f/A2 decreases withincreasing (xi + X2) and fi(0) at s = 0, T = 300 K and A = 2.25 x 10"11 m,respectively, and that the transition probability is pure phononic, when (xi +X2) = 0or fi(0) = 0. This shows clearly that the Mossbauer effect and the correspondingtransition probability result from the motions of the solitons and thermal phononsand the interaction between them in the molecular crystals.

(4) The transition probability decreases with increasing soliton velocity, v, sincethe parameter /z(T) is related to v. When the velocity of the soliton approachesthe sound speed vo, the Mossbauer effect transition probability decreases obviously.We can similarly calculate the effect of temperature on the transition probabilityof the soliton part at high and low temperatures and at s = 0.5. The results are1.0399 at T = 300 K and 1.05126 at T = 10 K, respectively. This shows clearlythat the transition probability of Mossbauer effect decreases as the velocity of thesoliton increases, compared to that in the case of s — 0, but its contributions to theprobability are still larger than that in the pure phonon state in such a case.

(5) In general, the magnitude of the transition probability depends on the inher-ent characteristics of the molecular crystals, (M,ro,vo, N, LJQ and ui), the proper-ties of the quasiparticles generated in collective excitation, (w9, q, v, and m), andthe environment condition of the molecular crystals, i.e., temperature. Therefore,the transition probability of different molecular crystals would be different.

Pang also calculated Mossbauer transition probability when the anharmonicvibrations of the molecules (9.18) arising from the temperature is taken into con-sideration in the Hamiltonian. In such a case, the exciton becomes a supersonicsoliton. Apply the above method, the supersonic exciton-soliton can be approxi-mately represented by

r . 2 « - 1 / 2 "I

^(M) = A(T)sechU(^-±j {x_vt) SR*-^\ (9.56)

where v > v0, A(T) is the amplitude of the supersonic soliton which depends on thetemperature. Thus the corresponding Mossbauer transition probability is obviouslydifferent from (9.55). The features of the Mossbauer effect caused by the nonlinearexcitation of the excitons are discussed in details in Pang 1987, 1989a, 1990, 1993e,1999, 2001c.


9.5 Nonlinear Excitation of Excitons in Protein

We now move on to discuss properties of nonlinear excitations and motions ofexcitons in protein molecules on the basis of an improved Davydov theory, proposedby Pang et al. (Pang 1987, 1989a, 1990, 1992b, 1993b, 1993e-h, 1999, 2001c, 2001e,Pang and Chen 2001).

As it is known, many biological processes are associated with bioenergy transportthrough protein molecules, where energy is released by the hydrolysis of adenosinetriphosphate (ATP). Understanding the mechanism of bioenergy transport in suchsystems has been a long-standing problem that remains of great interest today.As an alternative to electronic mechanisms, one can assume that the energy isstored as vibrational energy in the C=O stretching mode (amide-I) of a polypeptidechain. Based on the idea of Davydov (see Section 5.6), one can take into accountthe coupling between the amide-I vibrational quantum (exciton) and the acousticphonon in the amino acid lattice. Through the coupling, nonlinear interactionoccurs in the motion of exciton, which results in a self-trapped state of the exciton.The latter, together with lattice deformation of the amino acid molecules can travelover macroscopic distances along the molecular chains, retaining the wave shape,energy and momentum. In this way, bioenergy can be transported as a localizedexciton or soliton. This model of the bioenergy transport was first proposed byDavydov in the 1970s.

Davydov's idea yields a compelling picture for the mechanism of bioenergy trans-port in protein molecules and consequently has been the subject of a large numberof works (Bolterauer and Opper 1991, Borwn 1988 Brizhik and Davydov 1969,1983, Brown, et al. 1986, 1987, 1988, 1989, Christiansen 1990, Cottingham andSchweitzer 1989, Cruzeiro et al. 1985, Cruzeiro-Hansson 1992, 1993, 1994 Cruzeiro-Hansson and Scott 1994, Cruzeiro-Hansson and Takeno 1997, Davydov 1980, 1981,1982, 1991, Davydov and Kislukha 1973, 1977, Forner 1991,1992,1993,1994,1996,Forner and Ladik 1991, Hyman et al. 1981, Ivic 1998, Ivic and Brown 1989, Kerrand Lomdahl, 1989, 1991, Lawrence et al. 1986, Lomdahl and Kerr 1985, 1991,Macneil and Scott 1984, Mechtly and Shaw 1988, Motschman et al. 1989, Pang1986, Schweitzer 1992, Schweitzer Cottingham 1991, Scott 1982, 1983, 1984, Skrin-jar et al. 1984, 1988, Takeno 1984, 1985, 1986, Tekec et al. 1998, Wang et al. 1988,1989, 1991, Zekovic and Ivic 1999). Problems related to the Davydov model (seeSection 5.6), including the foundation and the accuracy of the theory, the quantumand classical properties, and the thermal stability and lifetimes of the Davydov soli-ton, have been extensively studied. However, considerable controversy has arisenregarding whether the Davydov soliton is sufficiently stable in the region of bio-logical temperature to provide a viable explanation for bioenergy transport. Manynumerical simulations have been based essentially on classical equations of motionand are subject to the criticism that they are likely to yield unreliable estimates forthe stability of the soliton since the dynamics of the soliton is not being determined


by the nonlinear Schrodinger equation. Simulations based on the |£>2) state, i.e.,|$(t)) in (5.70) in which B+(Bn) is the creation (annihilation) operator of excitonat site n, un is the displacement operator of the amino acid molecule at site n, Pn isits conjugate momentum operator, generally agree that the stability of the solitondecreases with increasing temperature and that the soliton is not sufficiently stablein the region of biological temperature (300 K). Since the dynamical equations usedin the simulations are not equivalent to the nonlinear Schrodinger equation, thestability of the soliton obtained by these numerical simulations is unreliable. Onthe other hand, simulations based on the |£>i) state, i.e.,

Wi) = 5>n(t)B+|0)e,exp I 52 [anq(t)a+ - a*nq(t)aq] 1 |0>,n [ q J

where |0) = |0)ea:|0)p/l, a^(aq) is the creation (annihilation) operator of the latticephonon, and anq(t) and Q*nq(t) are some undetermined functions, with Davydov'sthermal treatment, where the equations of motion are derived from a thermallyaveraged Hamiltonian, yielded a surprising result that stability of the soliton canbe enhanced with increasing temperature. Evidently, this conclusion is not reli-able because the Davydov procedure in which one constructs an equation of motionfor an average dynamical state from an average Hamiltonian, corresponding to theHamiltonian averaged over a thermal distribution of phonons, is inconsistent withstandard concepts of quantum-statistical mechanics in which a density matrix mustbe used to describe the system. There has been no exact fully quantum-mechanicaltreatment for the numerical simulation of the Davydov soliton. However, for itsthermal equilibrium properties, a quantum Monte Carlo simulation had been car-ried out by Wang et al.. In this simulation, correlation characteristic of solitonlikequasiparticles occur only at low temperatures, T < 10 K, for widely accepted param-eter values. This is consistent at a qualitative level with the result of Cottinghamand Schweitzer. The latter is a straightforward quantum-mechanical perturbationcalculation. The lifetime of the Davydov soliton obtained by using this method istoo small (about 10~12 - 10~13 sec.) to be useful in biological processes. This indi-cates clearly that the Davydov solution is not a true wave function of the systems.A systematic study with different parameters, different types of disorder, differentthermalization schemes, different wave functions, and different associated dynamicsby Foner et al. lead to a very complicated picture for the Davydov model. Theseresults did not completely rule out Davydov's theory, but they did not eliminatethe possibility of another wave function and a more sophisticated Hamiltonian ofthe system which might result in longer soliton lifetimes and good thermal stability.

Indeed, the question of lifetime of soliton in protein molecules is twofold. In theLangevin dynamics, the problem consists of uncontrolled effects arising from thesemiclassical approximation. In quantum treatments, the lacking of an exact wavefunction for the soliton has been the issue. The exact wave function of the fullyquantum mechanical Davydov model has not been known up to now. Different wave


functions have been used to describe the states of fully quantum-mechanical sys-tems. Although some of these wave functions lead to exact quantum states and exactquantum dynamics for the J = 0 state, they share the same problem as the originalDavydov wave function, namely the degree of approximation included when 7 ^ 0is not known. Davydov's wave function thus has to be modified. It was thought thatthe soliton with a multiquantum state (n> 2), such as coherent states of Brown etal., multiquantum state of Kerr and Lomdahl, and Schweitzer et al., two-quantumstates of Cruzeiro-Hansson and Forner, would be thermally stable in the regimeof biological temperature and could provide a realistic mechanism for bioenergytransport in protein molecules. However, the assumption of the standard coherentstate is unsuitable or impossible for biological protein molecules because there areinnumerable particles in this state and the particle number cannot be conserved.The assumption of a multiquantum state (n > 2) along with a coherent state is alsoinconsistent with the fact that the energy released in ATP hydrolyses (about 0.43eV) can excite only two quanta of amide-I vibration. On the other hand, numer-ical result obtained using the two-quantum model by Forner reveals considerabledifferences from one-quantum dynamics, i.e., the soliton with a two-quantum stateis more stable than that with a one-quantum state. Cruzeiro-Hansson consideredthat Forner's two-quantum state is not exact in the semiclassical case. Therefore,he constructed an "exact" two-quantum state for the semiclassical Davydov system,given by

N

|V(*)>= £ <P™({«ihW,t)B+B+t\0)ex.n,m=l

However, further investigation by Pang et al. concluded that the Cruzeiro-Hamssonwave function does not represent exactly the two-quantum state. An energy releaseof about 0.43 eV in the ATP hydrolysis does not support the model.

On the basis of the work by Cruzeiro-Hansson, Forner, and others, Pang im-proved and extended the Davydov model by modifying simultaneously the Hamil-tonian and the wave function of the systems. He included the coupling interactionbetween acoustic phonons and the amide-I vibrational modes in the original Davy-dov Hamiltonian, and replaced the one-quantum exciton state in Davydov's wavefunction by a quasicoherent two-quantum state. This resulted in an equation of mo-tion and properties of the soliton completely different from those in the Davydovmodel. It is believed that this model is able to resolve the controversy concerningthe thermal stability and lifetime of soliton in protein molecules.

The Davydov model is indeed too simple, both in its wave function and Hamil-tonian. It simply cannot properly describe the basic properties of collective exci-tations in protein molecules. Davydov's theory was applied to protein moleculesas an exciton-soliton model in an one-dimensional molecular chain. Although themolecular structure of the a-helix protein is analogous to some molecular crys-tals such as ACN, the a-helix protein molecule consists of three peptide channels.


Many properties and functions of the protein molecules are completely differentfrom those of ACN. Protein molecules act as a kind of soft condensed matter aswell as bio-self-organization with active functions, for instance, self-assembly andself-renewing. The physical concepts of coherence, order, collective effects, and mu-tual correlation are very important in protein molecules compared with other moregeneral molecular systems.

There is an obvious asymmetry in the Davydov wave function given above sincethe phononic part is a coherent state while the excitonic part is only an excitationstate of a single particle. It is thus unreasonable for the same nonlinear interactiongenerated by the coupling between the excitons and phonons to produce differentstates for the phonon and exciton. Taking these into consideration, Pang modifiedthe Davydov wave function according to the following,

( \ 2"5>(i)Z?+) \0)ex

n /

x e x p | ~\ Yl i^{t)Pn - irn(t)un] I |0)pfc, (9.57)

where un and Pn are the displacement and momentum operators of the aminoacid molecule at site n, <p(t), 0n(t) = ($(t)|un|$(*)) and vrn(t) = ($(t)|Pn|$(t)>are three sets of unknown functions, A is a normalization constant. We assumethereafter A = 1 for convenience of calculation.

Since the resonant dipole-dipole interaction between the neighboring amide-Ivibrational quanta (excitons) in neighboring amino acids with electrical momentof about 0.8-3.5D has been taken into account in the Davydov Hamiltonian, wemay as well consider the changes of relative displacement of the neighboring aminoacids arising from this interaction. Therefore, it is reasonable to include the in-teraction term X2(un+i - un)(B^+1Bn + BmBn+\) in Davydov's Hamiltonian torepresent correlations between collective excitations in protein molecules. Althoughthe dipole-dipole interaction is small as compared to the energy of the amide-Ivibrational quantum, the resulting change in relative displacement of neighboringpeptide groups cannot be ignored due to the sensitive dependence of dipole-dipoleinteraction on the distance between amino acids in the protein molecules. Themodified Davydov's Hamiltonian given by Pang is as follows.

H = Hex + Hph + Hint (9.58)

= Y, [eoB+Bn - J(B+Bn+1 + BnB++1)} + £ U | + \w{un - un_An n L -I

+ £ [xiK+i - un^)B+Bn + X2K+1 - un)(B++1Bn + B+Bn+1)] ,n

where £Q = /ô = 1665 cm"1 is the energy of the exciton (the C=O stretching


mode), Xi a nd X2 are the nonlinear coupling constants which represent the mod-ulations of the on-site energy and resonant (or dipole-dipole) interaction energy,respectively, of excitons produced by molecular displacements, M is the mass of anamino acid molecule and w is the elastic constant of the protein molecular chains, Jis the dipole-dipole interaction energy between neighboring sites. This Hamiltonianhas better symmetry and describes correctly the mutual correlations between thecollective excitations in protein molecules.

Obviously, the exciton wave function given in (9.57) is not an excitation stateof a single particle, but rather a coherent state, because

\<p(t)) « exp J5>(*)2?+ - ¥>(*)'£»] j |0>e«.

More precisely, the new wave function is a truncated standard coherent stateand retains only three terms in the expansion of the latter in the case of small(Pn(t) [i.e., \<pn{t)\ < 1]. \<p{t)) is thus referred to as a quasi-coherent state.It is not an eigenstate of the particle number operator, N = ^2nB^Bn, sinceN\(p(t)) = 2\<p(t)) - [2 + Y,n <Pn(t)B+]\Q)ex- But the number of quanta in this stateis determinate, and there are two excitons i.e.,

N = <<^)|iv>(i)) = £ |<^n(i)|2 1 + J2 kn(t)|2] = 2,

n L n J

where

$>n( t ) l a = l, [Bn,B+] = Snm.n

Thus, |v(t)) indeed represents a coherent superposition of multiquantum states.Therefore, this wave function is different from the one-quantum state proposed byDavydov, the two-quanta states proposed by Forner and Cruzeiro-Hansson, thestandard coherent state proposed by Brown et al., and the multi-quantum statesproposed by Kerr et al. and Schweitzer et al. The wave function given in (9.57) notonly gives the coherent property of collective excitations of exctions and phononsdue to the nonlinear exciton-phonon interaction, which results in a symmetric wavefunction for the states of the system, but also agrees with the fact that the energyreleased in ATP hydrolysis (about 0.43 eV) is only sufficient to create two excitons.The number of excitons in the Hamiltonian (9.58) is thus conserved.

Using (9.57) and (9.58), and the Heisenberg equations for the expectation valuesof the operators, un and Pn, in |<J>(t))

i/»^<*(t)|«n|*(t)> = {^{t)\[un,H]^{t)),

ihjtmt)\Pn\$(t)) = (*{t)\[Pn,H]\*(t)), (9-59)


we can obtain the equation of motion for /3n(t)

Mpn = Cj(0n+1 - 20n + /?„_!) + 2Xl(\<Pn+l\2 ~ \<Pn-l\2)

+2X2K(Vn+l - fn-l) + V n « + l - < + 1 ) ] . (9.60)

For the equation of motion for <p(t), a basic assumption is that |$(t)) is a solutionof the time-dependent Schrodinger equation

ih^Mt)) = H\*(t))-

Using (9.57) and (9.58), we can, after some tedious calculations, get

ih~Qf ~ £ ° ^ " ~ J(<Pn+l ~ Vn-l) + Xl(/Wl - Pn-l)<Pn (9.61)

+X2(Pn+l -Pn)(v>n+1 ~ <pn-l) + ^ W(t) - - ^(Pm^m - TtmPm) Vn,L m

where

w(t) = (m\HPh\m) - E f^ + iw(& - ^-o2l + E ^ « -In the continuum approximation, Pang obtained from (9.60) and (9.61) that

i h ^ A = R(tMx,t) - J r ' ^ l ^ - GP\<p{x,t)\Mx,t), (9-62)and

where

i?(i) = £o - 2J + ^ \w(t) - \ ^(Pmirm ~ *mPm) ,L m

s = V/VQ, and VQ = royw/M is the sound speed in the protein molecular chains.Equation (9.62) is a standard nonlinear Schrodinger equation. It shows that theprotein molecules are nonlinear quantum systems, in which exciton becomes a soli-ton that has the properties of microscopic particle in nonlinear quantum systemsdescribed in Chapters 3-6. The soliton solution of (9.62) then is

ip(x,t) = Jit?-sech \^(x-xo-vt) expji IgjÔ* ~ xo) ~ Ev^ j , (9-64)

with

»P- W ( l - « V ' " ( I"* 2 ) ' A-£o~2J- (9-65)

(9.63)


Although the forms of above equation of motion and the corresponding solitonsolution, (9.62) - (9.65), are similar to those of the Davydov soliton (see Section 5.6),they are significantly different because the parameters in the equation of motion andtheir representations have obviously different meanings. Immediate results of thisnewly proposed model include an increase of the nonlinear interaction energy Gpwhich is given by

a corresponding increase in the amplitude of the soliton, and a decrease in its widthdue to an increase of fxp,

|_ \XiJ \XiJ J

compared to the Davydov soliton. Here

w ^ - T T ^ and GD = - ^w(l - s2) w{\ - s2)

are the corresponding values in the Davydov model. The localized feature of thesoliton is enhanced, and its stability against quantum fluctuation and thermal per-turbations increased considerably as compared with the Davydov soliton.

The energy of the soliton in the improved model becomes

E = (m\H\m) = 2 [(e<> - 2J) + ^ - *&j] = Eo + \Msolv2. (9.66)

The rest energy of the soliton is given by

Eo = 2(e0 - 2J) - 8 ( X gJ~J 2 ) 4 = £s° + W, (9.67)

where

w 2(Xi+X2)4

is the deformation energy of the lattice. The effective mass of the soliton is

M 9m • 8(xi+X2)4(9s2 + 2-3s4)Mmol = 2mex + 3 u ; 2 j ( 1_ s2 )3 t ; o2 • (9-68)

In such a case, the binding energy of the soliton can be obtained and is given by

P 8(xi + X2)4

EBP = 3JW2—

= 8EBD[l + 4f^+6f-V+4f-V+f-Vl- 0-69)[ \Xi/ \XiJ \XiJ \XiJ J


It is obvious that EBP is larger than that of the Davydov soliton, EBD = —xi/SJ™2-The binding energy of the soliton given by the improved model is about sev-

eral ten times larger than that of the Davydov soliton. The increase in the solitonbinding energy is due to the two-exciton nature of the improved model and theadditional interaction term £nz2(/ jn+i - nn)(B++1Bn + B+Bn+i) in the Hamil-tonian, (9.58). However, we can see from (9.69) that the two-exciton nature playsthe main role in the increase of the binding energy and the enhancement of thermalstability for the soliton, compared to the additional interaction term, X2 < Xi • Theincrease in binding energy results in significant changes of properties of the soliton.To compare the various correlations, it is useful to consider them as functions of acomposite coupling parameter, such as that given by Young et al. and Scott, thatcan be written as

4n(7p = fc^2)!,where UID = \fwJM is the band edge for acoustic phonons (Debye frequency). IfAmap 3> 1, the coupling is said to be strong, and if 4n<rp <fC 1, it is said to be weak.Using the following widely accepted values for various parameters for the a-helixprotein molecule, i.e., J = 1.55 x 10~22 J, w = 13 - 19.5 N/m, M = (11.7 - 19.1) x1(T25 kg, xi = 62 x 10-12 N, x2 = (10 - 18) x 10~12 N, r0 = 4.5 x 1CT10 m, it canbe estimated that the coupling constant lies in the range of 4irap = 0.11 - 0.273.The corresponding range in the Davydov model is 4nap = 0.036 — 0.045. Therefore,the improved model is not a weakly coupled theory. Using the notation of Venzeland Fischer, Nagy et al, it is convenient to define another composite parameter7 = J/2HUD- In terms of these two composite parameters, Aitap and 7, the solitonbinding energy given by the improved model can be written as

EBP 8 / 4 7 T ( 7 P \ 2 I" 32 /47raP \ 2 l

~r = 3 \—) ' s = ex [ 1 + T {^r) \ • (9-70)

Using the above values for the various parameters, we can get 7 = 0.08. EBP/J

calculated using the above parameters is shown in Fig. 9.4 as a function of Aiujp.For the Davydov model, it can be shown that

EBD 1 /4TTCTZA2 , | \ 2 M v] A Xi

EBD/J of the Davydov model is also shown in Fig. 9.4 versus 4-KGD- From thisfigure, we can see that the difference between the soliton binding energies of the twomodels becomes larger with increasing 4na. Therefore, localization of the solitonis enhanced in the improved model due to the increased nonlinear interaction. Thesoliton binding energy increases due to the increase in exciton-phonon interactionin the improved model. This leads to the enhanced stability of the soliton againstquantum and thermal fluctuations.


Fig. 9.4 Soliton binding energy {Eg) obtained using the improved model and that of the Davydovmodel are shown as functions of the coupling constant, 4jra. The binding energy is given in termsof dipole-dipole interaction energy J.

As a matter of fact, the nonlinear interaction energy which is responsible for theformation of the soliton is

Gp = 8 ( a - J ) ) 2 w = 3-8xl0"21J'which is larger than the linear dispersion energy, J = 1.55 x 10~22 J. This meansthat the nonlinear interaction is so large that it suppresses the linear dispersioneffect in the equation of motion, leading to the exceptional stability of the soliton.On the other hand, the nonlinear interaction energy in the Davydov model is only

4v2

GD = z—^w = 1.18 x 1(T21 J,1 - sz

which is about three to four times smaller than Gp. The Davydov soliton is thus lessstable compared to the soliton in the improved model. Furthermore, the bindingenergy of the soliton in the improved model, EBP = (4.16 — 4.3) x 10~21 J in (9.66),is somewhat larger than the thermal perturbation energy, KBT — 4.13 x 10~21 J, at300 K and about four times larger than the Debye energy, KBQ = hujo — 1.2 x 10~21

J, (Here U>D is the Debye frequency). Therefore, transition of the soliton to adelocalized state can be suppressed by the large energy difference between the initial(solitonic) state and the final (delocalized exciton) state, which is very difficult tobe compensated by the energy of the absorbed phonon. The soliton is thus robustagainst quantum and thermal fluctuations. It has a large lifetime and good thermalstability in the biological temperature range. On the other hand, the binding energyof the Davydov soliton, EBD = Xi/3u>2 = 0.188 x 10~21 J, is about 23 times smallerthan that of the soliton in the improved model, about 22 times smaller than KBT,and about 6 times smaller than KBQ, respectively. Therefore, the Davydov solitonis easily destroyed by thermal perturbation and quantum transition effects, which


is why the Davydov soliton has a very short lifetime, and it is unstable at thebiological temperature of 300 K.

9.6 Thermal Stability and Lifetime of Exciton-Soliton at BiologicalTemperature

The thermal stability and lifetime of the exciton-soliton at 300 K in the proteinmolecules is crucial to the validity of the improved model. It is directly relatedto whether soliton can provide the mechanism for bioenergy transport in proteinmolecules. In this section, we will examine the thermal stability and lifetime ofexciton-soliton at biological temperature in more details.

Making use of (9.3), (9.58) can be written as

H = J2 [eoB+Bn - J {B+Bn+1 + B++lBn)] + £ huq (a+aq + £} (9.71)

+ ~m E [9MB+Bn + g2(q) {B+Bn+1 + B+Bn+1)] (aq + atqVnr°\

where

M=2^/^sinM; M=^^Lqîraq-i)• (9-72)

We now partially diagonalize the model Hamiltonian (9.71) and calculate the life-time of the soliton, (9.64), following procedure given earlier and results in (9.19) -(9.22).

Since we are interested in the case where a soliton is initially moving with avelocity v along the chains, it is convenient to carry out the analysis in a frame ofreference in which the soliton is at rest. We thus consider the Hamiltonian in thisrest frame of the soliton, H = H — vP, where P is the total momentum, and isgiven by

P = ^hq{a+aq-B+Bq),q

where

B+ = —!— Y^ eifinT°B+

In order to obtain a simple analytical expression, we make the usual continuum


approximation corresponding to (9.71). This gives

H = £ dx2 | (e 0 - 2J)<p+(x)<p(x) + H ^ ^

•j= E 2 ^ i ( Q ) + 292(q)} {atq + aq) jf dxeikx ip+{x)<p{x), (9.73)

where y>(a;) now represents the field operator corresponding to Bn in the continuumlimit (whereas it only indicated a numerical value previously). Here L = Nro,—IT < fcro < 7r, and uiq m y/w/Mro\q\, x — nro- Since the soliton excitation isconnected with the deformation of intermolecular spacing, it is necessary to take thisdeformation into account in transforming the phonons between the reference frames.Such a transformation can be realized by means of the following transformation onthe phonon operators

6* = a * ~ 7 w a 9 > b+q = a+q ~ 7Na*q' (974)

which describe phonons relative to a chain with a particular deformation. Here bq

(&+) is the annihilation (creation) operator of the new phonon. The vacuum stateof the new phonons is

which is a coherent phonon state, i.e., bq\0)ph = 0. Performing the canonical

transformation,

iP{x) = YiAiCi{x), / W = ^ q ( i ) 4 , (9.75)j i

where

fc;(x)Cj(x)dx = dlj, '£C3(J)Cj(x)=6(x-x'), f dx^x)]2 = 1,

and the operators A+ and A~£ are the creation operators for the bound statesCs(x) and delocalized state Ck(x), respectively. Pang partially diagonalized the


Hamiltonian (9.73) using (9.74) - (9.75), to get

H = W + ESA+AS + £ EkA+Ak + £ h(uq - qv)b+bq

k q

+7^ E ^ H - ?«)(#«,+«;6,)(i - A+A.)

- iE^^X^ + M^M-*-^), (9-76)

and

with

r h2V2 1Bt = 2 ô - 2 J - 2^T - ^(^ro)

2J ,

respectively. In (9.76),

F(k,k',q) = 2[9l(q) + 2g2(q)} f dxe^Cl(x)Ck(x)Jo

- »[»(.> + !.(,)] {l - ^ ^ ( J X ^ - ^ o l }ss ,F[fc, (A: + g),9]<5fc'jfc+<,,

F(k,q) = 2[9l(q) + 2g2(q)} f dxe^'C'e(x)C.(x)Jo

2TT . / igr0 N , r7r(fc-g)ro]V2//p V /XP - ifcr0 / [ 'PP J

where aq is determined by the condition,

(uig - ug)ag = (cjq + qv)a*,

which is required in order to determine the factor (1 - AfA3) in H in (9.76). Thus

(9.77)

(9.78)


we get

aq = ——r. 5TA/XZ—(w, + qv) csch ~— ,* w/z/>(l - s2) y 2fej, * V 2A<P /

and

W = \&J.

For this aq, |0)p/, is the coherent phonon state. However, unlike Ck(x) in (9.78)which is an unbound state, the bound state Cs(x) in (9.77) is self-consistent withthe deformation. Such a self-consistent state of the intramolecular excitation anddeformation forms a soliton which is stationary in its intrinsic reference frame.

For the soliton described by the state

l*> = ^y(^)2|0>e,|6)p/ l )

the average energy is given by

(9\H\9) = 2 (e0 - 2J - jj^j - -Jn2P. (9.79)

Evidently, the average energy in the soliton state |\t), (9.79), is equal to the solitonenergy EHO\ given above, or the sum of the energy of the bound state in (9.77), Es,and the deformation energy of the lattice, W, i.e.,

(9\H\9) = Esol =ES + W.

This is an interesting result. It shows clearly that the quasi-coherent soliton formedby this mechanism is a self-trapping state of the two excitons plus the correspondingdeformation of the lattice. However, it should be noted that |\f) is not an exacteigenstate of H, due to the presence of AÂS and A+A-k in H.

For discussion of the decay rate and lifetime of the soliton state, it is convenientto write H in (9.76) as Ho + V\ + V2, as suggested by Cottingham and Schweitzer,where

Ho = W + EsAfAs + J2 EkA+Ak £ fi(w, - vq)b+bq+k q

+ 7 ^ E f i H - vq)(aqb+ + a*qbq){l - A+As), (9.80)

Vl = "Tiff £ F{k'k + q'q) {bti + b*)Av Ak' (9"81)

" ^ kk'qV* = H ^ ' ^ - o + bq){A+Ak - A+A_fc). (9.82)

kq


Here, Ho describes the relevant quasi-particle excitations in the protein. This isa soliton together with phonons relative to the distorted lattice. The resultingdelocalized excitation belongs to an exciton-like band with phonons relative to auniform lattice. The bottom of the band of the latter is at the energy 4J/ip/3relative to the soliton, in which the topological stability associated with removingthe lattice distortion is included.

We now calculate the decay rate of the soliton by using (9.80) and V2 in (9.82),following the quantum perturbation of Schweitzer et al. We first derive a formulafor the decay rate of a soliton containing n quanta in a general system in which thethree terms in (9.57) are replaced by the (n + 1) terms of a coherent state,

exp $ > „ ( * ) £ + |0)«.. n J

After that, we obtain again the decay rate of the soliton with two-quanta. Thus,the above choice for Ho is such that \n) is the ground state of Ho, with energyW + nE's, in the subspace of n excitations, i.e.,

(n Y,BtBi n) = (n UtAs + J2AtA>>j n\=n.

In this subspace the eigenstates have the simple form

|n - m,kuk2,- • • ,km, { n , } ) - _ L = = ( A + ) n - m 4 j - 1 A+2 • • • A+jO)exxy(n — my.

n (dt)nq |n\n-m

q V n « -

where

n and m {m <n) are integers. The corresponding energy of the systems is

[ 2~\ n l

l - ( ^ ) \w + (n-m)E's + Ê'kj+Y/H^-n)nq.J J = l Q

Here E's is the energy of a bound state with one exciton, E'k is the energy of the anunbound (delocalized) state with one exciton. When m = 0, the excitation state isa n-type soliton plus phonons related to the chain with a deformation correspondingto the n-type soliton. For m = n, the excited states are delocalized and the phononsare relative to a chain without any deformation. Furthermore, except for small k,the delocalized states are approximations of ordinary excitons. Thus the decay ofthe soliton is nothing but a transition from the initial state with the n-type soliton

Nonlinear Quantum-Mechanical Properties of Excitons and Phonons bib

plus the phonons,

I") = 7=? I I H=(^)n|0)e,|6)pft ! (9.83)Vn! Y y/nq\

with energy

#.{»,} = W + nE's + Y, K<»q ~ ««Ki

to the final state with delocalized excitons and the same phonons,

\ak) = I J ^&<V(A+)»|0)e ! B, (9.84)q W

with energy

Ek{nq} = nE'k + H(uq - vq)nq

due to V2, in the perturbation interaction V = Vi + Vi- In this case, the initialphonon distribution is taken to be at thermal equilibrium. In the lowest orderperturbation theory, the probability of the above transition is given by

x^'|exp(^)v!exp(^f^)|nj). (9.85)

We can calculate the transition probability of the soliton resulting from theperturbation potential, (Vi + Vs>), using the first-order perturbation theory. Fol-lowing the procedure of Cottingham and Schweitzer, we estimate only probabilityfor the transition from the soliton state to the delocalized exciton states due to thepotential V2, which can be treated satisfactorily by perturbation theory since thecoefficient F(k, q) defined in (9.76) is proportional to an integral over the productof the localized state and a delocalized state, and therefore is of order 1/y/N. TheVL term in the Hamiltonian represents interaction between the delocalized excitonsand the phonons. The main effect of V\ is to modify the spectrum of the delocalizedexcitations in the weak coupling limit {JHP/KBTO C 1, To is defined below). Thisresults in a shift in the energies of delocalized excitons and phonons, and a finitelifetime. These effects are ignored in our calculation since they are only of secondorder in V\.

The sum over I in (9.85) is over an initial set of occupation numbers for phononsrelative to the distorted lattice with probability distribution P[ , which is takento be the thermal equilibrium distribution for a given temperature T. Using (9.80)


and (9.82), we can get

W = h2^TW E E E iffK*) + 2»2(*)1 [5i(fc") + 252(fc")]^ A fc' k"

(nfii)2 + (fc'r0)2 |_2nA*i ;J [2n/zi ;J

x J dt' f dt" exp | - 1 n Tn2 - | n ^ /i? J + nJ(fc'ro)2l (t1 - t")

x / exp i J2(uq - qv)b+bq{t' - t") (6+ + 6_fc)\ L «

I I ft V

x exp \i J2("q ~ qv)a+aq{t' - t") {b±v +bk,)\\,V i \ II >

where

9i(k)+2g2(k) = 2Xi^/^j-k[A(cos(r0k) - 1) + i(A + 1)sin(rofc)]

*2i(A + l)(rok)Xl][^,

M 1 - W (1 - S2 ) J ' A~Xl>

with

TV J A exp - ^ ^ / i ( a ; a - g t ; ) 6 + J |

P)) = —^ ji ! #TV | exp -/3 5]fi(wa-g«)6+6, \

Here /S = I/IVBT, A is the ratio of the new nonlinear interaction term to that in theDavydov model.

To estimate the soliton lifetime, we are interested in the long-time behavior ofdw/dt. By straightforward calculation, the average transition probability or decayrate of the soliton was obtained by Pang and it is given by

dW 2 7r2 ^ 2(rofc)2Sech2[7r(fc-fc')ro/2nMl]

xSR f dtexp | - t [nJ(fc'ro)2 +n(n2- | n ^ ^ + Rn(t) + n(t)l |

exp[i(o;t - fci;)t]exp^Wfc-Jfct;)]-! ' l J


where

, ,.s 4_ v \ak\2sin2[(uk-kv)t/2}

? n W n 2 i V ^ exp[/3fi(o;fc - kv) - 1] "

This is a general analytical expression for the decay rate of a soliton containing nquanta at any temperature within the lowest order perturbation theory.

From (9.86), we can see that Fn, Rn(t), £n(i) and fi - n\i\ given above allchange with increasing number of quanta, n. Thus, Fn is different for different n.In the following, we derive an explicit expression for the decay rate of the solitonwith two-quanta (n = 2). In this case, the integral in the decay rate expressioncan be evaluated explicitly, and R2(t) and £2(t) in (9.86) can be obtained usingapproximation method. The results are for v -> 0 and u)q -> y/w/M are given interms of the digamma function. We are interested in the long-time steady behavior.Thus we take the limit t -> oo. Then

R2(t) = -Ro [in (\uat) + 1-578+ |»7rl ,

KRpkBTt6(t) « J : ,

where

c 4(xi + X2)2 AM 2J/j,Pro 2fip pw _ tUva

irnw V w nfivo n \ M KB

We have used coth(w»i/2) « 1 in obtaining the above results. That is,

,. t ,.y. . TTJRO nRokBThm ^2(*) = -Vt, r] = -zz- = z •t-*oo pn n

For protein molecules, we have Ro < 1, To < T and RoT/To < 1. In this case,the decay rate of the soliton with two quanta was derived by Pang and it is givenby

r - r ^ L - 2 L . (r\2 ST (fcro)2|gl(fc) + 2g2(fc)l2sech2[(7rr-o/2/iP)(fc - k')]1 2 t%!o dt ~ VLPKN) ££ [n2

P + (fc'ro)2][exp(/3^) - 1]

1 {rf + [4fx2PJ/3 + 2(k'ro)

2J - hujk]2/h2}(1+R°V2

X (2.43wa)flo h2^ + [AupJ/Z + 2(k'ro)

2J - fiwfe]2

( 2 I 2 hr} J J

This is evidently different from that of the Davydov model with one quantum (n =

(9.87)


1) obtained by Cottingham and Schweitzer, which is of the form

_ 2irxt y ^ (fcr0)2 sin2(ATo)sech2[(7rro/2/X£))(fc - k')]

D ~ NhfMD \g 2Muk[n2D + (fc'ro)

2][exp(i3fiW,) - 1]

x (H£\*° &WD

Vug) Knl + [j»ys + j(k'roy - huk]' i y ' 8 8 J

where

nRgKBT D 2x1 [M D 2HD [M

Equations (9.87) and (9.88) are different not only in their values, but also in thefactors they contain. In (9.87), there is an additional factor,

and the factor {r]Dlâ)R° VD in (9.88) is replaced by

{-i 2 1 (l+-Ro)/2

in (9.87), as a result of the two-quanta nature of the wavefunction and the addi-tional interaction term in the Hamiltonian in the improved model. Unlike that inthe Davydov model, JJ, Ro and To in (9.87) are not necessarily small. Using theparameters of protein molecules given in Section 9.5, we find that r] = 6.527 x 1013

s"1, Ro = 0.529 and To = 294 K for the soliton in the improved model. Whencompared to RQ = 0.16, To = 95 K, and r]D = 2.096 x 1013 s"1 for the Davydovsoliton at T — 300 K, we see that the values of rj, Ro and To in the improved modelare about 3 times larger than the corresponding values in the Davydov model dueto the increases of \ip and of the nonlinear interaction coefficient Gp.

Using (9.87) and the commonly accepted values for the various parameters ofprotein given in Section 9.5, we can numerically compute the decay rate, F2, and thelifetime of the soliton, r = I/T2, for the a-helical protein molecules. For wavevectorsin the Brillouin zone and v — 0.2w0, values of T2 fall between 1.54 x 1010 s"1 and1.89 x 1010 s"1. This corresponds to a soliton lifetime, r, between 0.53 x 10~10 sand 0.65 x 10~10 s at T = 300 K, or T/T0 = 510 - 630, where r0 = ro/vo is thetime for the soliton to travel a distance of one lattice spacing at the speed of sound,which is equal to y/M/w = 0.96 x 10~13 s. In this period, the soliton, travelingat two tenths of the speed of sound in the chain, would travel several hundredsof lattice spacings. That is several hundred times of that of the Davydov solitonfor which T/T0 < 10 at 300 K. In other words, the Davydov soliton travels at onehalf of the sound speed can cover less than 10 lattice spacings during its lifetime.The soliton excitation in the improved model has a sufficiently long lifetime to be


a successful carrier of bio-energy. Therefore, the quasi-coherent soliton is a viablemechanism for bio-energy transport at biological temperature for the above rangesof parameters.

Fig. 9.5 Soliton lifetime T, relative to To, as a function of temperature T using parameterscharacteristic of the a-helical molecule.

We are interested in the relation between the lifetime of the quasi-coherentsoliton and temperature. Figure 9.5 shows the relative lifetime T/T0 of the solitonversus temperature T for a set of widely accepted values for the various parametersgiven in Section 9.5. Since it is assumed that v < VQ, the soliton will not travel theentire length of the chain unless T/T0 is large compared with L/TQ, where L = Nrois a typical length of protein molecular chains. Hence for L/r0 « 100, T/TQ > 500is a reasonable criterion for the soliton to be a possible mechanism of bio-energytransport in protein molecules. The lifetime of the quasi-coherent soliton shown inFig. 9.5 decreases rapidly as temperature increases. But below T = 310 K, it is stilllarge enough to fulfill the requirement. Thus the soliton can play an important rolein biological processes.

For comparison, we show simultaneously, in Fig. 9.6, log(r/ro) versus tempera-ture for the Davydov soliton and the soliton with a quasi-coherent two-quanta state.We find that T/TQ in the two models are very different. The T/TO of the Davydovsoliton is too small, and it can only travel a distance of less than ten lattice spac-ings at half the speed of sound in protein molecules. Thus the Davydov soliton isineffective for biological processes.

Dependency of the soliton lifetime on other parameters can also be studied byusing (9.87). Using parameters around the accepted values given earlier, it wasfound that the lifetime of the soliton increases with increasing (\i + X2) (Pang1987, 1989a, 1990, 1993e, 1999, 2001c).


Fig. 9.6 log(r/To) versus temperature. The solid line represents result of the new model, whilethe dashed line corresponds to that of the Davydov model.

9.7 Effects of Structural Disorder and Heart Bath on Exciton Lo-calization

The results discussed so far were all obtained analytically, in which the proteinmolecules were regarded as a periodic system and some approximate methods suchas long-wavelength approximation, continuum approximation, or long-time approx-imation, were used. Some average values as given in Section 9.5 were used forvarious parameters of the protein molecules in the calculations. However, in reality,a protein is not a periodic system, but rather it consists of 20 different amino acidresidues with molecular weights ranging from 75 mp (glycine) to 204 mp (trypto-phane). This corresponds to a variation between 0.67M and 1.80M, where mp is themass of the proton, M is the average mass of the amino acid molecules. Hence, thereexists structure disorder in proteins. Careri et al. demonstrated that a relativelysmall amounts of disorder in amorphous film of acetanilide (ACN), a protein-likecrystal, is enough to destroy the spectral signature of a "soliton". Therefore, itis necessary to investigate the influences of structural disorder on the stability ofsoliton in proteins at biological temperature. This will be the topic of this section.A numerical simulation based on the Runge-Kutta method will be used obtain re-lated results in which these disorder effects of the protein molecules are considered,without using approximation.

The transformation, an(t) —> an(t)exp[iR(t)t/h], enables us to eliminate theterm of R(t)an(t) in (9.61), where an(t) = <pn(t) is a complex function of timet, and can be represented by an(t) = a(t)rn + ia(t)in. Here a(t)rn and a(t)in

are real and imaginary parts of an(t). Equations (9.60) and (9.61) can then be


approximately written as

?iarn = -J(ain+1 + ain-i) + Xi(.Qn+i ~ Qn-i)ain

+X2(qn+i ~ qn){ain+i + ain-i), (9.89)

-twin = -J(arn+1 + arn_i) + Xi(.Qn+i ~ qn-i)arn

+X2(qn+i - qn)(arn+i + arn_i), (9.90)

in = j£ , (9-91)Vn = w(qn+1 - 2qn + qn_i) + 2xi{ar2

n+1 + ai2n+1 - arl^ - a i 2 ^ )

+4X2[arn(arn+i - orn_i) + ain{ain+l - ain-i)], (9.92)

where |an|2 = |arn|2 + |oin|2 and qn = j3n. Equations (9.89) - (9.92) are theequations of motion which determine the states of the soliton. We will obtaintheir solutions numerically using the fourth-order Runge-Kutta method. Obviously,there are four equations for every peptide group. Therefore, for a protein moleculeconsisting of TV amino acids, we have to solve a system of 4N equations. To use theRunge-Kutta method, we need first to discretize the equations, in which the timeis discretized and denoted by j , and the step size of the space variable is denotedbyh.

In the numerical simulation, Pang et al. used eV as the unit of energy, A forlength and ps for time. The following are details of the numerical simulation: atime step size of 0.0195 ps, the total energy E = ($(t)\H\$(t)) was conserved towithin 0.0012 eV, (a possible imaginary part of the energy can be developed duringthe simulation due to numerical inaccuracy which was kept to below 0.001 eV), thenorm was conserved up to 0.3 pp (parts per million). A fixed chain of TV units wasused and an initial excitation in the form of an(0) = Asech[(n—no)(xi+X2)2/4JW],where A is a normalization constant, was assumed. Pang et al. used N = 50 inthe simulation. For the lattice, qn(0) = nn(0) — 0 was applied. The simulation wasperformed using a data parallel algorithm and the MATLAB software.

9.7.1 Effects of structural disorder

Using average values of the various parameters given earlier in Section 9.5, Pang etal. first computed the solution of the above equations (9.89) - (9.92) in the case of afree and uniform periodic chain. The result is shown in Fig. 9.7. From this figure, wesee that the amplitude and energy of the solution remain constant throughout thecourse of motion. The collision behavior of two solitons, setting up from oppositeends of the chain, can be investigated and the results are shown in Fig. 4.4. Fromthis figure, we see that the two solitons go through each other without scattering.Therefore, the solution of the above equations are truly "soliton". We can thusconfirm that the exciton described by the above nonlinear Schrodinger equation isindeed localized as a soliton in the case of a free uniform periodic protein molecules.


Fig. 9.7 Soliton in a free and uniform periodic chain.

However, what would be the behaviors of the soliton when there are "structuraldisorders" in the protein molecules? To answer this question, we introduce struc-tural disorder into the systems and carry out the simulation. A random numbergenerator can be used to produce random sequences for the various parametersalong the protein molecular chains.

We first consider the influence of a disorder in mass sequence on the stabilityof the soliton. This disorder is modeled by a nonuniformity at given sites or adeviation from the uniform mass distribution M = 114mp = (1.17 — 1.91) x 10~25

kg per amino acid residue. In a first series of calculations we study the effect ofthe mass nonuniformity at given sites on the soliton. This mass disorder may becaused by amino acid side groups and local geometric distortion of molecular chains,imported impurity, or some other molecules bound to the protein (reactive centerssuch as heme groups). As an example, only the mass at site 49 was increased, allother masses were kept at M. For M49 = 100M, the soliton is still very stable.Very surprisingly, up to 100000M, no obvious perturbations and decays appear inthe motion of the soliton. These results are shown in Fig. 9.8. From these results,one can conclude that general disorder of mass sequence of the amino acid residuesat a given site does not disturb the soliton at all. This, however, may not be truein the case of vast impurity or under the influences of other disorders.

Fig. 9.8 Soliton for (a) M49 = 100M and (b) M49 = 100000M.


The influence of a random series of masses distributed along the whole chaincan be studied. Here we introduce a small parameter a.), to denote the variationof mass at each point in the molecular chain, i.e., Mk = a^M. The values of otkis randomly generated with equal probability within a prescribed interval. Theaperiodicity due to ctk, for example, 0.67 < a^ < 400, does not affect the stabilityof the soliton. However, when the mass variation is sufficiently large, for example,0.67 < afc < 700, the vibrational energy becomes dispersed, as shown in Fig. 9.9.The interval 0.67 < at < 400 over which the soliton can move unperturbed isevidently larger than the variation of masses of the natural amino acids (0.67 <dk < 1.80). Therefore, the soliton is very robust against mass disorder of naturalamino acids.

Fig. 9.10 Soliton with (a) Aw = ±45%w) and (b) Aw = 75%u).

We can also investigate the influence of fluctuations in the spring constant w,arising from the structural disorder, on the stability of the soliton. For a randomvariation up to ±45%-iD, no change is found in the dynamics of the soliton. For±55%u), the soliton velocity is only slightly reduced compared with the case of w.Finally, for ±75%u), the soliton disperses slowly and its propagation is irregular.The results are shown in Fig. 9.10. If in addition w is aperiodic, the soliton is stableup to ±40%w, and then at 45%UJ the slow dispersive phenomenon occurs.

The soliton, however, is more sensitive to the variation in J, resulting also from


structural disorder. For variation in J alone the soliton is stable for a change up to9%J, but it disperses at AJ = ±15% J, as shown in Fig. 9.11.

Fig. 9.11 Soliton at (a) A J = ±9%J, and (b) A J = ±15%J.

If (xi + X2) alone is aperiodic or is aperiodic together with a natural massvariation, the states of the soliton will be changed. (xi + X2) can be varied upto ±25%(xi + X2) without any change in the dynamics of the soliton. But for0.67M < M < 1M and A(xi +X2) = 25%(xi +X2), the soliton behaves differently.These are shown in Fig. 9.12.

Fig. 9.12 Soliton at (a) A(xi + X2) = ±25%A(xi + X2) and (b) 0.67M < M < 2M andA(xi +X2) = ±25%A(xi +X2).

In the case of changing the ground state energy Aeo> which arises from differentamino acid side groups and corresponding local geometric distortions due to im-ported impurities, it can be found that for an isolated impurity in the middle of thechain which results in a change of the ground state energy, Ae0 = e8n, the solitoncan pass the impurity only if e < 1 meV. In other cases, it is reflected or dispersed.In the case of a random fluctuation but Aeo = e\Pn\, £ < 1 meV, \(3n\ < 0.5, thesoliton can pass the chain. The results are shown in Fig. 9.13. For higher values ofs, the excitation disperses.

Finally, if all types of disorder, variation of mass, fluctuations in the springconstant, dipole-dipole interaction constant, coupling constant, and ground stateenergy, are taken into account, for example, Aw = ±10%w, A J = ±5%J, A(xi +


Fig. 9.13 The soliton for (a) Ae0 = eSn, £ = 0.5 meV, (b) Ae0 = e\Pn\, e < 1 meV, \/3n\ < 0.5,and (c) Aw = ±10%tD, AJ = ±5%J, A(xi + X2) = ±5%(xi + X2), 0.67M < M < 2M,Ae0 = e\0n\, e = 0.4 meV, |/3n| < 0.5.

X2) = ±5%(xi + X2), 0.67M < M < 2M and e = 0.4 meV (Ae0 = e|/?n|) occursimultaneously, it is observed that the soliton is still stable, as show in Fig. 9.13(c).Therefore the soliton can be said very robust against these structural disorders.

We have seen that the soliton is stable even when the structure disorders arelarge, for example, for a mass change at a given site up to Mn = 100000M, arandom mass nonuniformity in the range of 0.67 < afc < 400, a change in the springconstant up to ±40%u), fluctuations in the interaction constant J, coupling constant(xi +X2) and ground state energy e0 up to 10%J, ±25%(xi +X2) and Ae0 = e\f3n\,e = 1 meV, |/?n| < 0.5 and Aeo = sdn, e = 0.5 meV, respectively; or a combinationof fluctuations with Aw = ±10%u>, AJ = ±5%J, A(xi + X2) = ±5%(xi + X2),0.67M < M < 2M, Ae0 = e|/3n|, e = 0.4 meV, \0n] < 0.5. Therefore, the newsoliton is very robust against these structural disorders in the protein molecules.However, the actual degree of disorder in protein molecules is unknown. Also proteinmolecules are bio-self-organizations with high-order and coherent feature, whichare necessary condition for a protein to perform its biological functions. A largestructural disorder in a biological protein means a degeneration of the structureand the disability of the functions of the protein molecules which lead to diseaseof living systems. Therefore, larger structural disorders are not likely to occur inprotein molecules. Then, discussing effects of large disorder in all parameters on thestability of soliton may not be relevant and practical in normal biological proteinmolecules. Since proteins are not simply random heterpolymers. The amino acidsare not free particles, but are covalently bonded to the main polypeptide chains.


Its nonuniformity or aperiodicity is thus small, and so are the structural disorders.Therefore, the influences of the structural disorders on the soliton are expected tobe small. Nevertheless, the results given above indicate that soliton is stable in caseof larger structural disorders, and the new soliton is robust against such structuredisorders in protein molecules.

9.7.2 Influence of heat bath

Soliton are known to be sensitive to temperature in protein molecules. The ther-mal stability of Davydov soliton has been studied using different methods. In thefollowing, we focus our discussion on the effects of (1) a head bath to which theproteins are coupled to; and (2) of structural disorder due to a non-uniform massdistribution in the protein molecule. Both effects have been studied by Halding andLomdahl using classical molecular dynamics and a Lennard-Jones potential betweenthe peptide units of an a-helix. The work of Halding and Lomdahl is an example ofthe application of a classical thermalisation scheme to a classically described lattice.Lomdahl and Kerr, Lawrence and co-workers studied the stability of the Davydovsoliton at finite temperature and found that at 300 K the Davydov soliton is de-stroyed. The method of Lomdahl and Kerr will be used here to treat the influenceof the heat bath on the new soliton.

In accordance with Lomdahl and Kerr, Pang et al. considered a random-noiseforce Fn(t) and a dissipation force, MTqn, resulting from the interaction betweenthe heat bath at temperature T and the protein molecules, and included these forcesin the displacement equation of amino acid, (9.60), which now can be written as

Mqn(t) = w[qn+1(t) - 2qn(t) + qn-i(t)} + 2Xi [K+i | 2 - \an^\2]

+2X2 {a*n(t)[an+1(t) - on_i(t)] + an(t)[a*n+1(t) - <_!(*)]}

-MTgn + Fn(t), (9.93)

where F is a vibrational dissipation coefficient of amino acids. The correlationfunction of the random force can be represented by

(F(x,t)F(0,0)) = 2MTKBTS^t\

It is assumed that the random deviations obey the normal distribution with astandard deviation of y/a and a zero expectation value. That is,

N(F ) = * e~F"/2<r

[ n) V2^where a = 2MKBTT/T and r is a time constant, T is the reciprocal time constantof the heat bath. It can be shown that

Fn(t) = V6j2\Xnr(t)-\\.r=l "- J


Here Xnr(t) is a random number between 0 and 1. If we choose L — 12, then thedeviation of [Xur(t) — 1/2] is 1/12, and the standard deviation of Fn(t) is \[a. Thedomain of random noise force is |Fn(t)| < 6^/a.

Meanwhile, we can verify numerically that over a sufficiently long time the meankinetic energy is given by

(Tl-Mqn{t)) = \NKBT.

Even in the presence of damping and noise force, equations (9.89) - (9.92) have aconserved quantity ^2n \an(t)

2\ = constant.

Fig. 9.14 The state of the soliton at 300 K.

Fig. 9.15 The state of the soliton at 300 K for a time of 300 ps.

Applying the formulae given above, we can study the influences of dissipationforce and random noise force, arising from the heat bath, on the new soliton. How-ever, since the proteins now are in contact with a heat bath of about 300 K, onehas to find out whether the soliton motion will be destroyed by the thermal motionof the lattice. For a C=O vibrational energy of eo = 3.28 x 10~20 J and a temper-


ature of 300 K, the Boltzmann factor is 3 x 10~20, i.e., only three out of 10,000oscillators are excited in thermal equilibrium. Therefore, it can be safely assumedthat the heat bath affects the soliton motion only via the lattice, which exhibits aquasi-continuum of vibrational states. Through the coupling between the proteinand the heat bath, the heat bath also affects the oscillator system. Prior to the soli-ton commencement, the system is in equilibrium with the heat bath. The oscillatorsystem is in its ground state, while the lattice is in thermal motion, which can bedescribed by a linear combination of its normal modes. With the commencement ofthe soliton a non-equilibrium state is created. We can consider two extreme cases.In the first, the time that the soliton takes to travel through the protein is smallcompared to the time it takes the heat bath to re-establish equilibrium with thesystem, and in the second, the soliton velocity is small compared to the velocity ofequilibration. If the soliton velocity is high, the first case would be more realistic.

If the soliton starts at t = 0, the lattice energy fluctuations associated with theheat bath are larger by roughly three orders of magnitudes than the local latticeenergies associated with the soliton motion, but the soliton moves through the chaincompletely undisturbed at the biological temperature (300 K) as shown in Fig. 9.14.Here average values of the parameters of proteins were used. Therefore, despitethe large lattice energy fluctuations due to the heat bath, the non-linear couplingbetween the lattice and the amide oscillators (excitons) is still able to stabilize thesoliton. This agrees with the earlier analytic consideration. The state of the solitonin the cases of a long time (300 ps) and at a higher temperature of 310 K are shownin Fig. 9.15 and Fig. 9.16(a), respectively. However, at high temperature of 320 K,the soliton starts to disperse as shown in Fig. 9.16(b).

Fig. 9.16 The states of the soliton at (a) 310 K and (b) 320 K, respectively.

These results show clearly and sufficiently that the soliton in the improved modelis thermally stable at 300 K, and the lifetime of the soliton is at least 300 ps in whichthe soliton can travel over thousands of lattice spacings. The critical temperatureof the soliton is about 320 K. These are in agreement with the analytic results, andit not only shows that the analytic model is valid, but also establishes the solitonas a "possible" carrier of bio-energy transport in protein molecules.

The Davydov soliton does not have such properties. For comparison we show


Fig. 9.17 The states of the Davydov soliton at (a) 40 K, and (b) 300 K, respectively.

in Fig. 9.17 the results of the Davydov soliton obtained by Forner using the samevalues for the various physical parameters. The temperatures are T = 40 K and300 K, respectively. We see clearly from Fig. 9.17 that the Davydov soliton is notthermally stable at 300 K, and its critical temperature is about 40 K.

Finally, Pang et al. studied the thermal stability of the new soliton at 300 Kunder the influences of structural disorders. The results show that the new solitonis still stable at this temperature against the changes of parameters as those usedin Fig. 9.13 (c).

9.8 Eigenenergy Spectra of Nonlinear Excitations of Excitons

In this section, we calculate the eigenenergy spectra of the nonlinear excitations ofexcitons, or quantum vibrational energy spectra of amide-I in protein, based on theHamiltonian and dynamic equations in the theory of bio-energy transport discussedin previous sections. From the earlier discussion, we know that both Davydov'sHamiltonian and that in the improved model, (5.64) and (9.58), respectively arequantum mechanical by construction. Both analysis are based on the coherentstate ansatz and are therefore as "classical" as a quantum-mechanical treatmentcan be. An alternative approach would be to start from purely classical equations,and then subsequently apply the semi-classical or Bohr-Sommerfeld quantization.This has been done by Scott et al, Pang, and Chen et al. The advantage of thismethod is that it allows for an arbitrary number of amide-I quanta (excitons).

The starting point is to define normal displacement and momentum coordinatesQn and Pn for amide-I (exciton), for which the Hamiltonian is given by $3n(.P^ +Q2

n). The motion of the exciton can then be described in terms of a complexamplitude

An = uJ1

0/2(Pn-iQn).


In terms of An, the classical analogue of Hex in (5.64) is

tfex-cu* = 5 3 h > K | 2 - J(K+1An + A*nAn+1)} , (9.94)n

where w0 is the amide-I vibrational frequency. It is then assumed that the amide-Imode interacts with some (unspecified) low-frequency phonon mode qn of aminoacid with an adiabatic energy of

#ph-cias = 2Wclas z J 9 " 'n

The interaction energy can be taken as

-Hint-clas = ^Xclas / , gnl^nP-n

The total classical Hamiltonian is then

•Hclas = -Hex-clas + #ph-clas + -Hint-clas- (9.95)

Minimizing (9.95) with respect to qn yields

_ Xclas | . ,2

Wclas

In the continuum approximation, (9.95) reduces to

Hcias = Yl U l^ .1 2 - J(K+iAn + A*nAn+l) - \l\AnA , (9.96)

where 7 = Xdas/^cias is the nonlinear interaction constant. The equation of motioncorresponding to (9.96) is

[ijt - Wo) K | 2 + J(An+1 + An^) + ^j(\An\2)An = 0. (9.97)

In addition to the energy i?cias, equation (9.97) has another constant of motion,i.e. the particle number,

iv = 5>n|2.n

Equation (9.97) happens to be the discrete nonlinear Schrodinger equation for one-chain in the protein molecules. Therefore, the vibration of protein molecules isindeed a nonlinear problem as can be seen from the above. In Pang's improvedmodel, Xcias i n 7 is approximately replaced by (xi + X2)2ias-

Equation (9.97) is actually a special case of the following more general discreteself-trapping dynamic (DSTD) equation

i ^ + MA + jD(\A\2)A = 0,


where A — {Aa} is a complex a-vector, a corresponds to number of the modes ofmotion, £>(|yl|2) denotes the diagonal matrix, diag. (|^4i|2, |yl2|

2, • • • , |j4a|2), and

M is a real symmetric matrix. This equation has been extensively studied. Forthe a-helix protein molecules consisting of the three chains, the DSTD equation, or(9.97), becomes

(ijt-u^\A + MA + 1D{\A\2)A = Q (9.98)

where A = co\.(Ai,A2, A3, Ai} A5,A6,A7,AS, Ag) is a complex 9-vectors, C(|A|2)denotes the diagonal matrix, diag. ( l ^ 2 , \A2\

2, \A3\2, \A4\

2, \A5\2, \A6\

2, \A7\2,

\A8\2, \A9\

2) and M is a real symmetric matrix of order 9, which represents var-ious interactions between neighboring amide-Is, i.e.,

"0 J ei e2 £3 e3 e2 £i J "J 0 J sx e2e3 e3 e2 £iei J 0 J ei e2 £3 £3 £2£2 £1 J 0 J £1 £2 £3 £3

M = £3 £2 £1 J 0 J £1 £2 £3 • (9.99)

£3 e3 e2 Ei J Q J ex £2

£2 £3 e3 e2 £1 J 0 J £1

£1 £2 £3 £3 £2 £1 J 0 J

. J £1 £2 £3 £3 £2 £1 J 0 .

The dynamic equation of amide-I vibration quantum (exciton), with a order-9matrix (9.99), obtained here is similar to that obtained by Scott et al. from thefollowing equations for the three channels of a-helix protein.

dAi - ~ = -J(An+i - 2Anl + A n _ u ) - K1(Rn+hl - Rn_hl)Ani + LFL

+K2[(Rn-i,iAn-1,i - Rn+i,iAn+u) + Rni(An+Xti - An_ii()]

- 2 JAnl + NFN + PFP + QFQ + SFS + TFT

+UFV + VFV + XFX + ZFZ,

^ ± - w{Rn+u - Rnl + Rn_u) = K3(\An+u\2 - \An+u\

2)

+Ki[A*nl(An+hl - An-ltl) + Anl(A*n+ll - A*n_u)],

where

v 1 !~M is I [MKl~2xl0^h\l^Xu K2=2xlO"h\l^X2'

K 1 0 " v 1QU

K3 = ^Xl> Ki = ~2^X2

represent the nonlinear coupling between vibrations of the amide-I and displace-ments of amino acids. Here M is one third of the mass of a unit cell, w is the linear


force constant of the molecular chains, xi is the change in the vibrational energy ofthe amide-I per unit extension of an amino acid, and X2 is the change in the longi-tudinal dipole-dipole coupling energy per unit extension or displacement of aminoacid. In these equations, J, L, N, P, Q, R, S, R, U, V, X, Y and Z are variousdipole-dipole interaction energies between pairs of amid-I bonds. Their values are7.8, 12.358, 3.873, 1.592, 1.0082, 0.637, 0.472, 0.387, 0.196, 0.159, 0.116, 0.09 cm"1,respectively. The subscript j indicates unit cells along the helix, while I specifiesone of the three spines. The symbol "LFL" is a shorthand for the dipole-dipoleinteractions between laterally adjacent amide-I bonds. Thus

iAnA = L(Anfi + An-1>2) + ••• ,

iAnfl = L(An3 + Anl) + • • • ,

iAn3 = L(An,2 + Ai+i,i) H •

NFff,--- ,ZFz are defined similarly. In our discussion, the influences of smalldipole-dipole interactions, (L, N, P, Q, R, S, T, U, V, X, Z), on the matrix M are in-cluded in £i, e-i and £3. Therefore, equation (9.99) can indeed represent the dynamicproperties of the exciton, interacting with displacements of amino acid residues, inthe a-helical protein molecules.

Upon quantization using the method discussed in Chapter 6, the complex am-plitudes are replaced by the creation and annihilation operators of a harmonicoscillator, B+ and Ba, in the second quantization with the following properties,

Ba \m<*) = y/{rna + l)\ma + 1), Ba\ma) = y/m^\ma - 1),

and

[Ba,Bj~] = Saj.

The particle number and the energy corresponding to (9.99) then are representedby the operators

N = Y,(BZB<* + 2)> (9-100)

f 1 \ 9 9 9

H = I wo - -7 ) N - J £ B+Bj - £1 *£ B+Bh - £2 £ B+Bh

9 9

-£3 £ B+Bh - ^^2B+BaB+Ba, (9.101)

where h is taken to be 1 and both frequencies and energies are measured in thesame unit (cm"1).

Choosing a set of basic function, \m,k) — \rn^\vn^\m,-i)\m^)\m^)\rn^)\m-i)-|m8)|m9), where m (= mx + m2 + m3 + mi + m5 + m6 + m7 + ms + m9) is the total


quantum number, A; = 1,2, • • • , d(m), and d(m) is equal to the number of differentways that m particles can be distributed in the nine states. We are interested inthe common eigenfunction, |$m), of both operators. If we take

|*m) = Ci|m, 1) + C2\m, 2) + • • • + Cd{m)\m,d(m)), (9.102)

then the Hamiltonian equation H^m) = -Em|\tm) reduces to an algebra equationon the column vector, Cm = Col.(Ci, C2, • • • , Cd(TO)),

HmCm = EmCm, (9.103)

where Hm is a d(m) x d(m), real and symmetrical matrix with diagonal elements,Hmr = (m,r\H\m,r) and the off-diagonal elements HTS = (m,r\H\m, s). Here1 < r < d(m), 1 < s < d(m), and r ^ s.

Using (9.103), we can numerically obtain eigenenergy spectra of the nonlinearexcitations of the exciton in protein molecule, using the following accepted valuesfor the various parameters for the three channel a-helix protein molecules

Parameter Value0 (39 - 58.5) N / mM (3.51 - 5.73) x 10"2 5 kgX (56 - 62) PNJ 7.8 cm" 1

Xi (10 - 15) PNro 4.5 Aei 16.231 cm" 1

e2 13.951 cm" 1

£3 15.363 cm" 1

£p (0.205 - 0.210) eV

We can then obtain the parameters needed in our calculations, which are UJ\ =1693.98 cm" 1 , J 1 = 7.80 cm" 1 , 7 1 = 40.68 cm" 1 for the Davydov model andL% = 1712.08 cm" 1 , J2 = 7.8 cm"1 , 7 2 = 49.73 cm" 1 for the improved model. Thecalculated eigenenergies for m = 3 for both models are listed in Table 9.1.

From Table 9.1, we see that even though the energy spectra of the protein isquite complicated, its distribution shows the following trends. (1) The vibrationalenergy spectra consists of a series of manifolds or energy-bands, i.e. there areseveral energy levels corresponding to each vibrational quantum-number m. Forexample, the first three excitation states, m = 1 from 1610 - 1678 cm"1 , m = 2from 3179 - 3358 cm""1 and m = 3 from 4735 - 5001 c m ' 1 , consist of 8, 44 and164 energy levels, respectively. Hence, as m increases, the gap between energylevels, AE, decreases gradually. For instance, AE ranges from 6 to 23 cm" 1 form = 1, while it varies in the ranges of 0 — 14 and 0 — 11 cm" 1 for m = 2 and m = 3,respectively. (2) The vibrational spectra have strong local mode characteristics, i.e.,the gap between the energy-levels depends strongly on the nonlinear interaction, 7,


Table 9.1 The vibrational energy spectra of protein molecules withthree channels in cm"1.

M I Exp" Cal6 Calc I M I Exp° Cal" Calc

1 1611.01 1610.42 ~~T~ 1612.95 1612.011 1628.35 1627.64 1 1631.61 1630.111 1650 1654.37 1653.81 1 1662 1662.95 1661.981 1666 1668.23 1667.65 1 1679.27 1678.732 3150 3206.33 3179.40 2 3212.17 3203.192 3205 3213.60 3204.71 2 3224.25 3211.852 3225.39 3212.95 2 3226.57 3213.212 3216 3233.34 3216.84 2 3234.71 3218.192 3246.71 3242.48 2 3248.75 3242.452 3250 3252.57 3249.68 2 3259.67 3258.782 3260.85 3259.87 2 3263.57 3261.772 3264.66 3260.95 2 3265.73 3262.972 3267.91 3263.67 2 3267 3269.99 3267.392 3270.45 3269.43 2 3278.57 3277.712 3279 3279.97 3278.89 2 3280 3282.18 3280.212 3283.91 3282.84 2 3284.75 3283.972 3286.54 3285.44 2 3287.56 3286.492 3288.24 3287.44 2 3293.14 3290.492 3299.61 3298.96 2 3300.81 3300.092 3301.73 3301.15 2 3304.95 3302.132 3310.54 3309.47 2 3311.27 3310.212 3313.24 3312.91 2 3314.73 3313.372 3322.27 3321.54 2 3323.29 3322.492 3325.11 3323.56 2 3328.47 3327.962 3331.54 3329.16 2 3338.04 3333.912 3319.17 3345.11 2 3360.61 3358.583 4782.91 4735.46 3 4783.15 4735.963 4787.16 4736.91 3 4787.51 4737.083 4788.17 4737.54 3 4788.40 4737.643 4788.57 4738.21 3 4789.14 4738.543 4752 4789.68 4749.26 3 4803 4819.76 4805.313 4819.95 4805.30 3 4823.84 4806.843 4825.25 4808.56 3 4826.45 4809.183 4829.18 4812.86 3 4813 4829.88 4813.763 4841.69 4824.16 3 4842.36 4824.963 4842.91 4825.51 3 4843.02 4826.333 4846.16 4830.52 3 4847.97 4831.913 4848.71 4832.61 3 4848.86 4833.153 4850.95 4834.67 3 4851.66 4836.243 4852.92 4836.46 3 4855.38 4837.213 4857.12 4839.73 3 4841 4858.92 4841.943 4859.24 4842.45 3 4860.42 4843.913 4861.44 4846.93 3 4862.66 4847.943 4862.97 4849.86 3 4864.35 4851.623 4866.61 4852.32 3 4868.62 4852:76


Table 9.1 continued.

~M I Exp" Cal6 Calc I M I Expa Cal" Calc

3 4869.92 4854.47 3 4870.62 4855.863 4872.21 4855.96 3 4873.12 4857.093 4873.57 4857.94 3 4875.32 4858.363 4875.66 4859.72 3 4876.24 4860.123 4877.01 4861.55 3 4877.65 4863.253 4879.12 4866.28 3 4879.49 4866.313 4879.92 4868.07 3 4880.73 4869.293 4881.04 4869.45 3 4881.36 4869.693 4882.37 4970.11 3 4883.71 4870.823 4887.31 4871.84 3 4887.63 4874.313 4888.61 4874.81 3 4889.36 4876.253 4889.71 4875.91 3 4890.12 4876.833 4891.82 4877.52 3 4892.56 4877.913 4892.72 4878.72 3 4893.32 4878.963 4894.24 4879.31 3 4895.27 4879.613 4895.76 4881.73 3 4896.62 4884.663 4897.96 4885.56 3 4898.96 4886.573 4899.32 4886.93 3 4900.70 4887.133 4901.09 4887.29 3 4901.27 4887.793 4902.32 4888.42 3 4902.67 4889.923 4903.24 4890.49 3 4903.42 4891.823 4904.26 4893.42 3 4905.54 4894.323 4905.86 4996.62 3 4916.84 4897.833 4907.12 4898.56 3 4907.92 4899.373 4909.81 4899.81 3 4910.62 4901.833 4911.52 4905.21 3 4915.72 4906.323 4916.62 4907.52 3 4918.83 4909.513 4919.31 4913.52 3 4920.52 4914.723 4921.73 4914.89 3 4922.42 4915.673 4922.83 4916.08 3 4923.53 4916.213 4923.76 4918.42 3 4924.84 4918.933 4927.06 4919.92 3 4927.26 4921.833 4928.23 4923.66 3 4928.61 4924.263 4929.12 4924.96 3 4929.76 4925.363 I 4930.66 4926.92 | 3 | 4933.46 4927.30

"Experimental results from Careri et al. 1983, 1984, 1998, Careri and Eilbeck 1985, Eilbeck et al.1984, Scott 1990, 1992, 1998, Scott et al. 1985, 1989.^Results of the Davydov model.cResults of the improved model.


which can be seen from (9.101) and (9.103). This is due to the fact that 7 is muchgreater than J, £1, £2 and £3. (3) The local-mode degeneracies of energy levelsappear at higher-lying vibrational states which begin to occur at m > 2. Moreprecisely, there are degeneracies at 3242 cm"1 and 3259 cm"1 for m = 2, and form = 3, there are degeneracies at 4735, 4737, 4738, 4737, 4857, 4859, 4866, 4869,4870, 4877, 4878, 4879, 4887, 4916, 4918, 4924 cm-1. Therefore, the degeneracyincreases with increasing m.

A major advantage of the above method, (9.97) - (9.103) for computing energylevels is that it allows for treatment of more than one amide-I quanta (excitons), andits validity (in the zero dispersion limit) can be directly verified through (9.103). Itis clear from Table 9.1 that theoretically calculated values are in good agreementwith the experimental data obtained from infrared absorption and Raman scatteringexperiments in protein and ACN, and results obtained using the improved theoryare even closer to the experimental values compared with those of Davydov's theory,suggesting that the improved model is more appropriate for the protein molecules.Pang further studied the properties of infrared absorption of the protein molecules,using data given in Table 9.1 (see Pang 1992a, 1993c, 1993d, 1994b, 1997, 2000a,2001d).

9.9 Experimental Evidences of Exciton-Soliton State in MolecularCrystals and Protein Molecules

Energy transport in protein molecules is a very fundamental problem in life science.Because of the form of the exciton-soliton associated with the vibrational change ofthe C=O stretching and deformation of amino acids or peptide groups, existenceand motion of the exciton-soliton can be observed and determined by infrared ab-sorption and Raman spectra, and so on, in ACN and proteins. A lot of experimentswere carried out on ACN, protein, cell and animal tissue to verify the existence ofexciton-solitons in these systems. These experiments include infrared absorptionand Raman scattering in acetanilide and a-helical collagen proteins, specific heat ofacetanilide and protein, measurement of lifetime of thermal pulse or ground-staterecovery time in acetanilide, and infrared emission of human tissue, etc. Based onthese experiments, it can be concluded that the exciton-soliton could exist and isthermally stable at biological temperature 300 K.

9.9.1 Experimental data in acetanilide

In the early 1970s, Careri carried out a novel experimental investigation on proteindynamics. His intention was to circumvent problems arising in the study of realprotein by looking at hydrogen bonded crystals that might be regarded as "modelproteins". The basic idea can be understood by comparing the molecular structureof acetanilide and the structure of a typical polypeptide chain in natural protein as

Nonlinear Quantum-Mechanical Properties of Exdtons and Phonons 537

shown in Fig. 9.3. The similarity of bond lengths and angles of the peptide group(HNCO) and the same peptide chain of hydrogen bond • • • H-N-C=O- • • H-N-C=O-• • • H-N-C=O- • • suggests that the dynamic properties of ACN might provide cluesto the corresponding properties of natural protein molecules. This is also the reasonthat we first discuss the behaviors of ACN.

9.9.1.1 Infrared absorption and Raman spectra

In the measurement of Careri et al, the infrared spectra were obtained using threedifferent infrared spectro-photometers. For the study of temperature dependenceof the amide-I region, a Nicolet model 7000 Fourier-transform infrared spectropho-tometer was used. Spectra were collected for 100 scans using a band width of0.5 cm"1. The sample was thermostated using a closed-cycle helium refrigerator.Far-infrared absorption spectra were measured using a Michelson interferometer(model 720) equipped with a Golay cell. Samples consist of pellets obtained froma mixture of grounded ACN and polyethylene power. Pure polyethylene pelletswere used to measure background transmission. Raman spectra were excited by acoherent radiation model 52 argon ion laser operating at 4880 A or 5145 A, withstabilized output power of 20-200 mW. Incident light was filtered by proper choiceof interference filter and its intensity was monitored using a beam-splitter and asilicon photocell, scattered light was analyzed by a Jarrel-Ash model 25-300 Ramanspectrometer and detected by an ITT model Fw-130 cooled photomultiplier usingphoton counting electronics. They measured the infrared spectra of amide-I (1600-1700 cm"1), amide-II (1500-1600 cm"1), amide-III (1300-1500 cm"1), amide-IV orVI (500-700 cm"1) and amide-V (700-800 cm"1). The infrared spectra of amide-Iin 1600-1700 cm"1 and 2500-3500 cm"1, 4750-4850 cm"1 and 6200-6400 cm"1 areshown in Fig. 9.18. The absorption intensity vs. temperature is shown in Fig. 9.19.The Raman spectra of amide-I in 1630- 1700 cm"1 and low-frequency modes at 300K and 50 K are shown in Fig. 9.20 and Fig. 9.21, respectively.

According to analysis in Section 9.1, the eigenenergy spectra of amide-I are1665 cm"1, 1662 cm"1 and 1659 cm"1, which are assigned to the B2u, Biu, B3u

modes, respectively. The 1650 cm"1 mode in Fig. 9.18(a) and Fig. 9.20 should beassigned to the exciton-soliton. Thus, the exciton-soliton excitation, predicted bythe above theory indeed exists in these systems. From Fig. 9.19, we know that theabsorption intensity decreases in the form of e~^T with increasing temperature.This is consistent with the theoretical result of e" ' c + r T ' given in Section 9.4. Thecomparison between experimental (denoted by "*") and the theoretical values (solidcurve) is shown in Fig. 9.19(b). From Figs. 9.18 - 9.20, we see that the peak valuesof the infrared absorption of ACN agree with the nonlinear eigenenergy spectra ofthe exciton-soliton given in Table 9.1, obtained from the above bioenergy transporttheory. These interesting results provide experimental evidence to the existenceof exciton-soliton in ACN and confirm the validity of above bioenergy-transporttheory.


Fig. 9.18 Infrared absorption spectra of ACN in (a) 1600-1700 cm"1 and (b) 2500-3500 cm"1,(see Careri et al. 1983, 1998, Eilbeck et al. 1984, Scott 1990, 1992, 1998, Scott et al. 1985)

Fig. 9.19 (a) The absorption intensity vs. temperature for ACN, and (b) comparison of experi-mental and theoretical values.

9.9.1.2 Dynamic test of soliton excitation in acetanilide

The experimental measurement of the exciton-soliton dynamics is of interest. It isuseful to resolve the lifetime of the soliton since lifetime and mobility of soliton areimportant criteria for its usefulness as a means of energy transport. Fann et al.measured the relaxation time of the vibrational excitation of the 1650 cm'1 band


Fig. 9.20 Raman spectra in 1630-1700 cm"1 for ACN at 300 K and 50 K, respectively.

Fig. 9.21 Low frequency Raman spectra of ACN at 300 K and 50 K, respectively.

by transient-infrared-bleaching experiments which can set limits on the excitationlifetime. The source required for such measurements must be tunable around 6fim (1650 cm"1) with adequate spectral resolution to selectively excite the band(10 cm"1). In addition, pulses with picosecond duration and sufficient intensity tobleach the transition are necessary to observe the bleaching recovery with adequatetemporal resolution and signal-to-noise ratio.

The source Fann et al. choose to meet these criteria is the Mark free-electron

540 Quantum, Mechanics in Nonlinear Systems

laser, (FEL) which produces pulses in bursts (macropulses) about 1.3 [is in duration.The Mark III laser output is divided into pump (95%) and probe (5%) beams bya CaF2 wedge plate. Both pulses are focused by a single lens to a spot of 300-fj.min diameter on the sample which is held in a closed-cycle refrigerator with CaF2windows. The ACN was purified by repeated sublimation, crystalline domains weregrown by slow cooling from an ACN melt between CaF2 disks of 2mm thickness.A substantial increase in the probe-pulse transmission at the 1650 cm"1 band wasobserved when the pump pulse was simultaneous with the probe pulse. The trans-mission increases due to the pump roughly follows the 1650 cm"1 band shape at0 ps delay (pump and probe temporally overlapped), indicating that the effect isassociated with the 1650 cm"1 band. This show clearly that the exciton-solitonexcitation occurs in such a case in the acetanilide.

Fig. 9.22 Probe-bleaching vs. pump-probe delay at 1650 cm - 1 and 80 K ambient temperature.

Figure 9.22 shows the bleaching recovery dynamics at the band center as assem-bled from many scans. The fast component in Fig. 9.22 is limited by pulse-duration,while the slow component has a decay time of 15 ± 5 ps which can be associatedwith saturation recovery due to repopulation of the ground state of the 1650 cm"1

mode. The relaxation time or lifetime of 15 ± 5 ps indicates that the 1650 cm"1

band is strongly coupled to the lattice, and the 15 ± 5 ps is a typical relaxationvalue for a vibration line in the ACN. Fann et al. also measured the transmissionrecovery of the sample at 100 ps in which the probe decays to the "cold" base line.This is an important fact since it was thought that the 1650 cm"1 band shoulddisappear on deposition of energy into the 130 cm"1 optical phonons. Loss of the1650 cm"1 band due to such a heating should result in an increase in transmissionapproximately by a factor of 2 at 100 ps, which, however, was not observed. There-


fore, the time scale of such vibrational relaxations observed is inconsistent with thelifetime of 10"10 s predicted by the theory of Pang et al..

How could this happen? Further investigation is required in order to clarify thisissue. In practice, if the rate of energy flow into the 130 cm""1 optical phonon ismuch slower than in other materials, or the energy flows of 1650 cm""1 into the150-200 cm"1 optical phonon, instead of 130 cm"1 phonon, the ideal time scale of10""10 s could be observed in this experiments because the soliton-optical phononcoupling can accelerate the cooling speed. Therefore, the above experiment needsto be improved.

It was believed that a soliton should travel a distance of more than 100 aminoacids in its lifetime, as mentioned in Section 9.6. Therefore, it was thought thatthe lifetime of 15 ps for the exciton-soliton obtained by Fann et al. is too short toverify the usefulness of soliton in biological processes because it can only travel adistance of several tens of amino acids at subsonic speed in its lifetime. However, weshould remember that the acetanilide Fann et al. used was at room temperature.According to the result in Section 9.4, the exciton-soliton travels at supersonic speedin such a case, as shown in (9.55). Thus the solution can actually travel at least over150 amino acids during its lifetime of 15 ps. Then we can say that the experimentalresult of Fann et al. also supports the soliton theory of biological transport in ACN.

9.9.2 Infrared and Raman spectra of collagen, E. coli. and humantissue

9.9.2.1 Infrared spectra of collagen proteins

Collagen is a helical protein with three channels which is similar to the a-helixprotein molecules. Recently, Xiao, and Pang et al. measured the infrared absorptionspectra of the collagen. The sample was purchased from Sigma chemical Co. Ltd.and was used without further purification. It was placed between KBr windowsand transferred to a temperature cell in the spectrometer. Infrared spectrum wasrecorded on a Perkin Elmer spectrum GX-FT-IR spectrometer equipped with aDTGS detector. The measurements were performed at a resolution of 4 cm"1 inthe range of 400-4000 cm"1. To obtain an acceptable signal-to-noise ratio, 16 scanswere accumulated. Spectra were recorded in a variable-temperature cell with areported accuracy of ±1°C between 15°C and 95°C in intervals of 10°C.

The results obtained at 25° C in the this infrared absorption experiment areshown in Fig. 9.23. At high frequencies the spectrum is dominated by the amidewhere there are obviously two vibrational modes of amide-1,1666.01 and 1650 cm"1.Other amide bands are found above 1000 cm"1. For example, the amide II occursat 1542.09 cm"1, amide-III at 1455.89 cm"1, 1404.47 cm"1, amide-IV at 1335.8cm"1 and 1243.49 cm"1, and amide-V at 1081 cm"1. There are also rich spectrallines in the range of 2800 - 4000 cm"1. For example, 3209.01 cm"1, 3225.70 cm"1,3244.04 cm"1, 3218.19 cm"1 and 3286.33 cm"1, etc., which were not observedin other experiments. Results of a detailed study of temperature-dependence of


Fig. 9.23 Infrared absorption spectra of collagen proteins from 400 to 4000 cm"1 at 25°C.

absorption intensity of the amide-I region of the collagen in the temperature rangeof 5 - 95°C are reported in Figs. 9.24 and 9.25. We can see from these figuresthat the intensity of the band at 1650 cm"1 increases on cooling without apparentchange in frequency and shape, but it is weaken at 95°C. On the other hand, theamide-I absorption at 1666.11 cm"1 decreases on cooling. The peak intensities ofthe 1650 cm"1 and 1666.1 cm"1 bands as a function of temperature are shown in


Fig. 9.25. Clearly different temperature-dependence of the intensity can be seen forthe two bands. It is surprising that the absorption intensity of the band at 1666.1cm""1 increases linearly with temperature, but the intensity of the band at 1650cm"1 decreases exponentially with temperature, especially in the range of 5 - 45°Cwhich can be approximately expressed as exp[-(0.437 + 8.987 x 10~6T2)]. Thisexponential behavior, exp[—(0.437 + 8.98~6T2)] for the temperature-dependenceof the intensity of the infrared absorption of the band at 1650 cm"1 is especiallyinteresting. Alexander et al, Scott el al. and Pang obtained an exponential decreaseof the intensity of the 1650 cm"1 band with temperature, I oc exp[—(a + 6T2)], ina low temperature range (10-280 K), where T is temperature of the system, a andb are constants, and a linear increase of the intensity of the 1666 cm"1 band withtemperature, by using the soliton theory in ACN and the improved Davydov theoryin the a-helical protein, respectively. Interestingly, the theoretical results basedon the soliton theory for the a-helical protein resemble closely the experimentalresutls of the collagen shown in Fig. 9.25. This serves as an additional experimentalevidence to verify the existence of the nonlinear or soliton excitation in collagen.

Fig. 9.24 The infrared absorption intensities of the collagen in the region of the amide-I mode atdifferent the temperatures, (1) 95°C, (2) 85°C, (3) 75°C, (4) 65°C, (5) 55°C, (6) 45°C, (7) 35°C,(8) 25°C, and (9) 15°C.

The following conclusions can be drawn from these results. (1) Infrared absorp-tion spectra of the collagen are basically similar to that of ACN. (2) The 1666 cm""1,1671 cm""1 and 1650 cm""1 lines are always present in the infrared absorption of bothcollagen and ACN. The 1666 cm"1 or the 1671 cm"1 line is the vibrational frequencyof the amide-I, 1650 cm"1 is its anomalously new band. According to the analyticresults on ACN obtained by Careri et al, Eilbeck et al, and Scott, respectively,given in Section 9.1, we can conclude that the 1666 cm"1 or the 1671"1 spectral linerepresents the vibrational excitation of amide-I (or exciton), but the 1650 cm"1 lineshould be assigned to the soliton excitation in these systems, i.e., there are solitonmotion in collagen. (3) The dependence of intensity of the infrared absorption on


Fig. 9.25 Temperature dependence of strengths of (a) 1650 cm~1 and (b) 1666 cm~1 spectrallines in collagen in the region of 5 — 95° C.

temperature shown in Fig. 9.25 is basically the same as that obtained in ACN andprotein molecules by Alexander and Krumbansl, Scott el al., and Pang based on thesoliton theory. Therefore, results shown in Fig. 9.25 are direct verification of theexistence of soliton in these systems. (4) There are bands at 1680.31 cm"1, 1666cm"1, 1650 cm"1, 1624.94 cm"1, 3209.01 cm"1, 3225.7 cm"1, 3244.04 cm"1, 3262cm"1, 3278 cm'1, 3296.33 cm"1, 3316.2 cm"1, 3333.18 cm"1 and 3355.58 cm""1 inFig. 9.23 for collagen which are basically consistent with those at 1667 cm-1, 1662cm"1, 1653 cm-1, 1627 cm-1, 3204.71 cm"1, 3218.19 cm"1, 3242.48 cm"1, 3261.77cm"1, 3278.89 cm-1, 3298.96 cm-1, 3313.37 cm-1, 3333.91cm"1 and 3358.58 cm"1

in Table 9.1, respectively, obtained by the improved Davydov theory of bio-energytransport by Pang. This is not a coincidence. It shows that there exists solitonexcitation in these systems and the improved theory proposed by Pang correctlydescribe bio-energy transport in a-helical protein molecules.

9.9.2.2 Raman spectrum of collagen

Cai et al. measured the laser-Raman spectrum from acidity-I type fiber collagen.This protein is extracted from lungs of killed mouse with a weight of about 200 g.The samples of acidity-I type is obtained using the Kathryn's method and purified.In the test, they put the purified collagen in the lower part of a microscope ofSpeX1430-type laser-Raman instrument. The width of seam of Argon ion laser isabout 800 fim. The observed room temperature Raman spectrum of the collagenwas excited by a laser with a wavelength of 5145 A and 250 - 800 mW power. Theexperimental result is shown in Fig. 9.26 in the region of 1000 — 1800 cm"1 and2800 — 3000 cm"1, respectively. From this figure, we see clearly that the collagencan emit infrared lights at wavelengths 1670, 1650, 1562 cm"1 and so on, which arebasically consistent with that in ACN in 1600 - 1700 cm"1. As it is known, theline at 1670 cm"1 is an eigenfrequency of amide-I, whereas 1650 cm"1 should be


assigned to the exciton soliton in the collagens. Furthermore, these peak values ofthe Raman spectra approach the theoretical results in Table 9.1. It shows again thecredibility of the theory of bioenergy transport.

Fig. 9.26 Laser Raman spectra of collagen.

Fig. 9.27 Infrared spectra of tissue from a human finger.

9.9.3 Infrared radiation spectrum of human tissue and Ramanspectrum of E. col.

Pang et al. and Chi et al. measured the infrared radiation of tissue of human bodyat room temperature by optical multichannel analyzer (OMA) with multichannelsinfrared probe systems and a single-photon counter method, respectively. It wasfound in the experiments that the intensities of the emitted infrared lights of differ-ent tissues of human bodies are different and decrease in the following order fromthe strongest to the weakest, fingers, palms, cheeks, fore arms, upper arms, chestand abdomen. Fig. 9.27 shows the measured result for a tissue of a human fingerby the OMA and infrared probe systems in the wavelength of 2 - 6 fim. The widthof the seam of the OMA is modulated to 0.5 mm. In this experiment, the spectraof the substrate of the instrument was first measured which was followed by the


radiation spectra of the tissue of a human finger. The finger tissue was placed onthe seam. The environment light was screened during the measurement. The ex-perimental result shown in Fig. 9.27 were obtained by 11 scans, each lasting about0.99 seconds. It can been seen in Fig. 9.27 that human body can radiate infraredlights at wavelengths of 1 - 6 fim. The infrared spectra shown in Fig. 9.27 shouldbe assigned to the vibrations of the protein molecules, instead of chemical reactionor other biomolecules, because it is similar to those in Figs. 9.18, 9.20, 9.23 - 9.26,and that given in Table 9.1. Therefore we can conclude that the energy spectra inTable 9.1 are correct.

Fig. 9.28 Laser-Raman Spectrum of metabolically active E. Coli. courtesy of Webb

In 1980, Webb measured laser-Raman spectrum of the metabolically active E.Coli at low temperature. Results of this measurement are shown in Fig. 9.28, wherenine lines can be clearly seen.

Spectra radiated from protein molecules can also be obtained if the energy spec-tra given in Table 9.1, which is shown in Fig. 9.29 for m = 2, is used. The ninelines observed here are consistent with the experimental data of the laser-Ramanspectrum of E. Coli., given in Fig. 9.28, and of the Raman spectrum of ACN at50 K, given in Fig. 9.21. This again confirms the correctness of the energy spectragiven in Table 9.1, obtained by the theory of bioenergy transport.

Scott suggested that the energy would vary periodically between the three spinsof the a-helix protein with a phase shift of 120°, much like the voltage on a standardthree-phase power line. The frequency of alternation v is related to the transversedipole-dipole coupling energy, L, by v = L/h = 3.85 x 1011 Hz. Considering theovertones and the interactions of the moving soliton with a discrete lattice, a set ofinternal resonances {vi(th)} was suggested. In Webb's experiment on E. Coli, therewas also a set of spectral lines, denoted by {vi(exp)}, due to internal resonancesin the far infrared range between 1011 and 1012 Hz. It was interesting to notice


Fig. 9.29 The distribution of energy-levels for m = 2 and corresponding Raman spectra of pro-teins.

that {i/i(th)} ss {i/j(exp)}. This agreement seemed especially significant because{j/j(exp)} were measured without knowing the existence of the exciton-soliton, andthe theoretical model was constructed with with the knowledge of the molecularstructure of a-helix protein. Shortly thereafter it was suggested that Mie scatteringfrom cell density fluctuations (clumps cells) could provide an explanation for Webb'sobservations. In order to settle this issue an experimental program was organized atthe Los Alamos National Laboratory to measure again the laser Raman scatteringfrom metabolically active cells using the OMA to record the spectra in 1980s. Butthe result is not shown in Fig. 9.28. Therefore, this experimental result given inFig. 9.28 has yet to be reconfirmed.

9.9.4 Specific heat of ACN and protein

Careri et al. measured the specific heat of powdered crystalline ACN from liquidnitrogen to room temperature. The measured data were then fitted to the formula

C(T) = 4.59 x 1(T3T + 1.505 (9.104)

in units of Jg~1k~1, where T is temperature in °C.This temperature dependence of specific heat can be understood from the

exciton-soliton theory of bioenergy transport in proteins and ACN. As a matterof fact, using the Davydov approximation, the quantity W contained in the factor

JB(T,g)= * e"*-2mrg


in (9.46) is given by the following

W « Wnn±1 = £ ( 1 + vq)l3qn±lp;n - vq(3*qn±1pqn + Lq + 1 ) (\/3qn\2 + \(3qn±1\

2)

^\roSoWn?B'F{q,T)1

where

r = 2(xi+X2)2 „, = 7rr-0(xi+X2)2

0 8ro/?(l - s2) Ju;2 ' 4w2/?u2/i(l - «2)'

In the first order approximation, (9.46) can now be written as

d d2

ih—if(x,t) = (e0 - 2J)<p(x,t) - Jr2—ip(x,t) - G'{T)\ip\2y{x,t).

Its soliton solution is of the form

V(x, t)-]j 8J seen ^ 2J^ {x x0 ut)j exp ^ ^ (x - x0) - - y - j ,

where

r / m _ ^2(Xi+X2)2rg [ 1 „,„, T JG (T) " MW oX2(l-S

2) i1 " lB F{q'T)\ •The corresponding soliton energy is given by

Thus the specific heat arising from the motion of the exciton-soliton is

r _dE _ h\Xi+X2)AB'{l-bs*) [ _ 1 1 ^ ( g , T )v~dT~ 48w^2 J ( 1 _ S2)3 [x

4 ^ ^ W ^ ; j rfT •

At high temperature, .K^T 3> fkjq, we have

fi.Vo7T

Thus

C« =X B (a + 6T). (9.105)

where T is the absolute temperature. Using the accepted values of parametersfor ACN, we can obtain a = 5.15, b = 0.0199 K"1. Thus (9.105) is consistentwith experimental data given in (9.104). Meanwhile, Mrevlishvil and Goldanskill etal. measured the linear specific heat of various biopolymers including proteins andDNA. They obtained Cv = KB{a' + b'T), which resembles (9.105). The consistency


between the experimental and the theoretical results for the specific heat againdemonstrates that the above bio-energy transport theory is correct.

In short, from the experimental results in the ACN, collagen, E. Coli. and humantissues discussed above, we can conclude that exciton-soliton, described by thenonlinear Schrodinger equation, can exist, or can be localized in nonlinear systemssuch as ACN and a-helix protein molecules, and that the bioenergy transport theoryand the nonlinear quantum mechanical theory established by Pang is correct.

9.10 Properties of Nonlinear Excitations of Phonons

Prom previous sections in this chapter we know that exciton-soliton or localizationof exciton occurs due to exciton-phonon interaction. In such a case, the phonon isalso localized at the same time and moves as soliton in the systems. This can beverified from (9.11) - (9.12) or (9.60) - (9.63). To do so, we first note that from(9.14) - (9.15) and

, (xi+X2)2h2

* Mv2u2(l-s2)J'

where s = V/VQ, we can get

9 i r , , 2 , 2(x1 + X 2)3ft3 r 4a,2( l-g2)2 /?2

31— \ip{x, t) = 3 2T~3 u ^Ti ; vTu • (9.106)dx[rK »o/?J2(l - s2)wg [ ^2(Xi + Xa)2 J

Substituting (9.106) into (9.11), we obtain

„ ,,2 4(X l + X2)4ft4 [ 4^02(l-S

2)2/32 31

Utt - VOUXX = 4 2 , • 5T \U r^T ; r^~U . (9.107)Muj*r% J2/3(l - s2) [ h2{xi + X2)2 J

This is a ^-equation for u(x,t). It has a soliton solution, i.e., (9.15). Wecan conclude that the phonon has been localized. Obviously, the localization ofthe phonon is due to nonlinear exciton-phonon interaction in ACN and proteinmolecules because (9.107) reduces to a linear wave equation, uu — VQUXX = 0, ifXi = X2 = 0, here the lattice molecule performs harmonic vibration, and the solu-tion, u = j4sin(wi-#o), is simply a plane wave. Localization or nonlinear excitationof the phonons occurs only in nonlinear systems. The phonon is "self-trapped" asa soliton through interaction with the exciton. Its motion is described by the non-linear quantum mechanics.

If the lattice molecule or amino acid molecule is in an anharmonic state, itsHamiltonian is given by (9.18). In such a case, it can be shown that the phononalso moves as a soliton, or, the phonon can also be localized. We will discussproperties of this kind of phonon localization based on (9.18).


Inserting (9.3) into (9.18), we obtain

Hph = Y,hLJ*(ata<i + l ) (9-108)

+ SF(«i.9a)(°«i +°i,1)(a« +ai92)(°(9i+92) +°-(,1+«))'9192

where

x sin f - r o g i J sin f -r0q2 J sin -r o (gi + <?2) • (9.109)

From (9.108) and the phonon part, U\0)ph in (9.5), and the Heisenberg equations(9.7), we can get that

itwcq = hjqaq - ^2 F(k - q){ak + a*_k)(a*k_q + aq-k),k

iM*_q = -Fkjqa*_q + J2F(k~ ? ) K + a-*)("fc-, + <*«-*)•k

From these two equations, we can further get

(a*_, + a,) = -tf(a*_q + aq) + ^ ^ F(k ~ a" + "-,)("*-, + <*«-*)• (9-110)

Because

un(t) =ph (0\U\Rn\U\Q)ph = VlVNj2ui(t)ei9X,9

where (x = nr0) and

^ ) = <2A^>1 / a<a ' + tt-«>'


(«,)« + wj«, = - - ^ r sin I -rog I ^ s i n I -rQk I x

sin firofo - fc)l ufcu,-fce-i(fe-9)ro. (9.111)

Taking into account the dispersion relation, then in the long wavelength and thesecond order approximation

i nr\i i n \ 3 l / i 2 2 \

"<* 2 V M [2roq ~ 3! {2roq) \ = V09 I 1 - 24™ J '


(9.111) can be written as

(«,)« + vlq2uq - — vlrlq4uq = -%-gj^ $ ^ 9 ~ k)kukuq-k.k

In the continuum approximation, applying the following relation,

results in

(Pu _ 2(Pu _ «£r£5S = \rfd_ du 2

dt2 V° dx2 12 dx4 3M dx dx ' K • )

Letting Q(x,t) = —du/dx, then the above equation can be rewritten as

This is a nonlinear equation. It shows again that the phonon is a nonlinear mi-croscopic particle in such a case. Applying the boundary conditions, u(±oo) =Q(±oo) = 0, it has the following soliton solution

«<*,«) = 2 * £ ^ W M W l . {v>v°y {9'114)4 Ai ro \_roy/v

2/v£-l\

Thus

. x 2>M{v2-vlf/2 , [ y/2(x-vt) 1 .u(x,t) = K ° ; tanh V \ 'J, (v>v0) (9.115)

where u is the velocity of the phonon-soliton. The shape of this soliton is similarto the soliton solution (9.15), but (9.115) is a supersonic soliton (v > vo) whichsatisfies the nonlinear equation (9.113), while (9.15) is a subsonic soliton (v < v0),which satisfies the </>4-equation, (9.107). Therefore, the mechanism and propertiesof localization of the phonons in the two cases, (9.107) and (9.113), are obviouslydifferent.

Bibliography

Alexander, D. M. (1985). Phys. Rev. Lett. 60 138.Alexander, D. M. and Krumbansl, J. A. (1986). Phys. Rev. B 33 7172.Atkinson, K. E. (1987). An Introduction to Numerical Analysis, Wiley, New York.Benkui, T. and John, P. B. (1998). Phys. Lett. A 240 282.Bolterauer, H. (1991). in Davydov's soliton revisited: Self-trapping of vibrational energy,

eds. Christiansen, P. L. and Scott, A. C, Plenum, New York, p. 99.Bolterauer, H. and Opper, M. (1991). Z. Phys. B82 95.Brizhik, L. S. and Davydov, A. S. (1969). Phys. Stat. Sol. (b) 36 11.

(9.113)

(9.115)


Brizhik, L. S. and Davydov, A. S. (1983). Phys. Stat. Sol. (b) 115 615.Brown, D. W. (1988). Phys. Rev. A 37 5010.Brown, D. W. and Ivic, Z. (1989). Phys. Rev. B 40 9876.Brown, D. W., Lindenberg, K. and West, B. J. (1986). J. Chem. Phys. 84 1574; Phys.

Rev. Lett. 57 234.Brown, D. W., Lindenberg, K. and West, B. J. (1987). J. Chem. Phys. 87 6700; Phys.

Rev. B 35 6169.Brown, D. W., Lindenberg, K. and West, B. J. (1988). Phys. Rev. B 37 2946.Brown, D. W., West, B. J. and Lindenberg, K. (1986). Phys. Rev. A 33 4104 and 4110.Bullough, R. K. and Caudrey, P. J. (1982). Solitons, Springer-Verlag, New York, pp. 80-

160.Cai, G. P., Chen, L. L. and Yang, Q. N. (1992). Chin. J. Sickness of Labour-health 10

129.Careri, G., Buontempo, U., Caeta, F., Gratton, E. and Scott, A. C. (1983). Phys. Rev.

Lett. 51 304.Careri, G., Buontempo, U., Galluzzi, F., Scott, A. C, Gratton, E. and Shyamsunder, E.

(1984). Phys. Rev. B 30 4689.Careri, G., Gransanti, A. and Ruple, J. A. (1998). Phys. Rev. A 37 2703.Careri, G., Gratton, E. and Shyamsunder, E. (1998). Phys. Rev. A 37 4048.Carr, J. and Eilbeck, J. C. (1985). Phys. Lett. A 109 201.Caspi, S. and Ben-Jacob, E. (2000). Phys. Lett. A 272 124.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1996). Chin. Phys. Lett. 13 660.Chen, X. R., Gou, Q. Q. and Pang, X. F. (1997). Chin. J. Atom. Mol. Phys. 14 393; Chin.



Phys. 11 240; Chin. J. Comput. Phys. 16 346; Commun. Theor. Phys. 31 169.Chen, Y. J., Lin, L. M. and Ma, C. L. (1993). Chin. J. Biophys. 9 673.Chi, X. S., Pang, D. B., Li, Y. M., Yang, H. Y., Ye, D. O. and Zhu, G. X. (1994). Chin.

J. Biophys. 10 163.Christiansen, P. L. and Scott, A. C. (1990). Self-trapping of vibrational energy, Plenum

Press, New York.Cottingham, J. P. and Schweitzer, J. W. (1989). Phys. Rev. Lett. 62 1792.Cruzeiro-Hansson, L. (1992). Phys. Rev. A 45 4111.Cruzeiro-Hansson, L. (1993). Physica 68D 65.Cruzeiro-Hansson, L. (1994). Phys. Rev. Lett. 73 2927.Cruzeiro-Hansson, L. and Kenkre, V. M. (1994). Phys. Lett. A 190 59.Cruzeiro-Hansson, L. and Takeno, S. (1997). Phys. Rev. E 56 894.Cruzeiro-Hansson, L., Christiansen, P. C. and Scott, A. C. (1990). in Davydov's soliton

revisited: Self-trapping of vibrational energy, eds. Christiansen, P. L. and Scott, A.C, Plenum, New York, p. 325.

Cruzeiro, L., Halding, J., Christiansen, P. L., Skovgard, O. and Scott, A. C. (1985). Phys.Rev. A 37 703.

Davydov, A. S. (1973). J. Theor. Biol. 38 559.Davydov, A. S. (1975). Theory of molecular exciton, 2nd ed. Plenum Press, New York.Davydov, A. S. (1977). J. Theor. Biol. 66 379.Davydov, A. S. (1979). Phys. Scr. 20 387.Davydov, A. S. (1980). Zh. Eksp. Theor. Fiz. 78 789; Sov. Phys. JETP 51 397.Davydov, A. S. (1981). Physica 3D 1.


Davydov, A. S. (1982). Sov. Phys. USP. 25 898; Biology and quantum mechanics, Perga-mon, New York.

Davydov, A. S. (1985). Solitons in molecular systems, D. Reidel Publishing, Dordrecht.Davydov, A. S. (1991). in Davydov's soliton revisited: Self-trapping of vibrational energy,

eds. Christiansen, P. L. and Scott, A. C, Plenum, New York, p. 11.Davydov, A. S. and Kislukha, N. I. (1973). Phys. Stat. Sol. (b) 59 465.Davydov, A. S. and Kislukha, N. I. (1976). Phys. Stat. Sol. (b) 75 735.Eilbeck, J. C, Lomdahl, P. S. and Scott, A. C. (1984). Phys. Rev. B 30 4703.Eremko, A. A., Gaididel, Yu. B. and Vaknenko, A. A. (1985). Phys. Stat. Sol. (b) 127

703.Fann, W., Rothberg, L., Roberso, M., Benson, S., Madey, J., Etemad, S. and Austin, R.

(1990). Phys. Rev. Lett. 64 607.Fohlich, H. (1980). Adv. Electron. Electron Phys. 53 86.Fohlich, H.,ei al. (1983). Coherent excitation in biology, Springer, Berlin.Forner, W. (1991). Phys. Rev. A 44 2694; J. Phys.: Condensed Matter 3 1915 and 3235.Forner, W. (1992). J. Phys.: Condensed Matter 4 4333; J. Comput. Chem. 13 275.Forner, W. (1993). J. Phys.: Condensed Matter 5 823, 883, 3883, 3897; Physica 68D 68.Forner, W. (1994). J. Phys.: Condensed Matter 6 9089.Forner, W. (1996). Phys. Rev. B 53 6291.Forner, W. (1998). J. Phys.: Condensed Matter 10 2631.Forner, W. and Ladik, J. (1991). in Davydov's soliton revisited: Self-trapping of vibrational

energy, eds. Christiansen, P. L. and Scott, A. C, Plenum, New York, p. 11.Frushour, B. G.,ei al. (1975). Biopolymers 14 379.Gashi, F., Gashi, R., Stepancic, B. and Zakula, R. (1986). Physica 135A 446.Glanber, R. J. (1963). Phys. Rev. 13 2766.Goldanskill, V. I., Krupyanskii, Yu. F. and Flerov, V. N. (1983). Dokl. Akad. Nauk (SSSR)

272 978.Guo, Bai-lin and Pang Xiao-feng, (1987). Solitons, Chin. Science Press, Beijing.Halding, J. and Lomdahl, P. S. (1987). Phys. Lett. A 124 37.Ho, M. W., Popp, F. A. and Warnke, U. (1994). Bioelectrodynamics and Biocommunica-

tion, World Scientific, Singapore.Hyman, J. M., Mclaughlin, D. W. and Scott, A. C. (1981). Physica D3 23.Ivic, Z. (1998). Physica D113 218.Ivic, Z. and Brown, D. W. (1989). Phys. Rev. Lett. 63 426.Ivic, Z., Sataric, M., Shemsedini, Z. and Zakula, R. (1988). Phys. Scr. 37 546.Kabo, R. (1962). J. Phys. Soc. Japan 17 206.Kenkre, V. M., Raghvan, S. and Cruzeiro-Hansson, L. (1994). Phys. Rev. B 49 9511.Kerr, W. C. and Lomdahl, P. S. (1989). Phys. Rev. B 35 3629.Kerr, W. C. and Lomdahl, P. S. (1991). in Davydov's soliton revisited: Self-trapping of

vibrational energy, eds. Christiansen, P. L. and Scott, A. C, Plenum, New York,p. 23.

Knox, R. S., Maiti, S. and Mu, P. (1990). Search for remote transfer of vibrational energyin proteins, in Davydov's soliton revisited, eds. Christiansen, P. L. and Scott, A. C,Plenum Press, New York, p. 401.

Koeming, J. L. (1972). J. Polym. Sci-PD 59 177.Konev, S. V. (1965). Excited states of biopolymers, Science and technique, Minsk.Lawrence, A. F., McDaniel, J. C, Chang, D. B., Pierce, B. M. and Brirge, R. R. (1986).

Phys. Rev. A 33 1188.Lipkin, H. I. (1973). Quantum mechamics, North-Holland, Amsterdam.Lomdahl, P. S. and Kerr, W. C. (1991). in Davydov's soliton revisited: Self-trapping of


vibrational energy, eds. Christiansen, P. L. and Scott, A. C, Plenum, New York,p. 259.

London, R. (1986). The quantum theory of light, 2nd ed., Oxford University Press, Oxford.Macneil, L. and Scott, A. C. (1984). Phys, Scr. 29 284.Mechtly, B. and Shaw, P. B. (1988). Phys. Rev. B 38 3075.Motschman, H., Forner, W. and Ladik, J. (1989). J. Phys.: Condensed Matter 1 5083.Mouritsen, O. G. (1980). Phys. Rev. B 22 1127.Mrevlislivil, G. M. (1979). Usp. Fiz. Nauk 128 273 [Sov. Phys. USP. 22 433].Nagle, J. F., Mille, M. and Morowitz, H. J. (1980). J. Chem. Phys. 72 3959.Pang, X. F. and Chen, X. R. (2000a). Chin. Phys. 9 108.Pang, X. F. and Chen, X. R. (2000b). Commun. Theor. Phys. 32 437.Pang, X. F. and Chen, X. R. (2001). J. Phys. Chem. Solids 62 793.Pang, X. F. and Chen, X. R. (2002a). Commun. Theor. Phys. 37 715.Pang, X. F. and Chen, X. R. (2002b). Phys. Stat. Sol. (b) 229 1397.Pang, Xiao-feng (1986a). Chin. Acta Biochem. BioPhys. 18 1.Pang, Xiao-feng (1986b). Chin. J. Appl. Math. 10 278.Pang, Xiao-feng (1986c). Chin. J. Atom. Mol. Phys. 4 275.Pang, Xiao-feng (1987). Chin. J. Atom. Mol. Phys. 5 383.Pang, Xiao-feng (1989a). Chin. J. Atom. Mol. Phys. 7 1235.Pang, Xiao-feng (1989b). Chin. J. Low Temp. Supercond. 10 612.Pang, Xiao-feng (1990). J. Phys. Condens. Matter 2 9541.Pang, Xiao-feng (1992a). Chin. J. Light Scattering 4 125.Pang, Xiao-feng (1992b). J. Nature Sin. 15 915.Pang, Xiao-feng (1992c). J. Sichuan Univ. (Nature) Sin. 29 491.Pang, Xiao-feng (1993a). Acta Math. Sci. 13 437.Pang, Xiao-feng (1993b). Acta Phys. Sin. 42 1856.Pang, Xiao-feng (1993c). Chin. J. Biophys. 9 631.Pang, Xiao-feng (1993d). Chin. J. Infrared Millimeter Wave 12 377.Pang, Xiao-feng (1993e). Chin. Phys. 22 612.Pang, Xiao-feng (1993f). Chin. Phys. Lett. 10 381.Pang, Xiao-feng (1993g). Chin. Phys. Lett. 10 437.Pang, Xiao-feng (1993h). Chin. Phys. Lett. 10 517.Pang, Xiao-feng (1993i). Chin. Sci. Bull. 38 1572.Pang, Xiao-feng (1993J). Chin. Sci. Bull. 38 1665.Pang, Xiao-feng (1993k). J. Sichuan Univ. (Nature) Sin. 30 48.Pang, Xiao-feng (1994a). Acta Phys. Sin. 43 1987.Pang, Xiao-feng (1994b). Chin. J. Biophys. 10 133.Pang, Xiao-feng (1994c). Phys. Rev. E 49 4747.Pang, Xiao-feng (1994d). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing.Pang, Xiao-feng (1995). Chin. J. Phys. Chem. 12 1102.Pang, Xiao-feng (1996). Acta Math. Sci. (Suppl.) 16 1.Pang, Xiao-feng (1997). Chin. J. Infrared Millimeter Wave 16 288.Pang, Xiao-feng (1999). European Phys. J. B 10 415.Pang, Xiao-feng (2000a). Chin. Phys. 9 86 and 108.Pang, Xiao-feng (2000c). J. Phys. Condens. Matter 12 885.Pang, Xiao-feng (2001a). Commun. Theor. Phys. 33 323.Pang, Xiao-feng (2001b). Commun. Theor. Phys. 35 763.Pang, Xiao-feng (2001c). European Phys. J. B 19 297.Pang, Xiao-feng (2001d). Inter. J. Infr. Mill. Waves 22 277 and 291.


Pang, Xiao-feng (2001e). Physica D 154 138.Pang, Xiao-feng (2001f). Phys. Rev. E 62 6989.Pang, Xiao-feng (2001g). Phys. Sin. 28 143.Pang, X. F. (Pang Xiao-feng), (2002). Chin. J. Atom. Mol. 19 319 and 417.Pang, Xiao-feng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu.Pang, Xiao-feng (2004a). Commun. Theor. Phys. 41 165 and 561.Pang, Xiao-feng (2004b). Commun. Theor. Phys. 42 458.Pope, M. and Swenberg, C. E. (1982). Electronic Processes in Organic Crystals, Clarendon

Press/Oxford University Press, Oxford/New York, 1982.Popp, F. A., Li, K. H. and Gu, Q. (1993). Recent advances in biophoton research and its

application, World Scientific, Singapore.Potter, J. (1970). Quantum mechamics, North-Holland, Amsterdam.Sataric, M. and Zakula, R. (1984). II Nuovo Cimento D3 1053.Sataric, M., Ivic, Z. and Zakula, R. (1986). Phys. Scr. 34 283.Schweitzer, J. W. (1992). Phys. Rev. A 45 8914.Schweitzer, J. W. and Cottingham, J. P. (1991). in Davydov's soliton revisited: Self-

trapping of vibrational energy, eds. Christiansen, P. L. and Scott, A. C , Plenum,New York, p. 285.

Scott, A. C. (1982). Phys. Rev. A 26 578.Scott, A. C. (1982). Phys. Scr. 25 651.Scott, A. C. (1983). Phys. Rev. A 27 2767.Scott, A. C. (1984). Phys. Scr. 29 279.Scott, A. C. (1990). Physica D 51 333.Scott, A. C. (1992). Phys. Rep. 217 67.Scott, A. C. (1998). Phys. Lett. A 86 603.Scott, A. C , Bigio, I. J. and Johnston, C. T. (1989). Phys. Rev. B 39 12883.Scott, A. C , Gratton, E., Shyamsunder, E. and Careri, G. (1985). Phys. Rev. B 32 5551.Silinsh, E. A. and Capek, V. (1994). Organic Molecular Crystals, AIP Press, New York.Skrinjar, M. J., Kapor, D. W. and Stojanovic, S. D. (1988). Phys. Lett. A 133 489.Skrinjar, M. J., Kapor, D. W. and Stojanovic, S. D. (1988). Phys. Rev. A 38 6402.Skrinjar, M. J., Kapor, D. W. and Stojanovic, S. D. (1988). Phys. Scr. 39 658.Skrinjar, M. J., Kapor, D. W. and Stojanovic, S. D. (1989). Phys. Rev. B 40 1984.Spatschek, K. H. and Mertens, F. G. (1994). Nonlinear coherent structures in physics and

Biology, Plenum Press, New York.Stiefel, J. (1965). Einfuhrung in die Numerische Mathematik, Teubner Verlag, Stuttgart.Takeno, S. (1985). Prog. Theor. Phys. 73 853.Takeno, S. (1986). Prog. Theor. Phys. 71 395.Takeno, S. (1986). Prog. Theor. Phys. 75 1.Takeno, S. (1991). in Davydov's soliton revisited: Self-trapping of vibrational energy, eds.

Christiansen, P. L. and Scott, A. C , Plenum, New York, p. 56.Tan, Benkui and Boyd, J. P. (1998). Phys. Lett. A 240 282.Tekec, J., Ivic, Z. and Priulj, Z. (1998). J Phys.: Condensed Matter 10 1487.Vaconcellos, A. R. and Luzzi, R. (1993). Phys. Rev. E 48 2246.Venzel, G. and Fischer, S. F. (1984). J. Phys. Chem. 81 6090.Vladiminov, Y. A. (1965). Photochemistry and luminescence of protein, Science, Moskova.Wanger, J. and Kongeter, A. (1989). J. Chem. Phys. 91 3036.Wang, X., Brown, D. W., Lindenberg, K. (1988). Phys. Rev. A 37 3357.Wang, X., Brown, D. W., Lindenberg, K. (1989). J. Mol. Liq. 4 123.Wang, X., Brown, D. W., Lindenberg, K. (1989). Phys. Rev. B 39 5366.Wang, X., Brown, D. W., Lindenberg, K. (1989). Phys. Rev. Lett. 62 1792.


Wang, X., Brown, D. W., Lindenberg, K. (1991). in Davydov's soliton revisited: Self-trapping of vibrational energy, eds. Christiansen, P. L. and Scott, A. C , Plenum,New York, p. 83.

Webb, S. J. (1980). Phys. Rep. 60 201.Webb, S. J. (1985). The crystal properties of living cells ans seen by millinmeter microwaves

and Raman spectroscopy, in the living state-II, ed. Mishra, R. K., World Scientific,Singapore, pp. 367-403.

Xiao, He-Lan, Cai Guo-ping, Sun Su-qin and Pang Xiao-feng (2003). Chin. Atom. Mol.Phys. 20 211.

Xie, A., van der Meer, A. F. G. and Austin, R. H. (2002). Phys. Rev. Lett. 88 018102.Young, E., Shaw, P. B. and Whitfield, G. (1979). Phys. Rev. B 19 1225.Zekovic, S. and Ivic, Z. (1999). Bioclectrochem. Bioenergetics 48 297.

Chapter 10

Properties of Nonlinear Excitations andMotions of Protons, Polarons and

Magnons in Different Systems

In this chapter, we continue to discuss properties of excitations and motions of threetypes of microparticles, proton, polaron and magnon, in various nonlinear systems.We will see that their motions also obey nonlinear quantum mechanics.

10.1 Model of Excitation and Proton Transfer in Hydrogen-bonded Systems

There are many examples of hydrogen-bonded systems, which consist of a series ofhydrogen bonds, in condensed matters and living systems, such as ice, solid alcohol,carbon hydrates and proteins. These systems exhibit a considerably large electricalconductivity even though electron transport through the systems is hardly sup-ported. Through long period of studies, we have known now that this phenomenonis caused by proton transfer in these systems. However, understanding of protontransfer in such systems is a long-standing problem. Nonlinear dynamics and solitonmotion provide a possibility to resolve this issue. In view of their close connectionwith phenomena related to proton transfer across biological membranes, study ofsuch systems becomes even more important and is expected to provide insights tosome fundamental processes of life.

In the studies of proton transfer processes in hydrogen-bonded systems, it issuffice to consider one-dimensional chains which are referred to as Bernal-Fowlerfilaments. In the normal state of a chain, each proton is linked to a heavy ion(or oxygen atom in ice) by a covalent bond on one side, and a hydrogen bond onthe other. Therefore, there can be two types of arrangements of hydrogen bondedstates in these systems, namely the X—H- • • X—H- • • X—H- • • X—H- • • type and theH-X- • • H-X- • • H-X- • • H-X- • • H-X- • • type. Obviously the two systems shouldhave the same energy. The potential experienced by the proton in such a system canbe modeled by a double-well potential in which the two minima correspond to thetwo equilibrium positions of the proton between the two neighboring heavy ions (oroxygen atoms), as shown in Fig. 10.1. The barrier which separates the minima hasa height which is in general of the order of the oscillation energy in a covalent X-H

557


bond and is approximately 20 times larger than that in a hydrogen bond. Smallamplitude harmonic vibration about their equilibrium positions is assumed for theprotons in the hydrogen bonds.

Fig. 10.1 The double-well potential in the hydrogen-bonded system with H+ in one well (a) orthe other (b) of the potential.

If the protons in such systems are perturbed by an externally applied field suchas light or energy released by adenosine triphosphate (ATP) hydrolysis in proteinmolecules, then localized fluctuations of the protons occur, which result in changesin positions of the protons. Protons move and deviate from their equilibrium posi-tions, for instance, by a translation, a jump, a shift, or by hopping in the interbondsand intrabonds. This phenomenon of proton transfer along the hydrogen-bondedchains was observed experimentally, and the protonic conductivity along the chainswas found to be about 103 — 104 times larger than that in the perpendicular direc-tion. The motion of the protons may result in ionic and bonding (orientational orBjerrum) defects which correspond to the exchange and rotation of bonds, respec-tively. Thus, transfer of protons along the hydrogen-bonded chain is a result of thetransport of the two types of defects as shown in Figs. 10.2 and 10.3, respectively.It is possible that protons are transferred by jumping from one water molecule toanother along the hydrogen-bonded chain, and that a migration of hydroxoniumand hydroxyl ionic defects takes place in the intrabonds. When a proton movesfrom one end of a molecule to the other end, it may form a covalent bond with themolecule, for example, with an oxygen atom in ice, and the original proton moves toa neighboring molecule. Repetition of this process results in a continued motion ofthe proton along the chain. However, proton transfer cannot occur in one direction,but can be achieved with a re-orientation of OH groups by the second defect mech-anism, the Bjerrum defect. The motion of an orientation defect contains simplesuccessive rotations of OH groups, starting at one end of the chain and ending atthe other. As a result of these rotations, a pair of D and L defects (see Fig. 10.3)

Nonlinear Excitations and Motions of Photons, Polarons and Magnons 559

can be created and they can move to different ends in any internal part of the chain.A sequence of rotations of all of the molecules in a filament returns the chain to itsoriginal state. It follows that the motion of another proton can occur only after thepassage of a Bjerrum defect.

Fig. 10.2 Two ionic defects in a hydrogen-bonded system (ice).

Fig. 10.3 The two Bjerrum defects in a hydrogen-bonded system (ice).

Potential models of the positive ionic and bonding defects in such systems areshown in Fig. 10.4 (a) and (b), respectively.

The soliton excitation model of proton transfer was first proposed by An-tonchenko, Davydov and Zolotaryuk (the ADZ model) for crystal ice (Antonchenko


Fig. 10.4 The model with double-well potential curves for the two types of defect.

et al. 1983). The Hamiltonian of the system is given by

H = Hp + HOH + Hint

= E { ? KRn)t + <"l(*n+l ~ tf»)2] + U(Rn)} (10.1)n

+ E f [(««)? + n0< + nl(un+1 - u,,)2]n

n

where the double-well potential for the proton, due to the neighboring OH" ions,is given by

W.) = tfo (l - f | ) ,

with UQ being the height of the barrier of the double-well potential, RQ the distancebetween the local maximum and one of the minima of the double-well potential,and Rn the displacement of proton measured from the top of the barrier. The cor-responding conjugate momentum is pn = m{Rn)t. In (10.1), un and Pn — M (un)t

are displacement of the OH~ ion, and its conjugate momentum, respectively, Mand m are the masses of OH~ and proton, respectively, u>i is vibrational frequencyof the proton, fi0 and fti are characteristic frequencies of the OH~, x *s a couplingcoefficient between the proton and vibration of the OH~.

In the continuum approximation, and using the Hamilton equation in the non-linear quantum mechanics, we can get the following equations of motion for theproton and OH~ ion from (10.1),

Rtt - C20Rxx - wl ( I - ^ ) R + 2ÛR - 0, (10.2)

utt - n20u - v{uxx + j^(tf-Rl)= 0, (10.3)

respectively. Obviously, transfer of the protons and ionic motion are nonlinear


problems. In the case of x — 0, the following analytic solution can be obtained.

i? = T-Rotanh ^ i , / i= ^-27; 5\>

where s = v/Co, Co = u>ir0, and ro is the lattice constant. When v = vi, equations(10.2) - (10.3) have a soliton solution which is given by

D o . ,\Ho{x-XO-Vt)] 2 \llo(x-XO ~Vt)]R = zpRo tanh £-— , u = u0 sech !-—^ ,

L r o J L ro J

where

, - n ( R \ - I * (2U° X2R2Avi XRI n

If ^ ^ 0, and v ^ v, equations (10.2) - (10.3) do not have analytic solution. Thesolutions can only be found using approximation method or numerically.

Further investigations on the ADZ model lead to solutions for a far greater rangeof velocity values. The problem was addressed in a number of publications and avariety of theoretical extensions, including the one-component protonic chain witha new two-parameter, double periodic, one-site potential proposed by Pnevmatikoset al., have been developed. Shortcoming of the ADZ model is that the couplingbetween the protons and oxygen atoms (or heavy ions) is only one mechanism ofreducing the potential barrier, which protons have to overcome to move from onemolecule to another. This is included in the ADZ model by coupling the protonmotion with an optical mode of the heavy ionic sublattice. The nature of the pro-ton sublattice depends on the systems to be studied. In this model, an ionic defectappears as a kink or solitary wave in the proton sublattice, propagating togetherwith a localized contraction of the relative distance between neighboring oxygens.This excitation is referred to as a two-component solitary wave. In the ADZ model,the proton potential with the double-well ansatz plays an essential role in the de-scription of the motion of ion defects as topological solitary waves. The nonlinearinteraction generated by the coupling between the protons and oxygen atoms playsonly a secondary role by reducing the height of the barrier. Therefore, the prop-erties of the solitons are mainly determined by the double-well potential. Thus,this model is only effective for explaining the transfer of ion defects. The equationsinvolved are also very difficult to solve, and exact analytical solutions cannot beobtained. Furthermore, if realistic values for the parameters of hydrogen-bondedsystems are considered, the continuum approximation fails due to the narrowing ofthe domains of validity of the solutions with respect to the lattice spacing. This isthe case in ice in which the H3O+ or OH~ ions become almost point defects. Itis also very difficult to accept the one-component model proposed by Phevmatikos


et al. because the influence of the heavy ionic sublattice on the protons in such amodel was not considered in details. Therefore, the proton transfer in the hydrogen-bonded systems is still an open problem. A complete theoretical description of thecombined effect of the transfers of both types of defects had not been possible untila model was proposed by Pang and Miller.

Fig. 10.5 The one-dimensional lattice model for a hydrogen-bonded quasi-diatomic chain.

Pang and Miller proposed a model to study the dynamic properties of protontransfer resulting from the localized fluctuations of protons and structural deforma-tion of the heavy ionic sublattice due to the displacement of the protons. In thismodel, it is assumed that the hydrogen-bonded chain consists of two interactingsublattices of harmonically coupled protons of mass m and heavy ions (hydroxylgroups of ice, or complex negative ions, of mass M, as shown in Fig. 10.5. Eachproton lies between a pair of heavy ions, usually referred to as 'oxygens'. Theproton is connected by one covalent bond and one hydrogen bond to the two neigh-boring oxygens. Therefore, the potential energy of the proton in each hydrogenbond has the form of a double-well potential with the two minima correspondingto the two equilibrium positions of the proton. Obviously the double-well potentialis motivated, physically, by the simultaneous electromagnetic interaction of the twoneighboring oxygens with the proton.

If the proton can overcome the central barrier of the double-well potential andcan move from one well to the other, the relative positions of the proton and the twoneighboring oxygens have changed, and the positions of the covalent and hydrogenbonds are exchanged. Thus, ionic defects occur in the system. In such a case, theposition of the proton in the hydrogen bond is mainly determined by the double-wellpotential. The proton displacement is controlled by the elastic interaction involvedin the model. However, when the proton approaches the neighboring oxygen thecoupling interaction between the proton and the oxygen will be greatly enhanced inthe intrinsically nonlinear system. Thus, the relative position between the protonand the oxygen will also be significantly changed, i.e. the migration of the proton aswell as the deformation of the heavy ionic sublattice by stretching and compression

Nonlinear Excitations and Motions of Photons, Potarons and Magnons 563

are enhanced. This phenomenon may result in change of direction of the covalentbond between the proton and oxygen, i.e. rotation of the bond. In fact, as far asthe covalent bond is concerned, bond rotation in chemistry is simply carried outby relative displacements of the proton and the oxygens with charges. Thus theBjerrum defect occurs due to the coupling interaction. Therefore, the mechanismfor formation of this defect is different from that of the ionic defect mentioned above,although both are produced by changes in the relative positions of the protons andoxygens.

An electromagnetic interaction between neighboring protons, for example, thedipole-dipole interaction and resonant interaction, was included in the Pang-Millermodel, besides the above double-well potential and the elastic interaction due tothe covalent interaction and related actions. Thus, it is also natural to take intoaccount the changes in the relative positions of the neighboring heavy ions resultingfrom this interaction. Assuming again the harmonic model with acoustic vibrationsof low frequency for the heavy ionic sublattice, the Hamiltonian of system can bewritten as

H — Hp + Hion + H\nt

= E f n + \m"lRl ~ \rn^RnRn+1 + [/(#„)]

+ E [ H + ^ ( U " - U » - I ) 2 1 (IO-4)

+ V) xXl™ («n+l ~ Un-l) R2n + ™X2 ("n+1 ~ «n) RnRn+l ,

n ^ Jwhere

CTdW = (4 [ l - ( & ) * ] ' .

In (10.4), Rn and pn = m(Rn)t are the proton displacements and momenta, re-spectively, with Rn being defined relative to the mid-point between the nth andthe (n + \)th heavy ions or OH's in the static case. RQ is the distance between thecentral maximum and one of the minima of the double well, UQ is the height of thepotential barrier. Similarly, un and Pn = M(un)t are the displacement of the nthheavy ion from its equilibrium position and its conjugate momentum, respectively.Furthermore, xi = du^/du and \2 = dujf/du are coupling constants between theprotons and the heavy ionic sublattice, representing the changes in the vibrationalenergy of the protons and of the coupling energy between neighboring protons dueto a unit extension of the heavy ion sublattice, respectively, LOQ is the frequency ofharmonic vibration of the proton. The quantity (l/2)mcj^RnRn+i is the correlationinteraction between neighboring protons caused by the dipole-dipole interactions.LJQ and uji are the diagonal and off-diagonal elements of the dynamical matrix of the


proton, respectively, ft is the linear elastic constant of the heavy ionic sublattice.m and M are the masses of the proton and heavy ion, respectively. Hp in (10.4)is the Hamiltonian of the protonic sublattice with an on-site double-well potentialU(Rn), Hion is the Hamiltonian of the heavy ionic sublattice with low-frequencyharmonic vibration and Hmt is the interaction Hamiltonian between the protonicand the heavy ionic sublattices.

The Pang-Miller model is still based on coupling of two oscillators (proton andheavy ion), and in that sense, it is similar to the ADZ model. However, it differs fromthe ADZ model in the following, (i) Due to the large mass and the large numberof atoms or atomic groups, motion of the heavy ion is simply harmonic, with low-frequency acoustic vibration. In contrast, both acoustic and optical vibrations areallowed for heavy ions in the ADZ model, but the physical origins of this modelare rather vague. Generally, it is believed that optical and acoustic vibrationsare two different forms of vibration in nature. Therefore, the Pang-Miller modelfor heavy ion is more appropriate than the ADZ model, (ii) For protons lying inthe double-well potential, Pang and Miller adopted a harmonic oscillator modelwith optical vibration, but it includes an off-diagonal factor, which comes fromthe interaction between neighboring protons, and interaction with the heavy ions.Thus the vibrational frequencies of the protons are related to displacements ofthe heavy ioris. Therefore, the Hamiltonian in the Pang-Miller model has highsymmetry and a one-to-one corresponding relation for these interactions. However,in the ADZ model, the vibration of the proton is acoustic. This is not reasonablebecause the vibration frequency of the proton is very high compared to the heavyion due to its small mass and strong interaction. Therefore the Pang-Miller modelis more appropriate than the ADZ model. Moreover, the relation between theprotonic and interactional Hamiltonians in the ADZ model does not have a one-to-one correspondence, the physical meaning of the interaction Hamiltonian is alsovery vague or difficult to understand. It does not have a strict analytic solutionand therefore it is difficult to investigate the properties and mechanism of protontransfer in the systems in the ADZ model. Because the Hamiltonian in the Pang-Miller model not only includes the optical vibration of the protons, but also theresonant interaction between the protons caused by the electromagnetic interactionsbetween neighboring protons, and it also takes into account both the change of therelative displacement of the neighboring heavy ions resulting from the vibration ofthe proton and the correlation interaction between the neighboring protons, we canexpect that the Pang-Miller model can reveal some new results compared to theADZ model or other models.

10.2 Theory of Proton Transferring in Hydrogen Bonded Systems

When we use quantum theory to study the transfer of the protons, un, Pn andRn, pn should be regarded as operators. Following (9.2), we make a standard


transformation in the second quantization representation

l2muj0 , jhmuo ,fln = y—r— (an + aZ), pn = \]—^-(-i){an-aZ), (10.5)

V li y £

where i = ^/—l, a* (an) is the creation (annihilation) operator of the proton.Equation (10.4) becomes

H = £ |ôa+an - I - ^ ( " n + i - un)\ « + i + an+1)(at + an)+

f j |K+ i - »»-0(«i + »»)2 + [y»« + f («»« - »J2]} •Prom (10.6), we see that the motion of proton is a nonlinear problem because thereis a term in the fourth power of the operator a+ (or an) arising from the double-wellpotential and coupling interaction between proton and vibrational quantum of theheavy ions. Therefore, the characteristics of the proton is changed when comparedwith a bare proton. Because here we do not consider the spin of the proton, the wavefunction describing the collective excitations arising from the localized fluctuationof the protons and the structural deformation of the heavy ionic sublattice can bewritten as

I*) = \<P)\P) = j ; 11 + ^2<Pn(t)a+ |<V xL n

eXP I E Jh [ A l W P n " ^{t)Rn] \ | 0 U (10.7)

where |0)pr and \0)ph are the ground states of the proton and the vibrational exci-tation of the heavy ionic sublattice (phonon), respectively. <pn{t), Pn{t) = ($]Rn\$)and nn(t) = (<J>|Pn|$) are three sets of unknown functions. We assume hereafterthat A' = 1 for convenience of calculation except when explicitly mentioned. Thewave function of the proton,

\<P) = j ; l + 5>«(t)<£ |0)pr,L n

is not an excitation state of a single particle, but rather a coherent state, andcontains only one proton. From (10.6) and (9.59) we obtained

(Pn)tt(t) = /?(£„+! + /?n-l - 2/?n) + ^ ( | n - -uKlfn+i + V?n_i) + Xl(£n+1 - /8n_1) ^ <£„2 L w-ô J

+X2[(/?n+l - /?n)¥>n+l + (fti ~ Z^n-l^n-l]8fiJ70 . ., 6ftC/0

+ ^ 1 ^ 2 ^ l ^ + ^ ^ 2 ^ - (10-9)

In the continuum and long-wave length approximations, we can obtain the followingfrom (10.8) and (10.9).

<Ptt = e'ip + -ufrfoxx - 2(xi + X2)ro0x(x,t)«)), (IO.IO)

Afd2P(x,t) 2d

2p(x,t) (Xi + X2) d\<p(x,t)\*M Qt2 = Pr0 QX2 + Hr° — fa = f2(<P,P(x,t)), (10.11)

where ro is lattice constant of the heavy ion sublattice and

Obviously, the equations of motion for the proton and heavy ion of the systems ob-tained in the second quantization representation, equations (10.10) are </>4-equations.It is thus a nonlinear quantum mechanical problem and its motion can be describedby the nonlinear quantum mechanics.

Assuming that £ = x - vt, again from (10.10) and (10.11), we can get

ft-j&a-'X1**-"1'^ ( m 2 )

<ptt - £V + w? fxx ~ # M V , (10.13)

with

r _ , _9AXi±X2_ hUp 2rg(Xi+X2)2

l A ro ' 9~m^cJ0 MCg(l-8*)wo'

fi2 = !"?r2, s = £-, C0=r0^, (10.14)

where A is an integral constant, Co is the sound velocity in the heavy ionic sublat-tice. From (10.12) and (10.13), we see clearly that the proton moves in the formof soliton in the nonlinear systems, and there are two nonlinear interactions in thismodel, the double-well potential and the coupling between the proton and the heavyion. The competition and balance between the two nonlinear interactions can resultin two different kinds of soliton solutions, corresponding to two different kinds of de-fects (ionic and bonded defects) in the systems. This competition between the twointeractions is mainly controlled by the coupling interaction between the protons


and heavy ion, (xi + X2)) and proton velocity, v. When the coupling is weak, i.e.,when the distance between the proton and the heavy ion is large, the double-wellpotential dominates, namely, g > 0, e > 0. In this case, the soliton solutions of(10.12) and (10.13) at 0 < v < vi and v < Co can be written as

ip{x,t) = ±. /^tanhC, (10.15)

where

Equation (10.16) can be written as

0(x,t) = B<p(x,t), (10.17)

where

B= v^(Xi + X2)hr0 ivl - v2

MC02(l-s2)w0 V 9 '

When g < 0 and e < 0, 0 < v\ < v, or v > Co, the solutions of (10.13) are stillgiven by (10.15) and (10.16).

On the other hand, if the coupling interaction dominates relative to the double-well potential, i.e., g < 0, e < 0 and 0 < v < vi, v < Co, the solutions of (10.12)and (10.13) are

¥>0M) = T , ffisechC', (10.18)

V m" • • " ^ ^ ( • - S h ^ 1 " ^ " ' " ' 1 ^ ' ( m 9 )

where

When g < 0, s < 0 and Vi < v, or Co < v, solutions of (10.12) and (10.13) are stillgiven by (10.18) and (10.19). Therefore, there are different solutions for differentparameters. This indicates that properties of the protons depend mainly on thecoupling interaction between the proton and the heavy ions and on the velocity ofthe proton. Different (xi + X2) and v can lead to different values of e and g. Thus,we have different forms and properties of soliton solutions in different systems andfor different states of the proton.

(10.16)


For crystal ice, typical values of various physical parameters are r0 = 2.67 A,Ro = 1 A, Uo = 0.22 eV, X = hXl/2u0 = 0.10 eV/A, X' = hX2/2uj0 = 0.011 eV/A,Co = 2xl04m/s,t ; i = (7-9.5) x 103 m/s, m = mp, M = 17mp, w0 = (1-1.5) xlO14

s"1, wi = (4 - 5) x 1013 s"1. Using these values, the amplitude of the soliton in(10.15) for ice can be calculated and the result is

— {[2(^-o^)(l-^)-^1+X2),o][ 8hU0 2firo

2(xi+x2)2rM1/2 n q

[m2Rluj0 MCo2(l-S2)WoJ / ~u"y-

Its width is given by

= f 2{v\ - v2) 1 1 / 2

k ~ n { u$ - u20 + (4U0/mRl)(l - Zh/mBZuo) ~ 2A(Xi + X2H J ^ 8 " l r ° '

where

A ô(Xi+X2) IT

M C o2 ( l - S > o V f f '

This is an indication that the continuous approximation used in the calculation isappropriate to the ice.

To further understand the behaviors of the soliton solutions, we need to studycarefully the effective potential U((p) of the systems corresponding to (10.12) and(10.13), because the properties of the protons are determined by the effective po-tential,

U(<fi) = (-\e<p* + \gf^j + Uo. (10.20)

Obviously, the effective potential consists of a double-well potential and a nonlinearcoupling interaction between the proton and heavy ion, and depends on g and e.

(I) If £ > 0 and g > 0, the double-well potential plays a main role in determiningproperties of the protons. The protons and heavy ions become kink-antikink pairs,(10.15) - (10.16), which can cross over the intrabond barrier, to move from onewell to the other of the double-well potential. The coupling interaction between theproton and heavy ion is only of secondary importance, which only reduces the heightof the barrier, to allow the proton to easily cross over. This can be understood fromthe following. Consider the case of Xi = X2 — 0. The values of e and g may change,but as long as e > 0 and g > 0, it does not change the nature of the system. In thiscase, the effective potential has two degenerate minima

Umm = -^- + U0 (10.21)


at

£o(0 = ¥>min(0 = ±J->

respectively. Therefore, C/min and its "location", <A)(£), t n e barrier height, UQ, ofthe effective potential depend on g/e, or, on (xi + X2), v0, u>i, v and UQ. WhenXi = Xi - 0 a n d wi = wo = 0, £/min -4 0, <po(Q - <p(Ro). In such a case, theeffective potential of the system and U((p) in (10.20) consist of only the double-wellpotential, U(Rn) in (10.4), with U(ip) given by

TT, . 2hU0 A 3/J \ 2 2h2UQ 4 TT

which is not exactly the same as U(Rn) given in (10.4) because the coefficients aredifferent from those in U(Rn), due to quantum effect and the approximate methodused in the above derivation.

We can see from (10.20) and (10.21) that the <£o(0 increases and UQ decreaseswith increasing e and decreasing g, i.e., with increasing coupling constants. Thus,the height of the barrier decreases and the positions of the minima of the potential-wells become further separated in such a case. This means that the possibility forthe protons to cross over the barrier is increased with increasing coupling constants.The decrease of the barrier height, AC/o = UO—UQ, and the change in the equilibrium"position", A<£o(£) = <p(Ro) — <?o(f)i c a n be approximately given by

AC/o « Uo [ | - 1 + (22 - z2)(y + 1) - y] > 0,

A0o(0 ~ [ | ( l + | ) - | ] <P(Bc) > 0, (10.22)

respectively, where

^ ( i ? o ) - V - 8 r ~ ' y - 4J70MCg(l-r») > 0 '

Equation (10.22) shows that the values of the minima of the potential changes fromzero at <po(O = vCô) to negative at ^(f) > (fo{Q, *-e-> the larger the (\i + X2),the smaller <A>(£)I the lower the height of the barrier, and the more negative thevalues of the minima of the potential energy. Thus the possibility for the protons tojump over the barrier is enhanced greatly. It clearly shows that the solitons, givenby (10.15) and (10.16), in the case of e > 0 and g > 0 describes exactly the motionof the proton crossing over the barrier of the interbond double-well potential tojump from one molecule to another, resulting in ionic defects in the systems. Themotions of proton-kinks are accompanied by compression or rarefaction of the heavyion sublattice around the ionic defects. The physical meaning of the soliton <p(x, t)


with the plus sign in (10.15) is a localized reduction in the protonic density, (i.e.,expansion of the proton sublattice), arising from the motion of the kink-soliton.This effect leads to the creation of a negatively charged carrier and an extendedionic defect moving with a velocity v which is less than the speed of sound CQ.Therefore, the soliton corresponds to the OH~ ionic defect which appears in theBernal-Fowler filaments. The other soliton solution with the minus sign in ip(x, t)in (10.15) represents compression of the protonic sublattice and an increase of thelocalized proton density which leads to the creation of a positively charged carrierand an extended ionic defect. It corresponds to the HsO+ ionic defect. The solutions(10.15) and (10.16) thus represent proton transfer in the form of interbond ionicdefects accompanied by a localized deformation of the heavy ionic sublattice. Thesoliton is referred to as a kink I soliton.

It is also obvious that C/min and <£o(£) in (10.21) decrease with increasing velocityof the proton. It implies that the proton will be further separated from the originalheavy ions when its velocity is increased.

(II) In the case of e < 0 and g < 0 which corresponds to the soliton solutions(10.18) and (10.19), the coupling interaction between the protons and heavy ionsplays the main role in determining the properties of the protons. The protonsbecome another kind of soli tons which shift over the intrabond barriers by the quasi-self-trapping mechanism. In this case, the double-well potential plays a minor rolebecause the soliton solutions of (10.10) and (10.11) are still given by (10.18)-(10.19)when Uo = 0. The above effective potential is still twofold degenerate in such a caseand its minima are

Untn-y-i-Uo- 8 ^ ( X l + » ) a ( 1 " Z ) ( 1 + tf)"%> ( }

(here A = 0), located at

rt(0 = ±^«±ri(l-^) (! + !•), (10-24)

where

, _ /MC 02 ( l -s>3

*° V 2ro(Xi+X2)2 '

' - 4U° (, 3fi ^ _ w £ , _ 1 nZ ~ mulRl V ~ 2mX>uJ ug > °' V ~ y > "'

Prom (10.23) and (10.24), it is obvious that when (xi + X2) and v increase, $,(£)decreases. Therefore, strong coupling interaction and higher velocity of the protonbring the protons closer to the heavy ions, so that the distance between the protonand heavy ions decreases appreciably. On the contrary, tp'0(O increases, when (xi +X2) and v decrease. This means that the coupling interaction between them andthe velocity of the proton decrease with increasing distance between the proton and


the heavy ions. Since there are two new equilibrium positions of the proton in sucha case, the proton can shift from one side to the other side of the heavy ion bymeans of quasi-self-trapping and attraction interaction between the proton and theresidual negative charge of the heavy ion, which leads to bond rotation or Bjerrumdefect. Therefore, the solitons (10.18) and (10.19) in the case of e < 0 and g < 0represent a hopping motion of the proton over the interbond barrier of the heavyion, i.e., it represents the Bjerrum defect produced by the rotation of the X-H bond,arising from changes of the relative positions of the protons and the heavy ions. Insuch a case, there can be two protons between the two oxygen atoms and positiveeffective charge (the D Bjerrum defect), or no proton between them and negativeeffective charge (the L Bjerrum defect). This type of soliton is referred to as kinkII. The plus sign of <p(x,t) in (10.18) applies to the L Bjerrum defect, which leadsto the creation of a negative effective charge, while the minus sign in <p(x,t) in(10.18) applies to the case of the D Bjerrum defect which results in the creationof a positive effective charge. Thus, the Pang-Miller model supports two types ofdefects that occur in hydrogen bonded systems, i.e., kink I -+ I~ ionic defect, andkink II -> L Bjerrum defect and anti-kink I -» I+ ionic defect, anti-kink II ->• DBjerrum defect. The ionic defect is mainly produced by the double-well potentialthrough the mechanism of proton crossing over the interbond barriers in the mannerof translation, but the Bjerrum defect is a result of the coupling interaction throughquasi-self-trapping in the manner of lattice deformation and relative intrabond shiftof positions of the proton and the heavy ion.

In the transfer process, the protons cross over the interbond barriers in the formof kink solitons, and jump over the intrabond barriers in another soliton form. Thecoupling interaction also changes depending on the relative positions between theproton and the heavy ion. When the protons cross the interbond barriers, thecoupling interaction is small due to their long separation from the heavy ions, andit plays only a secondary role in determining the properties of the protons. Whenthe protons are near the heavy ions and when they shift over the intrabond barriers,the coupling interaction becomes so large that their positions relative to those ofthe heavy ions change considerably by means of quasi-self-trapping. In such a casethe coupling interaction determines the properties of the protons, which transformsinto another soliton form in the intrabonds. However, the above changes in theforms of proton transfer are not very sudden, but take place gradually. As a matterof fact, we see from (10.21) that the minima of the potential energy become moreand more negative with increasing (xi +X2)- When the latter becomes so large thatthe coupling effect is greater than that of the double-well potential, the potentialenergy minima reduce to that of (10.22) and (10.23). In this process, the changes ofthe velocity of the proton transfer in different regions affect the potential energy ofthe system. It can also influence the form of the proton transfer as discussed above.Therefore, the Pang-Miller model gives a clear description of the proton transferprocess in hydrogen bonded systems.


10.3 Thermodynamic Properties and Conductivity of ProtonTransfer

The theory for proton transfer given in the previous section must be validated byexperiments. To do so, it is necessary to study the thermodynamic properties ofproton transfer and calculate its conductivity. Pang was the first one who tackledthis problem.

Fig. 10.6 The kink-anti-kink solutions in hydrogen bonded systems.

From (10.15) and (10.16), we can see that if the nonlinear autolocalized excita-tion in the protonic sublattice is a kink (or anti-kink), there is also an anti-kink (orkink) soliton in the heavy ionic sublattice, which is a "shadow" of the kink (or anti-kink) as shown in Fig. 10.6. They propagate together along the hydrogen-bondedchains in pairs with the same velocity. In Fig. 10.6, curve 1 (3) corresponds to thekink (antikink) soliton in the protonic sublattice and curve 2 (4) corresponds tothe antikink (kink) soliton in the heavy ionic sublattice. The momentum of thekink-antikink pair can be obtained from

P=- f (—<px<pt + M(3x(3t) dx = PK + Pak = Msolv, (10.25)ro J \w 0 /

where Msoi = m*k + m*ak, with m*k and m*ak being the effective masses of the kinkand antikink, respectively, which can be obtained by inserting (10.15) and (10.16)or (10.18) and (10.19) into (10.25), v is the velocity of the kink-antikink pair. Alsoin (10.25),

PK = — —<Px¥>tdx, Pak = — / PxPtdxro J wo r0 J

are the momenta of the kink and antikink, respectively. The mobility and conduc-tivity of the proton can be obtained using (10.25).

Since proton transfer is associated with motions of ionic and bonded defectswhich are charged, when the proton-soliton is formed, a charge deviation from the


regular protonic charge distribution occurs in the systems. The soliton charge isrelated to the quantity d = R(oo) - i?(-oo), which is equal to ±2RQ for positive-negative ionic-defects, and ±(4vr - Ro) for positive-negative charge bonded-defects,respectively. Since the proton transfer is caused by the combined transition of theionic and bonded defects, we have q = qi + qs, where qi and qs are the partialcharges of an ionic and bonded defect, respectively. According to the results ofPnevmatikos et a/., qi = —otidi (where i is / or B). When a/ = as = a, thecoefficient is found to be a = q/4n. In most systems, a/ ^ as, and qi is expectedto be dependent on the dynamics of the heavy ions. Thus an electric current canoccur due to motion of the charged proton-soliton along one direction under anexternally electric-field. If we can obtain the corresponding electric conductivity,we can judge the validity of the theory given above by comparing the theoreticaland experimental results.

Having this in mind, we consider the conduction of the charged kink-antikinksoliton pair described above under a constant external force. Even though the sameforce is applied to both the protons and the heavy ions, the responses by the twosublattices may be different. Thus, we represent the effects by different fields Fxand Fi. Considering the dissipation effects, due to influence of environment, on themotions of the proton and heavy ion, the equations of motion, (10.10) and (10.11),are replaced by

<ptt = fi{<p,0)-î<pt - y ~ f ~ ~ >

Af = /2W)-r2/?-g, (10.26)

where Fx and F2 are the external forces on the proton and heavy ion, respectively,Fi and T% are the damping coefficients for the motions of the proton and heavyion, respectively. We assume that the effects of the external forces and dissipationeffects to the kink-antikink pair of (10.15) and (10.16) are small so that they leadonly to a small change to the velocity of the kink-antikink pair, but not to thewaveform. Furthermore, we assume that the forces Fx and F2 are function of timeonly. Applying the boundary conditions for tp(x,t) and u(x,t), we can obtain, from(10.25) and (10.26), the following equation of motion for the proton-soliton

dv 3yggKa-«*)F

dt + 7 2e(m + B*MmLJQ/h)' { '

where

mr1+B*r2M(muo/h) dv F = J™*,F Bp)

Since we assumed that the velocity of the kink-antikink pair is smaller than thesound speeds of the two sublattices, i.e., t ; < C o and v < «i, we can neglect theterms containing v, and regard g, <po, e, B and 7 as constants, i.e., g = go = const.,

(10.27)


<Po = <PQ = const., s = eo = const., B = Bo = const., 7 = 70 = const. Equation(10.27) then becomes,

| + W = 3 ^ [ m + ^ ( T ) ] - ' F , ( 1 0 , 8 )

which is an analogue of the equation of motion of a macroscopic particle withdamping in classical physics. It shows that the kink-antikink soliton pair behaveslike a classical particle.

Since the kink-antikink pair is charged, the electric field force acting on the pairdue to a constant electric field E can be written as

where <?i and q2 are the effective charges of the kink and the antikink. We can obtainsolution of (10.28) in the case of a steady current by approximating the mobility ofthe kink-antikink pair with

_ v_ _ 3(gi + B0^/muJo/nq2)viVmuj0g/h11 \E\ V2eo[mT1+B^(Tnuo/h)Mr2] ' {

The electrical conductivity of the kink-antikink pair in hydrogen bonded systemscan be obtained and it is given by

» 3no(gi + Boy/mujo/hq2)2viy/muog/h /-moma = a nnu = 7= , (10.oil)

y/2eo[mT1+Bl{mojolh)MT2] V

where no is the number of protons in an unit volume. Using parameters givenearlier and Ti ~ (0.6 - 0.7) x 1014 s"1, T2 ~ (9.1 - 13) x 1014 s"1, qi = 0.68e,q2 = 0.32e, n0 = 1022 mol"1 for ice (Onsager 1969, Bjerrum 1952, Schmidt et al.1971, Homilton and Ibers 1969, Bell 1973, Eisenberg and Kanzmann 1969, Eigenet al. 1962, Whalley et al. 1973, Weiner and Asker 1970, Pauling 1960, Peyrard1995), it can be found that /x = (6.6-6.9) x 10~6 m2/V-s and a - (7.6-8.1) x 10"3

(ft-m)-1.Experimentally, Eigen and Maeyer reported mobility as high as (10 - 20) x 10~6

m2/V-s for proton transfer in ice, while measurement by Nagle et al. yielded a lowervalue of (5 — 10) x 10~7 m2/V-s. The theoretical result based on the above modelfalls between these experimental values.

In order to further justify this soliton model of proton transfer in ice, we considerthe temperature dependence of mobility using the above soliton model, followingthe work of Nylund and Tsironis. In their work, the ice is placed in a heat reser-voir and a constant electric field is applied to accelerate the proton. The dampingeffect and a Langevin-type 5-correlated Gaussian stochastic force are considered inthe dynamic equation for the oxygen atom. Following the approach of Nylund andTsironis, we integrate numerically the dynamic equations for the proton and oxygenatom using the fourth-order Runge-Kutta method. We can then obtain the mobility

(10.29)


(or velocity) of the thermal kink soliton as a function of inverse temperature. Theresults for two different field values are shown in Fig. 10.7 using solid lines. Alsoshown in Fig. 10.7 (dashed line) are the results of Nylund and Tsironis which wasobtained using the ADZ soliton model. The most distinct feature of the mobility-temperature curves is the presence of two transition temperatures Tmax = 191 Kand Tm-in = 210 K, i. e., the mobility first rises as the temperature increases, until itreaches a peak at 191 K, and it drops subsequently with further increase in temper-ature, and reaches a minimum at 210 K, before rising again. The up-down-up trendin this range of temperature seems a general feature in the temperature dependenceof the soliton mobility and can also be observed for other values of electric fieldsand different barrier heights. This behavior is in qualitative agreement with exper-imental results obtained by Hobbs and Engelheart et al. in the same temperaturerange for crystal ice (see inset in Fig. 10.7). Therefore, temperature can enhancemobility at low and high temperatures, but a sharp drop occurs in the intermedi-ate temperature region. In addition to the similarity of temperature dependencesof the conductivity (experimental data) and the velocity (soliton model), anotherremarkable feature is the coincidence of the transition temperatures in the experi-mental and theoretical results. This coincidence provided evidence for the existenceof soliton in crystal ice. It supports the soliton model of proton transfer given aboveas well as the treatment of the electric properties of ice.

Next, we discuss the thermodynamic properties of the systems. The effectiveHamiltonian of the protons corresponding to (10.13) can be written as

ffeff = Jdx(±rf + \vl<fl + \^ ~ f V2) • (10.31)

The thermodynamics of the systems can be analyzed using a transfer integral tech-nique and statistical physical method. The classical partition function correspond-ing to (10.31) in terms of the field variables <p(x,t) and its conjugate <p*(x,t) canbe written as

Z(0,L)= [dp [due-?"*" = ZPZV, (10.32)

where L is the length of the system, T is its temperature, KB is Boltzmann constantand /? = 1/KBT. ZV and Z^ are are given by

Zp= [ dpe-PE* = (2nKBT)N/2 , Zv = [ dueT&E*, (10.33)

respectively, where

B' - £ d ^ *<•=£d* G * * + W - W) •and

Ep + Ev = Heff-

576 Quantum. Mechanics in Nonlinear Systems

Fig. 10.7 Kink soliton velocity as a function of inverse temperature for two electric-field values.The solid lines represent results obtained using the present soliton model. The dashed lines areresults of Nylund and Tsironis. The inset shows the experimental results.

The integrals given above can be evaluated exactly in the thermodynamic limit ofa large system containing N heavy ions (N —> oo) by using the eigenfunctions andeigenvalues of a transfer integral operator

j dzi-1e-f>eF"l"«''i-î(<Pi-i) = e-^Vito) , (10-34)

where F(<pi, <fii-i) relates the potential energy components E^. This calculation isperformed using the method of Krumhansl and Schrieffer, and Schneider and Stoll.V>i(</>) a n d Si in (10.34) satisfy the following equation

( V + - ^ ^ ) Ifcte) = {et - *)fc(*), (10-35)

with

We have assumed that

. _ evlm ~ RlK%T*-


Since the single-site potential in (10.31) is bounded from below, so is the eigenspec-trum, and we denote the lowest eigenvalue of the above Schrodinger equation by e0-Then, in the thermodynamic limit, we can approximately get

Zu = e-/3Neeo. (10.36)

The free energy per particle can be obtained which is

KrtTf = ~Y~ M2TTKBT) + eeo, (10.37)

where 6Q is determined from (10.35).Finally, we can find the internal energy per particle and the related specific heat

at constant volume. The results are

, Tdf~K T 4e2 (KBT\2 x 31e=f-TdT=KBT+T5g{-Ef) +0[(KBT)\,

Cv = = KB + ^(ie2KBT + O [(KBT)2] , (10.38)

respectively, where E^. — 2^2e3vim/Rog. Equation (10.38) gives the specific heatof hydrogen-bonded systems, including ice. It is linearly dependent on temperature,which is similar to (9.104) - (9.105) for the ACN and proteins.

10.4 Properties of Proton Collective Excitation in Liquid Water

In this section, we discuss collective excitation of protons in liquid water. We willapply the theory presented in earlier section to water, a hydrogen bonded system,to reveal the mechanism of magnetization of liquid water.

Water is familiar because it is closely related to growth and living of human be-ings, animals and plants. We can say that there would be no life without water inthe world. However, the properties of water are still not fully understood, althoughit has been studied for several hundreds years. Magnetization of liquid water isa prototypical example. It has been known that liquid water can be magnetized,i.e., when the water is exposed to a magnetic field, its optical, electromagnetic,thermodynamic and mechanical properties, such as density, surface tension, viscos-ity, melting temperature, solidification temperature, dielectric constant, conductiv-ity, refractive index, and spectra (ultraviolet, infrared and laser-Raman), will bechanged compared to non-magnetized water. These changes are due to magnetiza-tion of the liquid water.

Magnetized water has been widely used in industry, agriculture and medicine.For example, it can be used to expedite digestion of food and removing dirts inindustrial boilers, and so on. Even though many models, for instance, the resonantmodel by Ke Laxin, the model of hydrogen bond destruction, and physical-chemistryreaction of water molecules with ions, were proposed to understand this process,


the mechanism of magnetization has not been satisfactorily explained. In the res-onant model, Ke Laxin proposed that the magnetization of water is caused by aresonant effect among vibrations of the components including water molecules andhydration water, chelate and impurities under an externally applied magnetic fieldwith appropriate frequency, which leads to distortion of the hydrogen bonds, andchanges in structures and properties of the water molecules. On the other hand,Jiang et al. proposed that the water molecules form hydration ions with some ionsin liquid water. Thus the distribution and polarization and dynamic features ofwater molecules are changed. The externally applied magnetic field destroys thestructure and distribution of the ionic-hydrations through the Lorenz force. There-arrangements of the ionic-hydrations result in magnetization of water. In thisprocess, destruction of some hydrogen bonds are believed to be necessary.

However, these models are only qualitative, and cannot fully explain the mag-netization of water and its related properties. New ideas from nonlinear dynamicshave provided a solution to this important problem. In this section, we will reviewthe nonlinear theory of motion of protons and look into the mechanism of magneti-zation of liquid water. The model is based on the theory of proton conductivity incrystal ice discussed in Sections 10.1 and 10.2. In other words, it is a generalizationof the theory of proton conductivity in the ice, since liquid water is expected tobecome ice at 0°C through first-order phase transition.

10.4.1 States and properties of molecules in liquid water

States and distribution of molecules in liquid water are different from that in iceand vapor phases of water. It is known that the molecules in liquid water arepolarized, which can be easily demonstrated experimentally. For example, when acharged plastic pencil (by static electricity generated by friction contact) is broughtto a water stream from a tap, the water deviates from its original flowing directionand bends towards the pencil. This is a simple and direct demonstration that thewater molecules are polarized. Each molecule possesses a large dipole-moment.Thus, it can be expected that there exist many hydrogen-bonded chains formed bydipole-dipole interaction between molecules in the system.

The polarization of water molecules can be easily understood from the prop-erties of the first-order phase transition. It is well known that when ice becomeswater through a first-order phase transition at the melting point, 0°C, a certainamount of latent heat is released, due to changes of volume and entropy. As men-tioned earlier, ice is a hexagonal structure formed by hydrogen bonding betweenwater molecules and there is a large number of linear hydrogen-bonded chains in thesystem below 4°C, in which the ice lattice is extremely porous and contains many"vacancies" since the average number of nearest neighbor water molecules of eachmolecule (coordination number) is only about four. On melting, the ice lattice ispartially destroyed, and at the same time, some vacancies are filled and the density


of water becomes greater than that of ice. This is one of the principal anomaliesof water. With further heating, up to 4°C, the condensation process continues.When the temperature reaches above 4°C, the amplitude of anharmonic vibrationsincreases, and these molecules no longer vibrate around their equilibrium positionsand the ice lattice is completely destroyed. However, each molecule still has fourneighbors and hydrogen bonding still exist. The contribution of hydrogen bond-ing to the total energy of intermolecular interaction (11.6 Kcal/mole) is about 69Kcal/mole in such a state. However, because there are a lot of hydrogen bonds, themelting point (0°C) and the boiling point (100°C) of water are significantly higherthan those of other molecular liquids which are bound together by van der Waals'forces. For example, the melting point and boiling point of methane (CH4) are— 186°C and — 161°C, respectively. Obviously, the composition and structure of theintermolecular complexes and the number of associated hydrogen-bonded moleculesin the complexes (clusters), as well as density of water depend on temperature, andthe compositions decrease with increasing temperature due to disorder of thermalmotion of water molecules. Rough estimates give about 240 molecules in a clusterat room temperature, 150 at 37°C and 120 at 45°C.

Raman spectrum of liquid water can give us more information about the struc-tures and distribution of water molecules. Experiments by Jiang et al. and Panget al. showed that there are three Raman peaks in the range of 300 - 3700 cm""1,located at 300 - 900 cm"1 and 1600 - 1900 cm"1 and 2900 - 3800 cm"1, respec-tively. The peak in the range of 300 - 900 cm"1 is associated with the librationof water molecule. The peak at 1600 — 1900 cm"1 is a narrow hand, arising frombending vibrations of HOH bonds. The peak at 2900 - 3800 cm"1 is a wide band,containing four peaks. The two peaks at 3241 cm"1 and 3415 cm"1 correspond tosymmetric and antisymmetric stretching-vibrations of hydrogen-bonded OH bonds,and the two peaks at 3540 cm"1 and 3617 cm"1 represent symmetric and anti-symmetric stretching-vibrations of free OH bonds. Since the Raman spectra showchanges in the number of hydrogen bonds, the shape and intensity of these peaks inthe 2900 - 3800 cm"1 band can be used to investigate the nature of the hydrogenbonded chains in liquid water. It was confirmed by Raman spectral study in thisregion that (1) there are a lot of hydrogen-bonded chains of water molecules; (2) thenumber of the hydrogen-bonded chains and the number of water molecules in thechains decrease with increasing temperature; (3) there is a red-shift of frequenciesin the Raman spectra and infrared absorption spectra, when the water moleculesare hydrogen bonded with neighboring molecules.

10.4.2 Properties of hydrogen-bonded closed chains in liquid water

From the above discussion, we know that there are hydrogen-bonded chains inliquid water. Considering the special properties of liquid water, such as long-rangedisorder in the motion of molecules and the uncertainty of equilibrium positions of


the water molecules, it is reasonable to assume that some chains form closed loopsby linking head and tail of different linear chains. These loop configurations canconsist of various number of water molecules as shown in Fig. 10.8.

Fig. 10.8 Hydrogen bonds in a dimer (/3 = 110°), a linear hydrogen bonded (/3 = 0), and a cyclichydrogen bonded water molecule.

Let /? be the angle between the OH bond and hydrogen bond in the closedloops, as shown in Fig. 10.8. Its value depends on the number of water moleculesor the number of hydrogen bonds in the closed loop. For instance, in a dimer whichconsists of 2 water molecules, 0 is 110 degrees, and in the five membered ring it isapproximately 10 degrees.

Interactions between water molecules in the closed loops will cause furtherfrequency shift of the OH stretching vibrations, compared to that in free watermolecules. The maximum shift was observed experimentally in the case of /? = 0(linear hydrogen bond), the hydrogen-bonding energy of which is also a maximum.For 3540 cm"1, the maximum shift is about 400 cm"1, while for 3617 cm"1, it isabout 340 cm"1. Therefore, the value of j3 can be used as an indicator for forma-tion of closed hydrogen bonded loops in water. If a nonzero value of /? is obtainedexperimentally, we can conclude that closed loops exist in the liquid water.

Although the magnitude of the frequency shift depends on the number of watermolecules and the angle (/?) between the OH bond and the hydrogen-bond in theclosed loops, the functional relationship among them is very complicated. But it canbe derived based on the theory of proton transfer in hydrogen-bonded chains in icediscussed in Sections 10.1 and 10.2, and making use of the Badger-Bauer rule whichstates that the energy of a hydrogen bond is proportional to the frequency shift,Ai/, of the valency infrared vibrations of the OH group in a closed loop with respectto the vibrational frequency in the free molecule. From the structural propertiesof the closed hydrogen bonded systems, and the interaction and motion of protonillustrated in (10.4) and Fig. 10.5, we can obtain the ratio of the frequency shiftAu = I/Q — i/(0) at a given /3 to the maximum shift (Ai/)max = v0 — u(0), asa function of /? in the closed loops, where u0, v((5) and i/(0) are the frequenciesof intrinsic vibration, the valency infrared vibrations of the OH group for ft ^ 0

Nonlinear Excitations and Motions of Photons, Potarons and Magnons 581

(closed loop) and ft ~0 (linear chain), respectively. This ratio can be approximatelywritten as

Ai/ _vp- y(0) Vl + (1 - cos/?)[i/g - u2{0)]/u{0y(Al/)max (Al/)max

The variation of Ai//(A^)max with /?, given by the above equation is shown inFig. 10.9 (solid line), together with experimental data (symbol).

Fig. 10.9 The ratio of the frequency shift Ai/ = i/0 — v((5) to the maximum shift (Ai/)max =i/o — •'(O) as a function of /? for a ring chain.

This graph also reflects the dependence of the energy of the hydrogen bondon the angle (3. It is clear that the theoretical result is in good agreement withthe experimental data (Davydov 1982). It provides experimental confirmation thatthe theory developed by Pang et al. gives a fairly good description for the closedhydrogen bonded chains in liquid water.

10.4.3 Ring electric current and mechanism of magnetization ofwater

The hydrogen-bonded closed chain structure of liquid water is crucial for its mag-netization. The closed chains or loops have many different forms and types. It mayconsist of polymerizations of two, three, four, five, • • •, or n water molecules, asshown in Fig. 10.9. The form and type depend on the temperature of water. Thenumbers of water molecules in the closed chains decrease gradually with increaseof water temerature. Since the closed hydrogen-bonded chains are also channels ofproton transfer, we can generalize the proton transfer theory given in Section 10.4 toproton transfer in such closed loops. We assume that charged proton-solitons movealong the closed loops. Then an electric current can be produced by this motion ofthe proton-soliton around the closed loops of hydrogen bonds, when a liquid watercontaining many closed hydrogen-bonded chains is exposed to an external magneticfield, Hex. This is because the charged proton-solitons move around the hydrogen-bonded chains due to the action of the Lorenz force, F = qv x Hex/c, where v


is the velocity of the proton-soliton, c is the speed of light, q is the charge of theproton-soliton. The ring electric current / = qnv further induces a magnetic fieldHin (V x Hin = 4TT J/C) in the closed loops, where n is number of the proton-solitonsper mole. Here, the velocity v is still determined by (10.28), while the force F in(10.28) is the Lorenz force. For a closed loop with a diameter a, the magnitude ofthe induced magnetic field on the symmetrical axis is given by

27TQ2Jin ~ c(a2 + Z2)'

where Z is the distance to a point on the symmetry axis from the center of the ring.Therefore, if the external magnetic field and the structure of the closed loops areknown, we can determine the current J and the induced magnetic field Hin from(10.29). Obviously, there is an induced magnetic field in each ring. But they candiffer in magnitudes and directions due to different conformations, different numberof the protons and different number of hydrogen-bonds contained in the loops. Thus,these hydrogen-bonded chains, each consisting of a large number of water molecules,will arrange orderly via magnetic interaction, and form a locally ordered state. Thisleads to an ordered distribution of water molecules and magnetization of liquidwater. The arrangement of these closed loops depends on the interactions betweenthe induced magnetic fields of the loops and between the induced magnetic fieldand the external magnetic field. When the external field is very strong, all closedloops will be aligned along the direction of the external field, and the maximummagnetization is obtained. Based on the theory discussed earlier, the distributionand degree of the localized order of the magnetic-arrangements or magnetization ofthe water molecules can be determined, if we know the number of water moleculescontained in a loop, the numbers and distribution of loops, and the magnitudeand direction of the external magnetic field Hex. The effect of magnetization ofwater can then be quantitatively determined. However, the calculation is verycomplicated and can only be carried out numerically for certain distributions of theloops in liquid water.

Therefore, magnetization of water is nothing but a magnetic ordering of wa-ter molecules in the closed hydrogen-bonded chains, in the presence of an externalmagnetic field through magnetic interaction. This is a nonlinear and local orderingphenomena, and it is a collective effect of a large number of molecules, rather thanbehaviors of individual molecules. The physical fundamentals for magnetizationof water is the presence of a large number of closed hydrogen-bonded chains andring proton currents in the liquid water. Its theoretical foundation is the theory ofproton conduction which is a generalization of the proton transfer theory in ice. Adirect effect of magnetization of water is the change of states and distribution ofwater molecules in the liquid state and the density of the water molecules, whichlead to changes in other physical properties such as optical, electrical, mechanicaland thermodynamic properties. For example, the dielectric constant, susceptibility,


magnetoconductivity, refractive index, pH value, Raman spectrum of liquid watercan be affected. Such changes in physical properties of the liquid water have beenconfirmed experimentally. For example, it was found that (Xie 1983, Joshi andKamat 1965, Higashitani 1993, Ke Laxin 1982, Muller 1970, Liemeza 1976, Li 1976,Jiang et al. 1992, Pan and Xun 1985, Song 1997, Cao 1993, Binder 1984, My-neni et al. 2002, Davydov 1982, Koutselose 1995, Evans 1982, Evans 1983, Evans1987, Mouritsen 1978, Chikazumi 1981, Kusalik 1995, Dwicki 1997) the dielectricconstant, susceptibility, pH value, and magnetoconductivity all increase when liq-uid water is magnetized, and changes were also found in the Raman spectrum andthe infrared absorption. Pang et al. and Jiang et al. demonstrated that molecularstructure remains the same, the hydrogen bonds remain intact, the peak positions ofRaman spectrum and those of infrared absorption are unchanged, but their intensi-ties are changed when liquid water is magnetized, as shown in Figs. 10.10 and 10.11.The magnetization effect depends on the magnitude, direction and time of action ofthe external field, the larger the magnitude and longer the time in which the fieldis applied, the large the magnetization effect. Pang measured the refractive indexand dielectric constant of magnetized water. For pure water, its refractive indexis 1.3336 at 25 °C. For magnetized water, its refractive indeces are 1.3339, 1.3342and 1.3346 at applied magnetic field Hex = 1.8 T, 2.4 T, and 3.0 T, respectively.Therefore, the refractive index of magnetized water increases slowly with increasingapplied magnetic fields. Accordingly, the dielectric constant of magnetized waterincreased by about 1.4% when the applied magnetic field wass increased from 1.8T to 3.0 T.

Fig. 10.10 Comparison of Raman spectra of magnetized and nonmagnetized water.

Similar to any other magnetic systems, the magnetization in liquid water canbe saturated, in which the magnetization no longer change with further increaseof the external magnetic field once it reaches a certain value. In order to validatethis conclusion, Pang measured infrared absorption of magnetized water with anincreasing applied magnetic field, Hex, by a Nicolet Nexus 670 FT-IR spectrome-ter. The results are shown in Fig. 10.12. From this figure we see clearly that theintensity of absorption increases with increasing Hex. After the magnetic field hadbeen applied for five hours, the absorption reached the maximum value, confirmingthe saturation effect of magnetized water. The memory effect of magnetization in


Fig. 10.11 Raman spectra of magnetized (a) and nonmagnetized (b) water.

which the magnetization remains when the applied field is removed is also observedin liquid water. In Pang's experiment, the intensity of the infrared absorption de-creased gradually when the applied magnetic field was removed after the saturationmagnetization of water. The magnetized water can restore the normal state afterabout four hours, indicating that the magnetized water has a memory effect. Thesephenomena can all be explained using the same theory because magnetization ofwater is a magnetic ordering process of water molecules in these closed loops underthe action of the applied field. Evidently, this arrangement is directly related tothe magnitude, direction and time of action of the external field. When the appliedfield is so strong that all induced magnetization of the closed loops can be orderlyarranged along the direction of the applied field, the magnetization is saturated.In such a case the saturation magnetic field is given by the sum of the inducedmagnetic fields of all the closed loops, i.e.,

where N is the number of closed loops in the liquid water, the parameters a*, Jkand Zk are the diameter, induced current and magnetic field of the fcth closed loop,respectively.

When the applied field is removed, the magnetization cannot immediately dis-appear due to interactions among these induced magnetization, which is the originof the memory effect of the magnetized water. However, given sufficient time, thememory effect will eventually disappear. The duration of the memory effect can beobtained from (10.27). In fact, once the applied field is removed from the liquid wa-ter, we know from (10.27) that the velocity of the protons in the hydrogen-bondedsystems will be changed according to

dv


Fig. 10.12 The saturation effect of magnetized water. The effective strength of the appliedmagnetic field, Hex, is denoted by the time exposed. The curve labeled "0 min" represents thatof pure water (without magnetic field).

The solution of this equation is

v(t) = v{0)e-^ = v(O)e-^T,

where r = I/7 is the damping or relaxation time of the velocity of the proton.It shows that the velocity approaches to zero, when t > r — I/7. On the otherhand, we know from the above that the induced electric current J as well as theinduced magnetic-field H'in are proportional to the velocity of the protons, v. ThusJ and H'in will also approach zero, when the v approaches zero, and the interactionsamong these induced magnetic fields disappear gradually after a time of r = I/7.Therefore, r = 1/7 is the lifetime of the induced electric current J and inducedmagnetic field H'in in the absence of the external field. In other words, it is thememory time of the magnetized water. Evidently, this time is inversely proportionalto the damping coefficients, Fi and F^ of the medium (see (10.27)). In short,magnetization related phenomena in liquid water can be satisfactorily explainedby the proton transfer theory, indicating that the theory correctly describes themechanism of magnetization of water.

Based on this mechanism, we can obtain the following properties of magnetizedwater. (1) The basic requirement of magnetization of liquid water is the existenceof a large number hydrogen-bonded rings which consist of many molecules. Theconductive carriers of the ring electric current in the rings are protons, instead ofelectrons. Because the mass of an electron is about 2000 times smaller than thatof a proton, the velocity of protons is much smaller, and there are less number ofprotons than electron, the proton current in the rings is small. Furthermore, only afraction of water molecules form ring structures, the magnetization effect of wateris typically small.

(2) The external magnetic field Hex results only in ring proton current and


induced magnetic moments in the closed hydrogen-bonded chains. It does notchange the numbers of hydrogen-bonded water molecules. The hydrogen-bonds ofwater molecules are thus not destroyed due to magnetization, and the positions ofthe Raman and infrared absorption peaks remain the same. This is in agreementwith experimental results of Jiang et al., and Pang et al. which are shown inFigs. 10.10 and 10.11, respectively.

(3) Increasing the temperature of liquid water will increase the kinetic energy ofthe disordered motion of water molecules, resulting in breaks of the hydrogen bondsof water molecules, if the kinetic energy of the disordered motion is larger than thehydrogen-bonding energy. Therefore, the number of water molecules contained inthe hydrogen bonded closed loops decreases with increase of temperature and thenumber of disrupted hydrogen bonds increases also with increasing temperature.Thus, the magnetization effect of liquid water is very small at high temperature,and it disappears at about T = 100°C.

10.5 Nonlinear Excitation of Polarons and its Properties

The concept of polaron was first proposed by Landau to describe the motion of anadditional electron in a crystal lattice. It was suggested that the electron wouldpolarize the crystal and the crystal energy would be lowered in the process. Thelocalized electron and the lattice together are referred to as a polaron. Landau'sidea was discussed in details by Pekar, Frohlech and Holstein, and many others.However, many questions about polaron remain open.

We now take a look at the behavior of an additional electron in an one-dimensional polar crystal in the framework of Holstein model which takes intoconsideration possible appearance of soliton-like excitations of the additional elec-tron in such systems. Assume that the system consists of N identical diatomicmolecules, with atomic masses of mi and ra2, respectively, and located at theirequilibrium positions, Xn and xh of the nth molecular cell. We will consideronly the variations of the interatomic distance due to the longitudinal optical os-cillation with frequency uio, and ignore the longitudinal acoustic oscillations. TheHamiltonian operator for vibration of lattice molecules is given by

H* = - • £ * £ , • & + \M"» £ <+\Mw* £ U n U ^ ' (10-39)

n n n n

with

_ m i m 2 _ ( 2 ) (1 ) _ivi — , un — xn xn — a.

rrii + rri2

The last term in (10.39) represents the nondiagonal dispersion effect of the opticalphonons, with OJ\ <C UIQ. According to Fedyanin et al., the Hamiltonian of an


electron interacting with the lattice is given by

He + Hint = (eo + W)J2Nn-jJ2 (# 6«+i + bi+1bn) + V£NnNn+1 (10.40)n n n

in the second quantization representation, where W is the shift of the energy level EQof the electron at the nth site due to the influence of the electron at other sites (weassume there is only one electron level e associated with a possible bound state).Also in (10.40), J is the overlap integral (J > 0) and V is the strength of theelectron-phonon interaction (having the dimensions [energy] x [length]""1), 6+ (bn)is the creation (annihilation) operator of the additional electron Nn = b+bn.

For small V, the Hamiltonian Hph + He + Hlnt can be transformed to the well-known Pekar-Frohlich type which allows us to take into account the long-rangeinteraction between the electron and the polarization field caused by the lattice de-formation, whereas the Holstein Hamiltonian contains only short-range interaction.Applying the Bogolyubov variational principle, Fedyanin et al. transformed theHamiltonian H = Hph + He + Hint into the following form, with "separation" of thephonon and electron subsystems

H — Hph + He,

where

Hph=hJ2uq (a+a, + -J ,

He = (e0 + W - J ^ - j ) 5 > n - Je~F° £ (btbn+1 + b++1bn)

0 n

with

[ a , , a j j = 6qq>, [bn,b+,]+ = 8nn., yHe,Hph^_ = 0,

Nn = b+bn, u>2q = WQ + u\ cos qr0,

In the above,

Am A771

c _ , , 2 r ^n+1 ^n iamrn

A™_(^yn~mWi-2^y2

(10.41)


Here A is the determinant of the matrix constituted of u2 and UJQ, A™ is its minorobtained by eliminating the mth column and the nth row.

For 8 -¥ 0, the basic contribution to A™ comes from the diagonal terms (n = m)of

Thus, the parameters corresponding to the phonon subsystem uiq and V renormalizethe Hamiltonian He due to the application of the Bogolyubov variational principle.Obviously, the Hamiltonian given in (10.41) is a nonlinear operator. Therefore,the polaron undergoes a change from a linear elementary excitation to a nonlinearsoliton-excitation through the electron-phonon interaction.

Using the Heisenberg equation (9.7), Makhankov et al. showed that

ih(<fn)t = (eo + W - ^£j <pn(t) - Je~F° [<^+1(f) + pn_i(i)]

+ W^ [1 " V;+l (*)Vn+l (*) + ¥>;-! Wyn-l(t)]Vn(t). (10.42)

In this case, the trial wave function can be written as

\<p(t)) = U\0) = £ <pn(t)bt\0)po, (10.43)n

where |0)po is the ground state of the polaron. In the continuum approximation,Makhankov and Fedyanin obtained the following from (10.44) by neglecting termsinvolving derivatives higher than the second.

itupt =(eo + W- ^^J tp(x, t) - Je~F° [2<p + r20<pxx] (10.44)

+-^-2 [l - \<P\2 ~ r20\<px\

2 - | | ^ | 2 - rf (vlxv + <pxxf*)\ <p(x,t).

If we further retain only terms with orders less than the fourth with respect todispersion and nonlinearity in (10.44), we can get

-Jr2oe-F°<pxx - - ^ M V ( M ) - (10.45)

This is a standard nonlinear Schrodinger equation. Therefore, polarons can bedescribed by nonlinear quantum mechanics. The soliton solution of (10.45) is givenby

u>(x t) - rQeMi(kx-ut + 60)}^{X't] ~ 2WS cosh[(a! -vt- xo)/W.]' ( J(10.46)


and its energy is

E. _ eo + w + - £ L - « . - » + ^ - f £ ( ^ ) 2 , (10.47)

where

° ~ 2Jrl ' Ws ~ V2eF° '

The polaron-soliton (10.46) is localized in the region ss Ws. The polaron effect isdefined by the last two terms in (10.47). The third term describes the renormaliza-tion of the electron level shift due to its translation jumps. Note that there arisesa temperature renormalization of the conductive electron mass me = h2 /2r%J dueto J is replaced by Je~Fe at present. Therefore, the polaron now gets heavier, i.e.,Ms(9) > Ms(0), and the effective mass becomes temperature dependent in such acase. If 9 -»• 0, then Ms{6) > M5(0)(l + 3x#/8wo)> where x is a coupling constantbetween electron and phonon and is given by x = 4:V2/3Mh2u)Q. The same effect,i.e. increase of mass, was observed for 9 ^ 0 in cyclotron resonance experimentsby Rodriges and Fedyanin. Such a correspondence is naturally associated with thesimilar structures of the Pekar-FVolich and Holstein Hamiltonians.

For a polaron with a small radius, a canonical transformation method has beenwell developed. By means of this transformation, Makhankov and Fedyanin exam-ined the main contribution to the electron mass renormalization defined by its stronginteraction with the lattice vibration and obtained the Hamiltonian of a "residual"(weak) polaron lattice vibration interaction. In this case, the lattice Hamiltonian(10.39) is rewritten in terms of the normal coordinates Qq and £q via the unitaryoperator U — exp(-iL), where

L = * £ ^ ° 4 ' (10-48)

with

^ n

The U transformations of the normal coordinates £q viz, £q = U^qU^1 = £q +£* aresuch that £j may be regarded as the operator of lattice deformation. The overallHamiltonian, H, is then divided into three parts: Hph which describes harmonic os-cillations of lattice sites about their equilibrium positions, the polaron Hamiltonian


Hp,

Hp = (£o + W) Y,Nk - Je~F- £ (b++1bn + 6++16n+1)k n

n q q

where

* - ffiSf i : £ W*+S)s i - ['<»+"+a} »* ( i fe ) •and the Hamiltonian Hlni of the residual interaction of the polaron with the oscilla-tion of the deformed-lattice. The polaron motion described by (10.49) is adiabaticsuch that the lattice reconstructs completely with oscillations about the new equi-librium positions. Hmt describes the "friction process", i.e., the polaron motion isaccompanied by a change of the phonon number, and the polaron loses its energyand slows down. Such effects decrease the energy gain due to soliton formation.

Using \tp(t)) in (10.43), we now replaced the lattice deformation operator £° by

% = <¥(*)!*>(*)> = - ^ v ^ ^ > n ( i ) | 2 s i n ( 9 n + f ) •Then Hp is transformed into Hp which satisfies the following equation

ih^Mt)) - #>(*)>. (10.50)

From (10.50) and the relation H'p = Hp(^), we can obtain the equation satisfiedby <p(t) in the continuum approximation. Up to O(U!I/U1Q), the equation is

ihtpt(x,t) = (eo + W + s0- 2Je-FT)ip(x,t) - Je~FTrlipxx

-jj^MMlVOM), (10-51)

where

io = 2^lIlip{X't)l2dX-Equation (10.51) is similar to (10.45), and therefore, has similar soliton solution as(10.46) - (10.47).

The canonical transformation adopted here is valid for small V so that the renor-malization of the band width remains small, i.e. Je~Fr S> V2/MU>Q (This is thecondition for continuum approximation). If e~Fr <SC 1, we can neglect the elec-tron translational motion and arrive at the small radius polaron. For the electron-

(10.49)


acoustic phonon interaction, Whitfield and Shaw showed that the polaron statebecomes a soliton using the deformation potential technique.

Using the adiabatic approximation, Fedyanin et al. studied the properties of po-laron in the case where the kinetic energy of the lattice oscillation can be neglected.In this approximation, the electron is assumed to move rapidly in the slowly vary-ing field of the lattice deformation, which may take place if the time between twosuccessive translation leaps of the electron, n « h/J, is much less than the charac-teristic relaxation time of the lattice deformation, r-i sa UQ1, i.e., huo <C J- This isreasonable for a sufficiently wide electron band J. In this case, the self-consistentelectron amplitude, <p(x,t) in (10.51), satisfies the following equation

ih<pt(x, t) = (eo + W- 2J)cp(x, t) - Jr2yxx - ^f*, (10.52)

in the continuum limit.Equation (10.52) is similar to (10.45), but the coefficients are constant. Accord-

ing to the above approach, (10.45) - (10.47), the total energy of the electron andthe lattice in the adiabatic approximation is given by

MSV2 i / v2 \ 2

s~ 2 48J \Mi4) '

where Ms = H2j2Jr\ is the soliton mass which is the same as that of the elec-tron in conduction band in the given approximation, i.e., there are no tempera-ture renormalization of the energy shift due to the translation leaps (—2J) andthe soliton localization area W° = (2Jro/V2)Mwg- This implies the electron be-ing weakly connected with the lattice, since V -> 0 (W° -* oo), so that we havehw = eo + W — 2 J + ^mzv

2 and <p(x, t) as exp[i(u)t — kx), i.e., the soliton wavefunc-tion degenerates into a plane wave, in other words, the soliton is delocalized as anelectron.

In their study, Fedyanin et al. used the normal coordinate £g and its conjugatemomentum Pq, and introduced an energy functional E = (<p(t)\H\<p(t)), with \<p(t)}given by (10.43), which is a function of £q and Pq for a given >pn(t). The equationsatisfied by £q determines the motion of a conventional oscillator under a constantexternal force. The solution of the equation in such a case can be simply written as

Ut) = (o)cos(Wgt) - ^ / | £K(*)I2-' n

Inserting the above into the equation

~\<p(t)) = H\,p(t))


yields, in the continuum approximating, the following equation for ip(x, t),

itbpt(x, t) = (eo + B- 2J)<p{x, t) - Jrfoxx(x, t) - ^ y>(a, t), (10.53)

where

is the kinetic energy of lattice deformation consistent with the electron motion, and

2 r -|2£° = E M A ^ l>»(*)2lsin(^ + i )

q 1 L "

is the elastic energy of the lattice. Note that in the above adiabatic approximation,we have neglected the energy of harmonic oscillations about the shifted equilibriumpositions

which would lead to a constant correction to the excitation dispersion. Equation(10.53) is similar to (10.45) and (10.52), and thus has soliton solution similar to(10.46). It indicates that polaron in nonlinear system is a soliton and satisfies thenonlinear Schrodinger equation. It describes a quasiparticle of mass Ms = meFe,localized in the region of Ws = W®e~F". The energy of the quasiparticle, e + W —2Je~F$ + V2/(2MUJI), is less than that of the electron level, e+W + V2/2Mu%.Here

h2 nr0 UMU2 „ v2e ~fe#\

These properties of the polaron are independent of the method of calculation. Forsoliton satisfying the nonlinear Schrodinger equation (10.53), its "kinetic energy" is

E = -(m + Am)v2,

where

h2 4 V2 ( r0 \

with the latter being the temperature independent renormalization of the polaronmass. Using W° = 4JMUQ/V3, Am is given by

_(vyM^y_m ~ 24OJ*r2oj2 '


10.6 Nonlinear Localization of Small Polarons

Brown et al. proposed a theory of polaron formation based on the Frohlich Hamilto-nian in the transportless limit. It was found that two related processes are involvedin this formation. In the first process, energy interpretable as the polaron bind-ing energy is released by the exciton system into the normal modes of the lattice.Another process results in the dispersion of this energy throughout the lattice andresults in the development of a persistent deformation around the region occupiedby the exciton. This indicates that a polaron cannot be exist until the polaron bind-ing energy is dispersed and the lattice deformation is completed. The amount ofenergy released into the lattice during polaron formation and the time dependenceof this energy transfer are independent of the detail with which the initial bareexciton is distributed. Ivic and Brown further studied the nonlinear localization ofthe small polarons.

It is well known that a foreign particle or excitation in a deformable solid oftencauses local distortions of the host which can significantly affect the character anddynamics of the quasiparticles. For example, interstitial hydrogen isotopes in metalhydrides can cause large volume dilations; mobilities of photo injected charge carri-ers in organic molecular crystals such as anthracene and naphthalene exhibit noveltemperature dependences; excitonic spectra in organic molecular crystals such aspyrene and a-perylene are significantly affected by local distortions; and certainfeatures in Raman spectra of biological materials such as Z-alanine have been iden-tified with local vibrational modes. The quasiparticle transport in these deformablemedia was investigated by Ivic and Brown using the Frohlich Hamiltonian

H = J2 Enb+bn - ^2 Jmnbtfin + Yl huqa+aq+n nm q

Y,^g(x>t+X9naq)btbn, (10.54)qn

where Jnm is a transfer integral between different sites in the medium. For thetranslationally invariant acoustic chain model, they choose the phonon dispersionrelation uq and the dimensionless coupling function Xn to be

»q = us sin (hqr0\) , xl = &-«*• + 2 f i n ( g r o ) e - ^ ,V2 / yj2NMhujZ

respectively, where ro is the lattice constant, bn and aq are the annihilation operatorsof the small-polaron and phonon with wave vector q and frequency uiq, respectively.

Exact solutions corresponding to the Frohlich Hamiltonian are known only inthe limit of infinite effective mass, wherein the transfer integrals between differentsites in the medium are all zero. In general the solutions are found by the mostapproximate approaches which are either small-polaron theories, usually employ-ing perturbation methods based on a set of polaron basis states, or large-polaron


theories, usually employing variational methods to obtain nonlinear evolution equa-tions. Among the latter are theories which describe quasiparticle transport in termsof envelope solitons. Ivic et al. used the latter to display the unification of polaronand soliton theories of quasiparticle transport in the small-polaron regime based on(10.54).

Following the notation adopted by Ivic et al, we denote operators related tosmall-polaron by a tilde,

(\D(t)) = ^2&n{t)b+\O)»\0(t)),a

(\p(t)) = exp | £0,(i)S+ - #(*)fi+j |0>, (10.55)

where an = UanU+ and bn = UbnU

+ are annihilation operators of the dressed-phonon and the small-polaron, respectively, with

U = exp - £ (x**a+ - Xqn*aq) bmbJ . (10.56)

qm J

These D states contain the exact small-polaron states in the infinite-mass limit(fat) = 0).

The Hamiltonian (10.54) is not diagonal in the small-polaron states when J / 0.However, the interaction between small polarons (b, b+) and the dressed phonons(a,a+) is weaker than the interaction between the bare quasiparticles (b, b+) andbare phonons (a, a+). The microscopic dynamics can be well described by a factoredpolaron state \D(t)). Since the phase mixing part of /3qn(t) is chosen to be timeindependent, Ivic and Brown did not considered soliton, or polaron, formation frombare states, they considered instead the dressed polarons or solitons. To determinethe equations of motion of polarons, Ivic et al. applied the variational principle oftime-dependent quantum mechanics,

ft2 I d \6 dt( tp(t) ih— - H ip(t) ) = 0,

Jh \ at I

which was also used by others (Skrinjar, Kapor, and Stojanovic, Zhang, Romero-Rochin) and obtained the following evolution equations,

ihdJ^± +iHY^ {xlP*q{t) - tffat)] an(t) = ^2L, (10.57)

*4f I>^M<)l° = | § (10.58,

where (H) is the expectation value of the Hamiltonian, (10.54), in the state \D).Prom (10.57) and (10.58), we can eliminate the explicit phonon variables in the


latter, to obtain an integro-differential equation for the small-polaron probabilityamplitudes,

iH^T =(E-^ I*"!'K) &n(t) - £ Jl&n+lit) + &n-l(t)} +\ q J m

J dT^Gn^t - rJ^ lo^TjânW + ffnWfinW, (10.59)

where

J = Jexp -4£|x'|2sin2Qgr0)

is the renormalized tunneling matrix element, obviously it is reduced from the barevalue J by interactions with phonons, and

Gnj{t) = 2Y,XqnXY^q COS(CV),q

gn(t) = £ > , [xnPn^Y"*1 + xJ^,(0)e-"''] •q

The kernels Gnj(t) are related to the thermal part of the fluctuations gn(t), asdiscussed by Wang et al., Ivic et al. expressed a small-polaron Bloch state by

W)) == E «W exP {»' [kR" - ] *} *n 10) (10.60)

with

E{k) = ( E - ^ 2 \xq\2f>^q ) - 2Jcos{kr0). (10.61)

They chose

- , s - m f , r, lE(k)t]an(t) = u(k) exp t ikRn ^~ \

and /8g(0) = 0. Then (10.59) is linearized, indicating that the small-polaron, Blochstates are exact stationary solutions of the nonlinear evolution equation. In thecontinuum limit, Ivic et al. obtained the nonlinear Schrodinger equation given belowby searching for solutions with the D'Alembert property \a(y,r)\2 = \a(y — VT)\2,

and choosing /3q(Q) corresponding to a dressed soliton at zero temperature,

dn/lr f\ h2 rfi it2

i h ~ ^ = - — â(x,t)+E(0)a(x,t) - ^G(v)\a(x,t)\2Hx,t), (10.62)


where rh = H2/2Jrl, with J = J e "*^ ,

i

is the effective mass of the small-polaron,

E(0) =E-5{2- S)E0 - 2Je~52F

is the bottom of the energy band of the small-polaron, as given by (10.61), and

G(l,) = *£° _^L_is a velocity-dependent nonlinearity parameter. In the above, S is defined by

Pnq(t) = -6Xn + Pq(t) and Pnq(t)=6Xq

n.

The above result shows that the small polaron is a soliton which satisfies the non-linear Schrodinger equation. This soliton solution is

rr -i[E(o)/h]t i(kx-ut)a(x,t) = J£ Tr-r^ ^—, (10.63)

v ' V 2 cosh[/x(a; - vij\ v '

where

t I - t rhG(v) v2 . 1 . 2 v4 mG{vf

The plane-wave solutions (10.60) of this equation are not bare Bloch states, butare small-polaron Bloch states with a larger effective mass (m > m) and a lowerenergy [E(0) < E(0)] compared to a bare Block state, due to the dressing of thequasiparticle.

10.7 Nonlinear Excitation of Electrons in Coupled Electron-Electron and Electron-Phonon Systems

A coupled electron-electron or an electron-phonon system is often described bythe modified Hubbard model. The one-dimensional modified Hubbard model wasstudied in detail by Lindner and Fedyanin, and Makhankov et al. The modelHamiltonian is written in the form of

H = Hph + He + H^ (10.64)

where Hph and He are the Hamiltonians of the phonon and electron subsystems,respectively, while Hint describes the interaction between them. He corresponds tothe Hubbard model in the nearest-neighbor approximation. We assume that theground state is antiferromagnetic and that the lattice is divided into two equivalentferromagnetic sublattices A and B such that the resulting magnetic moment is zero


in the absence of an external magnetic field. Based on this, Makhankov et al.proposed the following for He,

He= J2 ^ i ( C C + C C ) + y E îVn,-ff-/iîVn,,, (10.65)n€A,j£B;<x n€A,B;<r n,a

where

_ / J , (j = n±l),Jnj \0, ( j ' /n i l ) ,

6+CT (fen.o-) is the creation (annihilation) operator for an electron with spin a ina Wannier state at site n, Nn>(r = b+tCrbn>(r is the number operator, J the tran-sition amplitude between nearest neighbors, V the repulsive interaction betweenelectrons in the same atom, fj, denotes the chemical potential. The lattice part ofthe Hamiltonian, Hph, is given in the harmonic approximation by

HPh = f E P n W 2 + f £(^"+1 " Rnf, (10.66)n n

here M is the atomic mass, w denotes the force constant of the system, Rn =Rn(t) = Rn + un(t) is the position of the nth atom and Rn is its equilibriumposition. In the linear approximation with respect to small deviations un(t), theinteraction between the electron subsystem and the lattice in (10.64) can be writtenas

Hint = / 5 > n , . - Un+l,a) [CC-1.T + C l X -ra,cr

<i,X.+Wi,J. (10-67)

where

denotes the matrix element of the local "force" between Wannier states of neigh-boring atoms, corresponding to the local potential V = V(x — Rn). Effects ofanharmonicity in the lattice vibrations and nonlinearity in the expansion of elec-tron integration force in this model have been studied and many interesting featuresof the microscopic particles (soliton) were revealed.

Since the ground state is antiferromagnetic, we are interested in the low-lyingexcitations, and in equations describing the probability amplitudes of spin up ordown states (<f>nt, <f>ni) at-the nth lattice site. We can eliminate terms of the type(un+i - un)Cn+C%_l5, etc. by the analogic method discussed in Section 10.5.To simplify the calculation, Makhankov et al. employed a method similar to theadiabatic approach related to the ground state, and examined the change of the state


|0) under the action of the evolution operator, which is defined on single-particlestates. Thus Lindner et al. and Makhankov et al. denned

<«,(*) = te(*)K,a(t)l¥>(*)> = <0|&£l(,(t)|0> = (0\Ubn>ITU\0). (10.68)

Making use of (10.66) - (10.68), we obtain the following equation for b£v(t),

iKM))t = [<„, fl]_ = V<,(tK_ff - M O ' ) (10-69)+ [J + /(tin - «n-i)]^+li<r(t) + [J + («„-! - «n)]^_lia(«).

The corresponding equation for fcf)(T is the same as (10.69) except that A and B areinterchanged. Applying (10.68) and (10.69), we can get

*«a(*))t = V<0|&£, (*)«„,_„ |0> - /!#*,(*) (10.70)+ [J + /(«„ - Un_!)]^+ l i f f (t) + [J + /(«„_! - «n)]^_l i a ( t)

and a similar equation for <t>fj^{t), which is the same as the above except that Aand B are interchanged. Applying the following decoupling procedure to (10.70),

<0K<,(t)C-*W,-<r(*)|0> "> (0K,.(*)|0)(0|^+_ff(i)|0)(0|6^_ff(i)|0)

= <,(*)C-.(*X-,W, (io.7i)where p = A, B, we finally get a system of nonlinear equations for the four complexfunction tf>â and the real function un. For the "pure" Hubbard model (/ = 0),similar nonlinear equations can be obtained for <j)n,<r{t) as well.

We now discuss the influence of spin ordering on the lattice vibrations.Makhankov et al. constructed the functional,

F(4>°nit,,Un) = (if(t)\H\V(t)) = - / i Y, ^".-l2 - E t J - J(«« - ""+1)]n€AuB;a n,cr

neAuB n n

From the Hamilton equations,

6FM(Rn)tt = Pn, Pn = -JZ~,

it follows that

M{Rn{t))tt = w[Rn+1(t) - 2Rn(t) + Rn^(t)} (10.72)

- i £ {<;w[^+i,.w - ^_i,ff(t)]- [î,a(«) - ^ i , ( * ) ] C w

- < ; ( * ) [ ^ l . f f W - ^n-l.«r(*)] + M&lA*) ~ ^n-l.ff(*)]0nlff (*)} •

This equation is a discrete analogue of the harmonic oscillator equation under theaction of an external force which does not vanish if variation in the initial spin


ordering takes place. The usual condition for antiferromagnetism {Nâ) = (-/Vjf±1 )now becomes 4>n-a = ^niii^ which causes the last term in (10.72) (—I) to vanish.

If we assume that the variation of magnetic order is described by a sufficientlysmooth function of coordinate along the chain, we can use the continuum approxi-mation

un(t) = Rn(t) -Rn^ (X(£, t) - flr0> <„(*) -> # (£ , t).

Then, expanding up to the second derivative and taking into account up to thecubic nonlinear terms, we have

Rn+i{t) -» r0 (x + xt + ^X« + • • • J ,

where (j)^(^,t) = 0 if n € A, and similarly $£ = 0, if n € i5. Makhankov et al.finally obtained the following equations for <j>a[x,t) and x(£,t)

HMZ,t))t = [{4>\{t,t))x + 2tf>_ff(£,*)][J + ro/(l - X€«,*))]+J |0_ ( r |

a ^ - /^ ( r , (10.73)

*(£,*) = ô2x««,t) + E §£(&<!>-<>)> (10-74)

respectively, where fg = 73^0-In the quasi-static approximation (or setting x(£) *) = xi^—vt)), w e c a n integrate

(10.74) to obtain

where

/ = (2J)2

M(w2 - «g)

with t)2 -C WQ a n ^ the constant K depending on initial conditions. Using (10.75),we can get from (10.73)

i(4>AZ,t))t = T(d>%(^t))^ + 2r-/£#(e,*)0-«r(£,*)L-<x(e,t)c J

-VMU) + V\4>-<r(tit)\*<l>a(t,t), (10.76)

where

T = J-r0I(l-K).

(10.75)


In terms of even and odd combinations of 0CT = {0 t , 0j.},

v{1) = <h + 4>u <P(2) = h - <Puwhere

h = 2 ' ^ = 2 'one gets

i<p?> = TVg - Utl2T - <5i,22T - n - ^ l [\<pU\* + | ^ 2 ) | 2 - 2\^\2] | pW

+ \\<P{i)\2 [5i,iV*{2) + *i.2V*(1)] } , (10.77)

where i = 1,2. Assuming that the average occupation number n = \</>•(• |2 + \<f>±\2

is constant and neglecting the term </>*$-„<!>-„ compared to \<j>â\2(j>a, we get two

decoupled nonlinear Schrodinger equations for (f^ and ip(2\ respectively, and theirsoliton solutions are

¥><*>(£,*) =v^sech y n ( Vr ~

J ) ( ^ - ^ ) e x p ^ - i w t t V (i = l,2),

where2 2

^^2T-ii-~, uj2 = -2T-fx+(V-I) + ^ ,

and i> is the soliton velocity. Therefore, electron in the Hubbard model with electron-electron and electron-phonon interactions satisfies the nonlinear Schrodinger equa-tion or the laws of nonlinear quantum mechanics. The soliton consists of a "drop"associated with a "bubble", and is localized in the region

Ws « J-(V - I),V T\i

which is accompanied by a lattice distortion wave that is also bounded in thesame region, and x(£,t) m (10.74) or (10.75) can be represented approximatelyby X K tanh[(£ — vt)/Ws] under appropriate initial conditions. However, for thepure Hubbard model, / = 0, T = J, and a half-filled band, i.e., n = 1, it followsthat wi = u>2 for fj, = V/2.

In the more general case, it is difficult to find the solution of (10.77). But if weneglect the last term on the right-hand side of (10.77), it becomes

i<p® + 8itlT<pM - &,22V« - \ v (p\vW |2 - Mi+1> | 2 ) v>W = 0, (10.78)


where i = 1,2 and

7? = 1 - 2 ^ ' V( 2 )=^( 1 ) -

This is a vector nonlinear Schrodinger equation. In the case of rj = 1 (or I = 0) whichcorresponds to the situation when the electron-phonon interaction is "switched off",Makhankov et al. proved that (10.78) is integrable. They were able to obtain theLax pair, examine its symmetry property, and construct the soliton solutions forgiven values of r).

10.8 Nonlinear Excitation of Magnon in Ferromagnetic Systems

Presence of magnons in ferromagnetic crystals can change the potential energy ofinteraction between atoms. In the case of a ferromagnetic chain, a deformation iscreated by excitation of the magnons which can result in localization of magnonthrough interaction between the spin deviation and the deformation of the lattice.Thus the magnon becomes a soliton in such a case, it can move without changingits shape. Nonlinear excitations in ferromagnetic chains were studied by Pushkarovand Pushkarov. They used the following Hamiltonian.

H = T + V - \ l l J(W« - un+s){S+S-+s + S~S++S)71(5

- \ E J > " - Un+s)SznS*n+s. (10.79)

nS

Here un is the position of atom n in the chain, S runs over the nearest neighborsof a given atom, S^ are the cyclic spin components given by the relations S^ =S%±iSn, J{un — un+s) and J(un — un+s) are exchange integrals which are functionsof \un - un+s\,

m

is the kinetic energy of the atoms of mass m. The harmonic approximation isassumed for interatomic interaction and the potential energy is given by

mvl s-^ 2

n

where vo is the sound velocity and ro the lattice constant.At low temperature when only a few spin waves are excited, the operators S^

and S^ can be expressed in terms of the Bose operators an and a+ as,

5+ = V2San, S~ = A/25a+, Szn = 5 - a+an. (10.80)


In the bilinear approximation with respect to the Bose operators, we can obtain thefollowing by inserting (10.80) into (10.79).

H = T + V-^Y1 J K - w"+*) - f E { J(u" - u-+^nS n6

x [a+an+s + ana++s] - J{un - un+s) [a+an + a++5an+5] } . (10.81)

Prom

iK^\<p) = H\<p)

and (10.43), we obtain

i h ^ = \T + v - y Yl J(u« - w"+*) I v»L n6 J

- 5 J2 J(un - un+s)(pn+s +SJ2 J(un - un+5)<pn. (10.82)(5 5

We have assumed in (10.82) that

J(\un - un+i\) + J(\un - un_i|) w 2J,

J(|un - un+i|) + J(\un - un_i|) « 2 Jo - Ji(un +i - wn_i),

^ J(un - un+i) w 2J0-/V - Ji ^ ( u n + i - un_i),nS n

where

Ji = - p — > 0.dun

Equation (10.82) can then be written in the following form

i h ^ =\T + V- JOS2N + ASE(«n+i - «n-0 + 2(Jo - ^)S v>n

- J(ipn+1 + ipn-i - 2tpn) - JiS(un+1 - Un-x). (10.83)

Taking into account the explicit expressions of T and V as well as the Hamiltonequations for un and Pn = m(un)t, we get

m{un)tt = mvl{un+1 + un_i - 2un) - Ji5(|<pn+i|2 - |^n-i|2)- (10.84)

In the long-wave and continuum approximations, equation (10.84) takes the form

—d^ = v°-dx-2 ^Tto1^01 • ( 1 0-8 5 )


In the continuum approximation (10.83) becomes

m ^ A = [r+v _ joS,N + _ M £ | _ + 2 ( Jo _ J)s] vM

with

(2J!g)2 i + s2 /•«> 4

where s = v/uo- Therefore, the magnon is a soliton in nonlinear systems, whichsatisfies the nonlinear Schrodinger equation, and it can be described by the nonlinearquantum mechanics. The soliton solution of (10.82) is

where

Hv _ Jrnvl 2a=2lS> M = ^ ^ ( 1 - S ) '

TlV= 2{JlS)i 1 + s2

3JS(mvl)2 (1 - s2)3 '

hu = T + V- J0S*N + ( 2a y 5 ' + 2(Jo - J)S + ^ .

mu5(l-s2 ) 4J5Here xo and x' are arbitrary constants, xo appears as a result of the translationinvariance of the problem and together with the velocity v can be determined bythe initial conditions. 9 depends on the initial phase of the wave function.

The energy of the system with a magnon-soliton is given by

ftV (JiS)4 (2 + 3s2 - s4)s2 / 1 f t o w,

where

hw0 = - JoSN2 + 2(J0 - J)S - r(J^lS)

24.2.

In (10.87), the total energy /in; is given as a sum of the rest energy hu>0 and ki-netic energy. The first two terms in HUIQ represent the ground state energy of theanisotropy spin system being considered and the last one describes the rest en-ergy due to the solitary excitation. In the case of small velocities, (v2 <£ v2,), the

(10.86)


soliton moves as a particle with an effective mass m* = m*,(l + Am*/m*,), wheremj = h2/2JS is the effective mass of a free magnon and

Am* = v x ' m*0m/L2i$

is the change in mass due to localization of the magnon, i.e., the transformation ofthe magnon into a "magnon-soliton". If the velocity v equals to zero, the solitonenergy has the absolute minimum,

Emin = -J0S2N + 2(J0 - J)S - 7 ^ l 5 )

24 , 2 .

We see that the energy of the system with a soliton at rest is smaller than the energyof free magnon. Hence such a state is more favorable. As the velocity v increases,the energy also increases. At v > v/y/b, the contribution of the last term in (10.87)predominates over the last one in HU/Q.

Chain elongation A occurs due to spin deviation. Prom (10.85), we obtain that

A _ [°° du, _ 2hS*~ J^dx**- mvl{l-s*y

Thus, A increases with velocity v. In the case of a stationary soliton, we haveA = 2J1S/mvl.

Pushkarov and Pushkarov studied nonlinear excitations of magnons in ferromag-netic systems with a biquadratic exchange and anharmonic lattice deformation usingthe same method, and obtained many useful results. Davydov and Kislukha exam-ined low-lying excitations of anisotropic Heisenberg ferromagnet in one-dimensionalsystems described by the general Hamiltonian

H = -»HX £ S* + 2 J £ UsxnS

xn+1 + r,S*nS«n+1 + S*nS*n+1 - \) , (10.88)

n n ^

where B = (0,0,Hz), S = 1/2. For a system of N spins which are rigidly fixed atlattice sites, if —1 < (£,77) < 1, i.e., for any possible combination of SJ and S?(compared to S£), we have the so-called "easy axis" Oz model. When £ = 77, thereis isotropy in the base (x-y) plane. If 77 = 1, £ r\ (or alternatively f = 1, f ^ 77),we have the "easy plane" model {y-z or x-z, respectively). Finally, setting £ = 0(or r) = 0) gives rise to the Ising model in a transverse field. The simplest variantof (10.88) is an exact "Ising model" when £ = 77 = 0. £ = 77 = 1 yields the isotropicHeisenberg model, which includes also the case of £ » 1 (or 77 » 1), if the last termSn^n+i m (10.88) can be treated as a small perturbation. However, in the studyof low-lying excitations related to nonlinearity, this small term in (10.88) must beincluded.


Applying the Jordan-Wigner transformation for Sf and 5 |

a+ = ( - ^ - 'S fS? • • • 5*_15+> SI = S+S- - i ,

an = (-2)n-1S!Si--S^_lS-, S± = SZ±iS%, (10.89)

where a+ and an are the Fermi operators {an, a^} = <$mn, {an, an} = {a+,a+} = 0,we obtain, instead of (10.88), the Hamiltonian in terms of the Fermi operators

H = \»NHZ - (nHz + 2 J) Y,N* + \j{£ + i) £ > + an+1 + at+1an)-k n

\j(S - V) E(an-i«n - an-ian) + 2JîVnATn+1, (10.90)n n

where Nn = a+an. The nonlinearity is now due to the term S^S^+1. The anomalousterm —(1/2)J(£ —77) Y^ni

at.-ian ~ an-\-\o-n) is defined by the difference between thex and y components. Note that model (10.88), like (10.90), corresponds to a systemof rigidly fixed spins.

From the Heisenberg equation and (10.90) and (10.43), one can obtain

ihpt(x,t) = -Aip(x,t) + -J(Z + ri)rZ<pxx(x,t) + -J(Z-ri)r0<p*x + 2J\ip{x,t)\2ip{x,t),

where

A 9 T ft V-Hz izj\ - 9 M

Note that A can be positive (e.g. for J > 0, |£| < 1, and \q\ < 1) or negative(£ » 1, 77 « 1). In the case of J > 0 and 0 < f < 1, 0 < 77 < 1, A > 0, d > 0,letting h = ro = 1 and introducing new variables, </> = <p(z,t)/</d, t' = tA, andx' — y/4d/(£ + rj)x, we have

i<t>r -4> + 4>x<x> + oufe, + \()>\2<l> = 0 , (10.91)

where

VMC + r?)'

Equation (10.91) is still a nonlinear Schrodinger equation, and should have a soli-ton solution. If a < 1, a<j>* could be treated as a small perturbation. Based onthis equation, Makhankov et al. discussed structural stability of microscopic parti-cles (soliton) described by the nonlinear Schrodinger equation given in Chapter 4.Fedyanin et al. also obtained an approximate solution of (10.91) using an alterna-tive approach. They assumed <j> = y/2exp(-it')F(x',t') and derived the following


equation by inserting it into (10.91).

iFv + \FX,X, + \F\2F = -\aF:, exp(ii'). (10.92)

Equation (10.92) has the following solution at a = 0,

Fo = 2v0exp \i ( — z0 + #0 ] sechz0,L \"o /J

where z0 = 2i/o(x'-x'o), with x'o = 4fit'. If a ^ 0 but very small, we can assume thatthe parameters, i/0, /x0, x'o and 00, change only due to this perturbation. Followingthe perturbative approach of Karpman and Maslov, Fedyanin et al. obtained thefirst order correction in a to the time-dependences of the variables, v, /i, x' and 0which are given by

a f°° sinhz . ._» ,„ , . ,2 J_oo cosh z

n r°° i F*

9V = 2 ^ , - 4(M2 - z,2) + 3 r fcosh-1 z - ™*£±) (iF;)Q{z)dz,

2u J_0O \ cosh z Jwhere

<?(z) = exp [i(-Hz-g + 2*')] .

However, note that

ut» ss aft |e2i t /"°° iF*F;dz| = 0.

We then have

/ i~- -a i /gs in2* ' , x' = -^cos2 i ' , 0 = 4v%t'.

The perturbed solution of (10.91) can be written as

<Kx,t) = Aosech | ^ [a;' - cos(2t')]l exp Li (l - ^ W ^ | ^ sin(2t/)l ,

where Ao = 2%/l — w. This indicates that the magnon is still a soliton in this case.The soliton solution of (10.91) can be obtained using other approaches.

From the above discussion, we can conclude that the magnon is localized, asa soliton, due to nonlinear interactions in nonlinear ferromagnetic systems. Themotion of the magnon-soliton can be very well described by the nonlinear quantummechanics. Existence of the magnon-solitons has been verified by neutron scattering


and NMR experiments in CuGeO3 by Ronnow et al., Enderle et al., and Revaratet al., respectively. The readers are referred to the paper by Mikeska and Steiner(1991) for further details on magnetic soliton.

10.9 Collective Excitations of Magnons in Antiferromagnetic Sys-tems

Collective excitation and motion of magnons due to magnon-phonon interactionsor magnon-magnon interaction in Heisenberg antiferromagnetic systems have beenextensively studied by Pang and co-workers and many other scientists. The resultsshow that the characteristics of the collective excitation in these systems are quitedifferent from those in ferromagnetic systems. In this section, we present someresults obtained by Pang et al. on collective excitation in anisotropic Heisenbergantiferromagnets with magnon-phonon and magnon-magnon interactions.

Assuming the double-sublattice model, the Hamiltonian of the Heisenberg anti-ferromagnet can be expressed as

H = T + V + \Y,J2 [Un+sS*nS*n+5 + r,n,n+sSynS«n+d + Jn<n+sS^n+s)n S

+\ltib [ti,i+'S'S!+6 + ViJ+sS]S*+s + JS,J+SS]S!+S\ , (10-93)i s

where

n n

are the kinetic and potential energies of lattice oscillations, respectively, with mbeing the "mass" of a spin, r0 the lattice constant, and VQ the sound velocity in thecrystal which we set equal to unity in subsequent calculations, S*, N (K = x, y, z)is the spin component at site n(j) in the fc-direction. Applying the transformation,^n(j) = (^n(j) + ^n(j)) anc^ making use of the Dyson-Maleev representation of thespin operators

St = V2s(l-^f)a, S; = yfiSa* (l - ^f) , S'a = S - a+a,

St = V2Sb^(l-b-^y S^ = V2s(l-b-^)b, S>a=b+b-Si

where a+ (a) and b+ (b) are the creation (annihilation) operators of the Heisenbergmagnon field on the two sublattices, respectively. Taking into account the symmetryof the sublattices A and B and the fact that the A and B sublattices are neighbors


of each other, the Hamiltonian (10.93) can be approximately written as

A A B B

H&T + V- J0NS2 + S]T^ Jn,n+satan + S£)£ J^n+sb+bjn 8 n 8

s A A

n 8

s A A

+ 2" X] X ^ n ' n + l 5 + J?n.n+«)(On6n+« + a+6^+(5)n (5

i s

x A B

n <5

+a+on + ( 5a+an + 6++<56n+ia+6n+(5)

1 A A

~« ^ ^(^n.n+i + Vn,n+s) (aiananbn+8 + o,nbn+sb^+gbn+sn 8

+aibn+5aUn + anb++5b++sbn+s), (10.94)

where the last four terms are anomalous terms resulting from magnon-magnoninteractions. They can be neglected if we are interested only in effects of magnon-phonon interaction. We further assume that the wave function of the collectiveexcitation state of the quasi-particles in the system is of the form

A A

\f(t)) = j l + 5>o n( t)a+ + 5>w(t)&+ |0), (10.95)[ n j

where |0) is the vacuum state (ground state). In the continuum approximation, i.e.,

T T T dUf

But

d 1 2 d2 , . , . ,Vn,f±i & <Pnf ±rQ—(pn'f + -ro-,—2<pnlf -\ , (n = a,b;n = b,a),


etc., we can get the following from the Heisenberg equations for ipf,

ih((pf)t = 2SJ0<pf + S(£> - r)o)<Pf + S(£o + %)</>/ + ^(& - Vo)rZ<pfxx

g+ (& + Vo)ri<p%x - S(& - m)roufx<pf (10.96)

-S(Si + r]l)roufx<p) - 2J1Sroufx<pf,

where

iff (X,t)= ipaf (x, t) + <pbf (x,t).

Considering the symmetry of sublattices A and B, and the fact that the sublat-tices A and B are neighbors, from the Hamilton equation

-M(uf)H = -^(<p{t)\H\<p(t))

where M is the mass of a lattice point (atom, for example), Uf its displacement, aclassical quantity, we can get

-Muftt « -K'rlufxx + JiSrod^l2)* + j ( 6 - m)Sro(\tpf\2)x, (10.97)

where K' = mv$/r$ is the force coefficient, Uf is defined as Uf = uaf+Ubf. We havealso assumed that Ma = Mi, = M. Equations (10.96) and (10.97) form a completeset of equations for the collective excitations in a Heisenberg antiferromagneticsystem with magnon-phonon interactions.

For simplicity, we consider here only the anisotropic antiferromagnet with £ = T](other cases, of course, can be discussed in the same way). In this case, J > £ andJ < £ correspond to the easy magnetic axis (Oz) and the easy magnetic plane (xOy)in an antiferromagnet, respectively. Equation (10.96) reduces to

ifupt = 2/oSV + 2£0Stp* + S£orl<pxx - 2JlSrQuxip - 2^Sroux(p*. (10.98)

Using (10.97) and its conjugate equation, and performing the transformation

<p± = ip±ip*, we obtain

<Ptt-A0vxx-B0<p-Co\>p\2v = 0, (10.99)

where

Ao = ^ > 0 ,

Bo = ^ [452(J02 - e0) - 8S2ro(Jo Ji - &fc)C] ,

r _ MJiS3rUJoJi - Zoti)


and C is an integral constant. It is clear that the magnon is still a solitonin nonlinear antiferromagnetic systems. It satisfies the </>4-equation. There-fore, it can be described by the nonlinear quantum mechanics. In the case of-u2 < (AQB0)/(A0 - v2), there are non-topological soliton solutions to (10.99) ifCOI{AQ - v2) > 0, i.e., if Jo > £0, and at the same time either (a) JOJX > £0£i,v < mia.[y/K/mr0,2£0Sr0/h], or (b) J0Ji < £ofi, 2^0Sr0/h < v < y/K/Mr0. Thenormalized solitary wave is then

<p=M sech (?£*) ««*•- «>, (10.100)V vv s \ Ws J

where

k' = — W = 4 ( A ° ~ ^ 2 _ 4>BQ _ cPApC2

Ao' s Coro ' (A0-v

2) 1 6 ( ^ 0 - ^ 2 ) 2 '

Here v is velocity of the soliton. If v2 — Ao > 0, i.e., CQ/(AQ — v2) < 0, equation(10.99) has the following topological soliton solution

*-\l&.**{Hr)*'~°- (10-101)It can be seen from the above conditions that localized soliton can be excited

by the magnon-phonon coupling only for the easy magnetic axis antiferromagnet.This was not observed before. No parallel has been obtained in ferromagnet. Inthe case being discussed, the coupling of the longitudinal lattice oscillations withthe magnon results in a remarkable change in the transverse exchange integral ofthe antiferromagnet. It is the nonlinear interaction caused by the coupling that isvital to the formation of the soliton. In this case, the velocity of the soliton satisfiesv < mm[y/K'/Mr0,2£0Sr0/h}.

We have so far only considered the collective excitation caused by magnon-phonon interaction. In fact, when magnon-magnon interaction in the system be-comes too strong to be neglected, a new nonlinear interaction source will contributeto the collective excitation. The formation process and the properties of the collec-tive excitation will change accordingly if this interaction is taken into consideration.The Hamiltonian of the system in this case is still given by (10.94), but the interac-tion term now includes direct interactions between neighboring magnons and othertwo-magnon effects, i.e., influences of a magnon upon the transfer of other magnonsand upon the magnon "resonance", etc..

Pang et al. employed the following quasi-average field approximation to treatthe effects of the anomalous correlation terms in (10.90) upon the soliton formation,and the quasi-particle energy in the collective excitation, that is

a-tananb++s = {a+an)anb++s + (an&++(5)a+an - {a+an)(anb++s),atanatbt+s = (atan)a+a++s,


Then the Hamiltonian of the system (10.94) becomes

A A B B

tf = £0 + S E E Jn,n+sa+an + S £ £ Jjj+sbfbjn S j 8

1 A A

+ OS E y~^n'"+'J ~ 7ln,n+s){anbt+6 + O.+ bn+S)1 n S

A A

+ 2SY1 E^" ' n + < 5 + Tln,n+6)(anbn+6 + aj&^+j)n 6

A A

- E E Jn,n+S((aUn)bi+sbn+6 + {b++sbn+S)aUn) (10.102)n 6

* A A

- j j E E ^ n - n + l 5 ~ J?"-"+'5) [«anan) + (bn+5bn+5))(anb++s + a+bn+s)+n S

((anbt+s) + (a+bn+5))(atan + b++sbn+5)}

A A

n <5

whereA A

E0 = T + V- J0NS2 + E E Jn,n+S(aUn)K+sbn+s)+ (10.103)n S

\ E E(^ .»+« ~ W + a ) [((a+on) + <6+M6»+«»((a»6++4> + <<£&»+*»] •n *

Similar to the derivations of (10.96) - (10.98) we can obtain

<Ptt ~ A<pxx +B<p- g\<p\2lo, -o = -t»o, 5 = Co H Z2 •

Therefore, equation (10.104) has the same soliton solution as (10.100) or (10.101).

Simply replacing C o , Ao, and Bo in (10.100) and (10.101) by g, A, and B, re-

spectively, we get the solutions. Therefore, taking into account magnon-magnon

interaction only changes the amplitude and velocity of the soliton and does not

alter the fundamental nature of the collective excitation. The magnon-magnon

interactions enhance the effects of the nonlinear interactions, thereby prompt for-

mation of more stable solitons. This is because g is always greater than Co if solitons

can exist. Furthermore, Jo S> J i and £o S> £i- We can see t ha t in the presence of


magnon-magnon interactions, they are only the soliton solution of the type (10.100)or (10.101) in the velocity range of v2 < A. In this case

2 _AB_ _ r g V _ AB _ rl A r „, 2 _ 2 2y|2A-v* 16(A - v*)* " A - v* 16(A-v*)[9 + 8b{J° v ^)\ •

Pang and Xu further studied nonlinear excitations of magnons in antifer-romagnetic molecular crystal, such as Ni(C2H8N2)2-NO2(ClO4) (NENP) andNi(C3HioN2)2N02(C104) (NINO), and antiferromagnet with conservation of or-der parameters. It was found that the nonlinear excitations of magnons can still bedescribed by (10.99) or (10.104). The only differences are the coefficients in theseequations. Therefore, magnons in nonlinear antiferromagnetic systems obey thelaws of nonlinear quantum mechanics.

It should be noted that even though magnon-phonon coupling and magnon-magnon interactions have same effects on the formation of solitons, there are dif-ferences between the two types of interactions. In the first place, the mechanism oflocalized nonlinear collective excitation caused by the first type is the breaking ofkinetic symmetry, that is, it is caused by the interaction between magnon and lat-tice oscillation. In a steady state, the soliton and the localized deformation whichdepend on lattice oscillations propagate together with the same speed along theantiferromagnetic chain. The mechanism of excitation caused by the second typeis the spontaneous breaking of symmetry brought about by the magnon-magnoninteractions in the single-axis anisotropic antiferromagnet. However, the collectiveexcitations resulting from both mechanisms have the same characteristics, i.e., thestructural anisotropy. As it was already mentioned earlier, the two mechanismsmay cancel each other in isotropic antiferromagnetic chains, and no soliton can ex-ist as in the case of ferromagnetic chains. It also indicates that soliton excitationof magnons in such systems is determined by the anisotropy of the system. Aslong as anisotropy exists in a given system, magnon-phonon coupling and nonlinearinteractions between magnons will make the magnons "self-trapping" in a range ofdimension 2Wa in the one-dimensional chains and a stable soliton is excited. Whenthe anisotropy changes, the amplitude, the momentum, and the number of solitonsall change accordingly.

Formation of solitons due to nonlinear interactions in anisotropic antiferromag-netic chain lead to many interesting physical phenomena. Indeed, anomalies havebeen observed in experiments. Attempts have been made to explain them usingthe magnetic soliton model, even though analytical expressions in place of (10.100)and (10.101) have not been obtained. (The readers are referred to work by Mikeskaand Steiner 1991) For example, Boucher et al. used the soliton concept to ex-plain the phenomenon of nuclear spin-lattice relaxation (NSLR) in antiferromag-netic chain (CH3)4NMnCl3, even though theoretical expression for magnetic soli-tons has not been obtained. Through measurement, Boucher et al. obtained theratio T^"1 of NSLR of N15 in antiferromagnet, as a function of external field H (2


kAm"1 < H < 80 kAm"1) and temperature T (2 K < T < 4.2 K), and observedthat Tj"1 diverged exponentially with H/T at a certain temperature. With theanalytical results for the soliton given above, we can explain the excitation and thebehavior of the soliton in such systems, which in turn validates the theory.

It should also be pointed out that effects of external fields are not included inour discussion. If any such field is present and if it is in the direction along theeasy magnetic axis, then its effect, due to the opposite magnetization directionsof the two sublattices, will be equivalent to a periodic external field of period 2TQwhich strengthens the discreteness of the lattice and thus invalidates the contin-uum approximation. However, if the direction of the external field is perpendicularto the antiferromagnetic spin direction, the continuum approximation will still bevalid. For this reason, earlier experimental and theoretical studies were concen-trated mainly on transverse fields, rather than longitudinal fields.

Bibliography

Aliotta, F. and Fontana, M. P. (1980). Optica Acta 27 931.Antonchenko, V. Ya., Davydov, A. S. and Zolotaryuk, A. V. (1983). Phys. Stat. Sol.(b)

115 631.Bell, R. P. (1973). The proton in chemistry, Chapman and Hall, London.Bernal, J. D. and Fowler, R. H. (1933). J. Chem. Phys. 1 515.Binder, K. (1984). Applications of the Monte Carlo Method, Springer-Verleg, Berlin.Bjerrum, N. (1952). Science 115 385.Bogoliubov, N. N. (1950). Ukraimian. Mat. J. 11 1950276.Bontis, T. (1992). Proton transfers in hydrogen bonded systems, Plenum Press, London.Boucher, J. P., Pynn, R., Remoissewet, M., Regnault, L. P., Endo, Y. and Renard, J. P.

(1980). Phys. Rev. Lett. 45 486.Boucher, J. P., Pynn, R., Remoissewet, M., Regnault, L. P., Endo, Y. and Renard, J. P.

(1990). Phys. Rev. Lett. 64 1557.Braum, J. M. and Klvshar, Yu. S. (1990). Phys. Lett. A 149 119.Braum, J. M. and Klvshar, Yu. S. (1991). Phys. Rev. B 43 1060.Brown, D. W. and Ivic, Z. (1989). Phys. Rev. B 40 9876.Brown, D. W., Lindenberg, K. and West, B. J. (1986). J. Chem. Phys. 84 1574.Brown, D. W., Lindenberg, K. and West, B. J. (1987). J. Chem. Phys. 87 6700.Brown, D. W., Lindenberg, K. and West, B. J. (1988). Phys. Rev. B 37 2946.Brown, D. W., West, B. J. and Lindenberg, K. (1986). Phys. Rev. A 33 4110.Buttiker, M. and Ladauer, R. (1981). Phys. Rev. A 23 1397.Cao, Changliang (1993). Physics Sin. 22 361.Chikazumi, S. (1981). Physics in high maginet fields, Springer-Verlag, Berlin.Chochliouros, I. and Pouget, J. (1995). J. Phys. Condensed Matter 7 8741.Currie, J. F., Krumhansl, J. A., Bishop, A. R. and Trullinger, S. E. (1980). Phys. Rev. B

22 477.Davydov, A. S. (1982). Biology and quantum mechanics, Pergamon, New York.Davydov, A. S. and Kislukha, N. I. (1973). Phys. Stat. Sol. (b) 59 465.Davydov, A. S. and Kislukha, N. I. (1976). Sov. Phys.-JETP 44 571.Derlin, J. J. (1990). Int. Rev. Phys. Chem. 9 29.


Desfontaines, H. and Peyrared, M. (1989). Phys. Lett. A 142 128.Devreese, J. T. (1972). Polarons in ionic crystals and polarsecal counductors, NorthHol-

land, Amsterdam.Dwicki, J. C.,et al. (1997). J. Am. Chem. Soc. 99 7403.Eigen, M. and De Maeyer, L. (1958). Proc. Roy. Soc. London, Ser. A 247 505.Eigen, M., de Maeyer, L. and Spatz, H. C. (1962). Physics of ice crystals, Coll. London.Eisenberg, D. and Kanzmann, W. (1969). The structure and properties of water, Claren-

don, Oxford.Enderle, M.,et al. (2001). Phys. Rev. Lett. 87 177203.Engelheart, H., Bullemer, B. and Riehl, N. (1969). in Physics of Ice, ed. N. Riehl, B.

Bullemer and H. Engelheart, Plenun, New York.Evans, M. W. (1982). J. Chem. Phys. 76 5473 and 5480; ibid 77 4632.Evans, M. W. (1983). J. Chem. Phys. 78 925.Evans, M. W. (1987). J. Chem. Phys. 87 6040.Fedyanin, V. and Yakushevich, L. (1976). JINR pl7-96-27, Dubna.Fedyanin, V. and Yakushevich, L. (1980). JINR pl7-80-69, Dubna.Fedyanin, V. and Yakushevich, L. (1981). Phys. Lett. A 85 100.Fedyanin, V. and Yushankhai, Y. (1978). JINR pl7-11996, Dubna.Fedyanin, V. K. and Makhaankov, V. G. (1979). Phys. Scr. 20 552.Fedyanin, V. S. and Yushankhai, V. (1978). Teor. Mat. Fiz 35 240.Fisher, A. J., Hayes, W. and Wallance, D. S. (1989). J. Phys.: Condensed Matter 1 5567.Fraggis, T., Pnevmatikos, St. and Economon, E. N. (1989). Phys. Lett. A 142 361.Frohich, H., Pelzer, H. and Zienau, S. (1950). Plilos. Mag. 41 221.Frohlich, H. (1954). Adv. Phys. 3 325.Goldanskill, V. I., Krupyanskii, Yu. F. and Flerov, V. N. (1983). Dokl. Akad. Nauk (SSSR)

272 978.Granicher, H. (1963). Phys. Kond. Materie 1 1.Habbard, L. (1963). Proc. Roy. Soc. A 276 238.Halding, J. and Londahl, P. S. (1988). Phys. Rev. A 37 2608.Higashitani, K. (1992). J. Colloid and Interface Science 152 125.Higashitani, K. (1993). J. Colloid and Interface Science 156 90.Hobbs, P. V. (1974). Ice physics, Clarendon Press, Oxford.Hochstrasser, D., Buttner, H., Defontaines, H. and Peyrared, M. (1988). Phys. Rev. A 38

5332.Hochstrasser, D., Buttner, H., Defontaines, H. and Peyrared, M. (1989). J. Physique. Coll.

50 3.Holstein, T. (1959). Ann. Phys. 8 325 and 343.Homilton, W. C. and Ibers, J. A. (1969). Hydrogen bonding in solids, Benjamin, New

York.Ivic, Z. and Brown, D. W. (1989). Phys. Rev. Lett. 63 426.Jiang, Yijian, Jia Qingjiou, Zhang Peng Cheng and Xu Lu, (1992). J. Light Scattering 4

7216.Joshi, K. M. and Kamat, P. V. (1965). J. Indian Chem. Soc. 43 620.Karpman, V. I. and Maslov, E. M. (1977). Sov. Phys. JETP 46 281.Karpman, V. I. and Maslov, E. M. (1978). Sov. Phys. JETP 48 252.Kashimori, Y., Kikuchi, T. and Nishimoto, K. (1982). J. Chem. Phys. 77 1904.Kawada, A., McGhie, A. R. and Labes, M. M. (1970). J. Chem. Phys. 52 3121.Ke Laxin, B. N. (1982). Magnetization of water, Beijing, Measurement Press.Klivshar, Yu. S. (1991). Phys. Rev. A 43 3117.Koutselose, A. D. (1995). J. Chem. Phys. 102 7216.


Kristoforov, L. N. and Zolotaryuk, A. V. (1988). Phys. Stat. Sol. (b) 146 487.Krumhansl, G. A. and Schrieffer, J. R. (1975). Phys. Rev. B 11 3535.Kuper, C. G. and Whitfield, G. D. (1963). Polarons and excitons, Plenum Press, New

York.Kusalik, P. G. (1995). J. Chem. Phys. 103 10174.Laedke, E. W., Spatschek, K. H., Wilkens, M. and Zolotaryuk, A. V. (1985). Phys. Rev.

A 32 1161.Landau, L. D. (1933). Phys. Z. Sowjetunion 3 664.Li, Benyuan (1976). Physics Sin. 5 240.Liemeza, J. (1976). Z. Phys. Chem. 99 33.Lindner, U. and Fedyanin, V. (1978). Phys. Stat. Sol. (b) 89 123.Lindner, U. and Fedyanin, V. (1978). Phys. Stat. Sol. (b) 95 k83.Makhankov, V. G. (1981). Phys. Lett. A 81 156.Makhankov, V. G. and Fedyanin, V. K. (1984). Phys. Rep. 104 1.Makhankov, V. G., Makhaidiani, N. V. and Pashaev, O. K. (1981). Phys. Lett. A 81 161.Marcheson, F. (1986). Phys. Rev. B 34 6536.Mazenko, G. F. and Sahni, P. S. (1978). Phys. Rev. B 18 6139.Mei, Y. P. and Yan, J. R. (1993). Phys. Lett. A 180 259.Mei, Y. P., Yan, J. R., Yan, X. H. and You, J. Q. (1993). Phys. Rev. B 48 575.Mikeska, H. J. (1979). J. Phys. C 11 129.Mikeska, H. J. and Steiner, M. (1991). Adv. Phys. 40 pl91-356.Mittal, R. and Howard, I. A. (1999). Physica D 125 79.Mouritsen, O. G. (1978). Phys. Rev. B 18 465.Mrevlislivil, G. M. (1979). Sov. Phys. USP. 22 433.Mrevlislivil, G. M. (1979). Usp. Fiz. Nauk 128 273.Muller, K. (1970). Z. Chem. 10 216.Myneni, S.,et al. (2002). J. Phys. Condensed Matter 14 L213.Nagle, J. F. and Morowitz, H. J. (1978). Proc. Natl. Acad. Sci. USA 75 298.Nagle, J. F. and Nagle, T. S. (1983). J. Membr. Biol. 74 1.Nagle, J. F., Mille, M. and Morowitz, H. J. (1980). J. Chem. Phys. 72 3959.Nelmes, R. J. (1980). Ferroelectrics 24 237.Nylund, E. S. and Tsironis, G. P. (1991). Phys. Rev. Lett. 66 1886.Onsager, L. (1969). Science 166 1359.Oraevskii, A. N. and M. Yu. Su, Dakov, (1987). Zh. Eksp. Teor. Fiz. 92 1366 [Sov. Phys.

JETP 65 767].Pang, X. F. and Chen, X. R. (2002). Commun. Theor. Phys. 37 715.Pang, Xiao-feng (1990). Acta Math. Phys. Sin. 10 47.Pang, Xiao-feng (1993). Phys. Stat. Sol. (b) 189 237.Pang, Xiao-feng (1994). Theory of nonlinear quantum mechanics, Chongqing Press,

Chongqing, pp. 554-573 and pp. 574-583.Pang, Xiao-feng (2000). Chin. Phys. 9 86.Pang, Xiao-feng (2000). J. Shandong Normal Univ. Sin. (Nature) 15 43.Pang, Xiao-feng (2002). Prog. Phys. Sin. 22 214.Pang, Xiao-feng (2003). Phys. Stat. Sol. (b) 239 862.Pang, Xiao-feng (2003). Soliton physics, Sichuan Sci. and Tech. Press, Chengdu, pp. 372-

482 and pp. 714-771.Pang, Xiao-feng and Feng, Yuan-ping, (2003). Chem. Phys. Lett. 373 392.Pang, Xiao-feng and Feng, Yuan-ping, (2003). Chin. Phys. Lett. 20 1662.Pang, Xiao-feng and Muller-Kersten, H. J. W. (2000). J. Phys.: Condens. Matter 12 885.Pang, Xiao-feng and Zundel, G. (1997). Acta Phys. Sin. 46 625.


Pang, Xiao-feng and Zundel, G. (1998). Chin. Phys. 7 70.Pan, Zhongcheng and Xun Chengxian (1985). Chin. Med. Phys. 7 1985226.Pauling, L. (1960). The nature of chemical bond, Cornell University, Ithaca.Pekar, S. (1946). J. Phys. USSR. 10 341 and 347.Peyrard, M. (1995). Nonlinear excitation in biomolecules, Springer, Berlin.Peyrard, M., Pnevmatikos, St. and Flytzanis, N. (1986). Physica D 19 268.Peyrard, M., Pnevmatikos, St. and Flytzanis, N. (1987). Phys. Rev. A 36 903.Pimentel, G. and McClellan, A. (1960). The hydrogen bond, Freeman, San Francisco.Pnevmatikos, St. (1988). Phys. Rev. Lett. 60 1534.Pnevmatikos, St., Flytzanis, N. and Bishop, A. R. (1987). J. Phys. C. Solid State Phys.

20 2829.Pnevmatikos, St., Savin, A. V., Zolotariuk, A. V., Kivshar, Yu. S. and Velgakis, M. J.

(1991). Phys. Rev. A 43 5518.Pushkarov, D. and Pushkarov, Kh. (1977). J. Phys. (c) 10 3711.Pushkarov, D. and Pushkarov, Kh. (1978). Phys. Stat. Sol. (b) 90 361.Pushkarov, D. and Pushkarov, Kh. (1979). Phys. Stat. Sol. (b) 81 703.Pushkarov, D. and Pushkarov, Kh. (1979). Phys. Stat. Sol. (b) 93 735.Pushkarov, D. and Pushkarov, Kh. (1984). Phys. Stat. Sol. (b) 123 573.Revarat, Y. F.,et al. (1996). Phys. Rev. Lett. 77 1861.Rodriges, K. and Fedyanin, V. (1981). JINRp. 17-81-169, Dubna; JINRp. 17-81-8, Dubna.Ronnow, H. M.,et al. (2000). Phys. Rev. Lett. 84 4469.Scheider, T. and Stoll, E. (1975). Phys. Rev. Lett. 35 2961.Scheider, T. and Stoll, E. (1978). Phys. Rev. Lett. 41 1429.Scheider, T. and Stoll, E. (1980). Phys. Rev. B 22 5317.Schmidt, V. H., Drumeheller, J. E. and Howell, F. L. (1971). Phys. Rev. B 4 4582.Schuster, P., Zundel, G. and Sandorfy, C. (1976). The hydrogen bond, recent developments

in theory and experiments, North-Holland, Amsterdam.Sergienko, A. I. (1987). Phys. Stat. Sol. (b) 144 471.Sergienko, A. I. (1988). Sov. Phys. Solid State 30 496.Sergienko, A. I. (1990). Sov. Phys-JETP 70 710.Skrinjar, M. J., Kapor, D. W. and Stojanovic, S. D. (1988). Phys. Rev. A 38 6402.Sokolov, N. D. (1955). Usp. Fiz. Nauk. 57 205.Song, Dongning (1997). J. App. Math, and Mech. 18 113.Tsironis, G. P. and Pnevnatikos, St. (1989). Phys. Rev. B 39 7161.Walrafen, G. E. (1970). J. Chem. Phys. 52 4276.Walrafen, G. E. (1986). J. Chem. Phys. 85 6964.Wang, X. D., Brown, D. W., Lindenberg, K. and West, B. J. (1986). Phys. Rev. A 33

4101.Weberpals, H. and Spatschek, K. H. (1987). Phys. Rev. A 36 2946.Weiner, J. H. and Asker, A. (1970). Nature 226 842.Whalley, E., Jones, S. J. and Grold, L. W. (1973). Physics and chemistry of ice, Roy. Soc

Canada, Ottawa.Whitfield, G. and Shaw, P. (1976). Phys. Rev. B 14 3346.Xie, Wen Hui (1983). Magnetized water and its application, Science Press, Beijing.Xu, C. T and Pang Xiao-feng (1996). Chin. J. Atom. Mol. Phys. 12 508.Xu, C. T and Pang Xiao-feng (1998). Chin. J. Atom. Mol. Phys. 14 219.Xu, C. T and Pang Xiao-feng (1999). Chin. J. Atom. Mol. Phys. 16 506.Xu, C. T and Pang Xiao-feng (2000). Chin. J. Atom. Mol. Phys. 17 99 and 513.Yomosa, S. (1982). J. Phys. Soc. Japan 51 3318.Yomosa, S. (1983). J. Phys. Soc. Japan 52 1866.


Zhang, Q., Romero-Rochin, V. and Silbey, R. (1988). Phys. Rev. A 38 6409.Zolotariuk, A. V. (1986). Theor. Math. Fiz. 68 415.Zolotariuk, A. V., Spatschek, K. H. and Laedke, E. W. (1984). Phys. Lett. A 101 517.Zolotaryuk, A. V. and Pnevmatikos, S. (1990). Phys. Lett. A 143 233.Zolotaryuk, A. V., Peyrard, M. and Spatschek, K. H. (2000). Phys. Rev. E 62 5706.Zundel, G. (1972). Hydration and intermolecular interaction, Mir, Moscow.

Index

D operator, 375 Bekki, 422, 438TV-soliton solution, 130, 132 bell-type non-topological soliton, 116<£4-equation, 75 Belova, 208<£4-field equation, 84, 144, 340, 378 Bender, 370</>6-nonlinear Schrodinger equation, 448, Bernal-Fowler filaments, 557

464 Bernstein, 3064>%-&e[d equation, 283 Besley, 459</>i-field equation, 283 binding energy, 241

Bishop, 356Abdullaev, 143, 426-428, 431, 433-436, Bjerrum defect, 558, 563

438 Bjorken, 323, 324, 326Ablowitz, 88, 330, 394 Blow, 136absorption coefficient, 488 Bluman, 110acetanilide, 471 Bogoliubov, 323, 324, 326adiabatic approximation, 402, 428, 431, Bogolubov dispersion, 441

435, 436, 452, 488, 592 Boivin, 300, 302ADZ model, 559 Born approximation, 183, 200, 424, 488Alexander, 473, 492, 544 Bose-Einstein condensation, 35Alonso, 197 liquid helium, 72Anderson, 146 of exciton, 33Anderson localization, 200, 423 Boucher, 612anomalous dispersion region, 254 bound state, 283antifluxon, 62 Brachet, 74Antonchenko, 559 breather, 226Aossey, 150, 153 bright soliton, 136auto-Backlund transformation method, Brizhik, 264, 265

379 Brown, 503, 505, 593, 594average velocity, 170 Burgur's equation, 240

Burt, 245, 247, 321, 324Backlund transformation, 90, 330, 379

elementary, 90 Cai, 544Badger-Bauer rule, 580 canonical commutative relation, 99Barashenkov, 438-440, 442-444 canonical conjugate variables, 101Barenghi, 74 canonical quantization, 95, 99Barthelemy, 273 Careri, 520, 536, 537, 543, 547BCS theory, 24 Carr, 306

619


Carter, 218, 219 D defect, 558Causality, 321 D'Alembert, 595center of mass, 118 damping function, 184

acceleration, 118 dampling coefficient, 178equation of motion, 119 dark soliton, 136position, 121 dark soliton solution, 242velocity, 120, 121 Davidson, 249

characteristic curve, 413 Davydov, 250, 258, 342, 343, 345, 386,characteristic direction, 332 387, 501, 559, 604characteristic equation, 332 Davydov model, 473, 475characteristic line, 332 improved, 142Chen, 405, 407, 409, 428, 430, 529 Davydov soliton, 526, 528Cherednik, 274 De Oliveira, 447Chi, 545 Debye temperature, 491Chiao, 399 decay rate, 516, 517Chu, 134 Degtyarev, 419circulation, 73 density operator, 181circulation of velocity, 30 Desem, 134Cohen, 402 destroy period, 239coherent excitation, 249 Desyatnikov, 459, 464, 466, 467coherent length, 24 Dianov, 273coherent state, 40, 295 diffusion, 178Cole-Hopf transformation, 240 coefficient, 178, 187, 188collective coordinate method, 99 equation, 240collective excitation, 607 parameter, 184

in antiferromagnetic system, 607 disordered system, 423in liquid water, 577 dispersion effect, 233, 236

collective-coordinate method, 180 dispersion equation, 236collision, 126, 154 dispersion relation, 236compatibility condition, 105 dispersive effect, 253complementary principle, 12 dispersive medium, 236completely integrable system, 105 displacement current, 60condensation dissipative term, 61, 64

Bose-Einstein, 35 Dodd, 283of momentum, 35 Doebner, 83of position, 35 Doktorov, 452

conservation Doran, 136of mass, 117 double-well potential, 241

Cooper, 367, 368, 370 Duncan, 370Cooper pairs, 24 Dyson-Maleev representation, 607Cottingham, 502, 513, 515, 518 Dziarmaga, 185, 186, 188coupling constant, 181Cowan, 444 effective mass, 144, 145Crespe, 452 effective potential, 145critical current, 26 Ehrenfest's theorem, 406critical dimension, 114 Eigen, 574critical magnetic field, 25 eigenequation, 87critical temperature, 23 eigenvalue, 87Cruzeiro-Hansson, 503, 505 Eilbeck, 306, 473, 543Cui, 269 elliptic wave, 76

Index 621

Emplit, 136 time-dependent, 66Enderle, 607 Ginzburg-Landau parameter, 25Engelheart, 575 Gisin, 83Eremko, 480 Goldanskill, 548Ermakov-Pinney equation, 463 Goldstone mode, 125Euler-Lagrange equation, 102, 171 Gordon, 74, 135, 136, 147exciton, 33 Gorshkov, 143, 144, 147, 353, 356

binding energy, 263 gravitational force, 76Bose-Einstein condensation, 33 Green 330self-trapping, 263 Grimshaw, 352

Gross, 74Poner, 502 Gross-Pitaerskii equation, 46, 72Forner, 503, 505, 529 Gross-Pitaevskii-Anandan equation, 75Faddeev, 401 ground state, 37Falkovich, 256

Fann, 538-541 Halding, 526Fedyanin, 586-589, 591, 596, 605, 606 H a l l e f f e c t 3 1

Firth, 464 Hartree approximation, 159Fischer, 508 Hasegawa, 150, 252, 272, 351, 422flux quantum, 24 Heisenberg, 291fluxon, 62 Heisenberg antiferromagnet, 607rogel, 359, 364 u • u r i. cm_ f, , ' Heisenberg ferromagnet, 604Foldy, 411 , ... . or._ . , , _ hereditary operator, 85Fomin, 110 , . , , ,. . ...cv-, ,. , 9t-n , m high-order dispersion, 444Kronhch, 250, 593 „.,,TT_ i_ c j-nr i.- nrsr. mo Hubert transformation, 128Fraunhofer diffraction, 209, 212 „.„ . „ „ _ '„ i, i i, ,• .i oi 1 Hill equation, 413, 414

Fredholm alternative theorem, 311Frish, 74 H l r o t a ' 3 8 7

frozen magnetic flux, 24 H i r o t a m e t h o d ' 3 3 1 ' 375> 3 7 8

function transformation, 331 Hobbs, 575fundamental soliton state, 299 h o l e s o l l t o n ' 1 3 6

Holstein, 586, 587, 589Gaididei, 459-464 Hubbard model, 596Galilean invariance, 198 h u m a n issues, 545Galilean transformation, 114, 450 Husimi, 405Galilei invariance, 315Galilei transformation, 315, 334 Ichikawa, 350Garcia-Ripoll, 411, 413, 414 Ikez i> 1 5 3 ' 1 5 4

Gardner, 330 impurity mode, 206Gashi, 496, 497 Infeld, 459Gauge transformation, 316 infinitesimal transformation, 333Gelfand, 110 influence functional, 182Gelfand-Levitan-Marchenko (GLM) initial stability, 162

equation, 342 instability interval, 413Genenbauer polynomial, 248 integrable model, 173Germanschewski, 459 interacting many-particle system, 438GGKM transformation, 330 interactionGinzburg, 74 between microscopic particles, 146Ginzburg-Landau equation, 40 kink-antikink, 145

steady state, 48 kink-kink, 145


of microscopic particle with random Kuznetsov, 115field, 434

interaction potential, 144 L defect, 558interference function, 321 L'yov, 256intrinsic nonlinearity, 243 Lagrangian function density, 72inverse scattering method, 201, 264, 330, Lamb, 153, 154, 384, 386, 387

340, 345, 415, 437, 456 Landau, 263inverted quadratic potential, 409 Landau damping, 350iso-transformation, 313 Landau levels, 32Ivic, 496, 497, 593-595 Larmor frequency, 400

Larraza, 267Jacobian amptitude, 66 Lawrence, 526Jacobian sine function, 66 Lax, 87, 88, 212, 330Jiang, 578, 579, 586 Lax equation, 126Jordan, 83 Lax operator equation, 87Jordan-Wigner transformation, 605 Legendre elliptical integral, 66Josephson current, 27, 58 lens transformation, 461Josephson effect, 26, 28 Lieb, 315Josephson relations, 60 Lindner, 596, 598Jost coefficient, 201, 217, 437 linear fluid equation, 332Jost function, 138, 201, 340, 347 linear heat conduction equation, 334

linear potential, 407Kabo, 490 linear-quadratic potential, 410Kalbermann, 188-191, 193, 209 Liouville equation, 332Kalbermarnn, 189 liquid helium, 28Karlsson, 462 Lisak, 146Karpman, 135, 146, 201, 346, 350, 352, Liu, 405, 407, 409, 428, 430

399, 444-447, 606 localization energy, 241Kartner, 300, 302 localization of exciton, 258Kaup, 346, 351 localization of microscopic particleKdV soliton, 150 experimental verification, 267Ke Laxin, 578 initial condition, 263Kerr, 254, 503, 505, 526 Lomdahl, 503, 526Kerr coefficient, 254 Lonngren, 194, 195Kerr effect, 252, 254 Lorenz broadening, 420kink soliton solution, 242 Lucheron, 367Kislukha, 604Kivshar, 201, 205, 206, 208, 346-348, 351, macroscopic quantum effects, 23

415, 417, 419, 420, 423, 425 Madelung's transformation, 105Klyatzkin, 432 Maeyer, 574Kodama, 351, 420, 422 magnon, 601Konno, 382 magnon-magnon interaction, 607Konopelchenko, 459 magnon-phonon interaction, 607Konotop, 212-215 Maissner effect, 24, 25Krokel, 136 Makhankov, 145, 162, 164, 165, 168, 313,Kruglov, 464 438-440, 442-444, 588, 589, 596-599,Krumbansl, 473, 492, 544 601, 605Kruskal, 154, 330 Malomed, 346, 350, 415, 417, 419, 420,Kumei, 110 453-455, 458Kundu, 444 Manakov, 313

Index 623

Manakov's formula, 216 completeness, 277Marchenko equations, 136 fundamental principles, 84Markovian-Wiener process, 187 fundamental theory, 89Maslov, 346, 352, 606 hypotheses, 84mass conservation, 106 self-consistency, 277mass density, 85 validity, 277Mathieu's equation, 463 nonlinear recursion operator, 86mean field method, 431 nonlinear Schrodinger equation, 144, 145,Michinel, 464, 466 194, 200, 202, 251, 255, 264, 299, 331,Microcausal, 321 334, 336, 344, 345, 350, 352, 355, 375,microscopic particle 382, 426, 427, 433-435, 437, 438, 444,

acceleration, 122, 123 447, 451-453, 459, 460, 478, 495, 506,diffusion, 178 588, 601, 605group velocity, 123 bell-type non-topological solitonmass conservation, 106 solution, 116mass density, 85 eigenvalue problem, 315motion, 124 fluid-dynamical form, 106stability, 161 fourth-order, 446transport property, 178 generalized, 84, 448versus macroscopic particle, 188 high-order dispersion effect, 422, 455

Mielnik, 83 quintic nonlinearity, 173Mikeska, 607, 612 stochastic, 432Miles, 268 third-order, 446Miller, 562 two-dimensional, 388, 411Mitschke, 273 with V(x') = ax', 122Miura, 330 with V0(x') = a V 2 , 122Mollenauer, 272, 273 with a driving field, 422Morris, 444, 448 nonlinear superposition, 91Moussa, 452 nonlinear superposition principle, 89Mrevlishvil, 548 nonpropagating surface water soliton, 269

normal dispersion region, 254Nagle, 574 Nozaki, 350, 422, 438Nagy, 508 Nylund, 574-576Nakamura, 387, 391Noether theorem, 110 Okopinska, 370Nogami, 405, 406, 408, 410, 411 Onsager, 74non-integrable model, 173 operator, 85non-reflective potential, 343 order parameter, 36non-stationary medium, 427 liquid helium, 72nonlinear Dirac equation, 291 Ostrovsky, 143, 144, 147, 353, 356nonlinear excitation Ott, 350

magnon, 601polaron, 586 Packard, 74

nonlinear fluid equation, 332 Painleve equation, 393nonlinear Fourier transformation, 94 Pang, 31, 268, 306, 310, 359, 473, 474,nonlinear Klein-Gordon equation, 84, 281, 476, 480, 483, 491, 492, 497, 500, 503,

352, 354 506, 516, 517, 521, 526, 529, 541, 544,nonlinear operator, 85 545, 562, 579, 586, 607, 610, 612nonlinear perturbation theory, 100 Pathria, 444, 448-450nonlinear quantum mechanics Pekar-Prolich, 589


perfect anti-magnetism, 24 reductive perturbation theory, 441perfect conductivity, 23 reflection coefficient, 204, 213, 214, 424Perring, 287 for scattering by point impurity, 202persistent interaction, 244 for scattering by two impurities, 203persistently interacting fields, 324 Reid, 413perturbation relativistic effect, 74

conservative, 174 relativistic theory, 281perturbation method, 335 Revarat, 607perturbation theory, 345 Reynaud, 273Pestryakov, 250 Rica, 74Petrov, 464 Riccati differential equation, 213Petviashvili, 350 Riemann invariance, 332phase shift, 149 Ripoll, 413phase velocity, 236 Roberts, 74Pinney, 463 Rodriges, 589Pismen, 74, 350 Rogers, 379Pitaevskii, 74 Ronnow, 607Pokrovsky, 459 Rose, 370polaron, 586 Rowlands, 459polynomial currents, 245 Rubinstein, 286Pomeau, 74Pomeranchuk effect, 29 S-I-S junction, 56proton transfer, 557 S-N-S junction, 56proximity effect, 54 saturable model, 465pseudo-conformal transformation, 115 scale transformation, 114, 442Pushkarov, 601, 604 scaling relation, 370

scattering data, 127, 136, 342quadratic potential, 407 scattering matrix, 138quantization scattering of microscopic particle

condition, 157 by N point impurities, 204magnetic flux, 24 by an isolated point impurity, 201methods, 95 by impurities, 200

quantum diffraction by two point impurities, 202in superconducting junction, 27 Scharf, 221, 222, 224, 225, 227

quantum fluctuation, 299 Schochet, 370quantum fluxon, 62 Schwarz, 74quantum Hall effect, 31 Schweitzer, 502, 503, 505, 513-515, 518quantum magnetometer, 25 Scott, 306, 310, 473, 475, 508, 529, 531,quantum nonlinear Schrodinger equation, 543, 544, 546

155 self-condensation, 242quasi-coherent state, 477, 505 self-consistency, 281quasi-probability density, 158 self-focus, 255quasiperiodic perturbation potential, 221 self-focusing, 242, 257Quiroga-Teixeiro, 464 self-focusing effect, 256

self-interaction, 242, 243, 245random inhomogeneous medium, 436 rational current, 246Rangwala, 379 transcendental current, 246Rao, 379 self-localization, 241rational current, 246 energy, 241Redekopp, 387 of water molecules, 269

Index 625

self-similarity solution, 333 squeezing state, 296self-similarity transformation, 333 stability, 161self-trapping, 242 interval, 413

mechanism, 258 structural, 161, 164Sergeev, 350 statistical perturbation method, 436Shabat, 88, 134, 136, 163, 216, 242, 264, Steiner, 607, 612

313, 330, 340, 347 stochastic nonlinear wave equation, 433Shafheitlin, 265 stochastic parametric soliton resonance,Shaw, 591 435Shchesnovish, 452 Stolen, 272Shepard, 367, 370 structural stability, 162, 164Simmons, 370 Sulem, 110, 255Sine-Gordon equation, 124, 283, 331, 333, superconducting junction

334, 337, 364, 376, 383, 384, 387 quantum diffraction, 27breather solution, 340 superconductivitydissipative term, 61, 64 critical temperature, 23generalized, 84 Davydov's theory of, 37kink soliton solution, 62 superfluid liquid heliumkink solution, 338 relativistic motion, 76Lamb solution, 63 superfluidityone-dimensional, 61 of 4He, 29perturbed, 225, 356 superpropagator, 182stationary, 168 symmetrytwo-dimentsional, 63, 64 spontaneous breakdown, 38

Sine-Gordon model, 206, 209Skinner, 150, 153 Tajiri, 378, 387-389, 392Skryabin, 464 Takagi, 397, 399, 400Skyrme, 287 Takeno, 489Smith, 273 Takhtajan, 401SNIS junction, 59 Talopov, 459soliton, 83 Teixeiro, 466

binding energy, 507 Thurston, 150bright, 136 time-dependent parabolic potential, 411dark, 136 Tones, 370effective mass, 507 Toyama, 405, 406, 408, 410, 411energy, 507 transition probability, 515, 516hole, 136 translation mode, 365in optical fibers, 272 transmission coefficient, 424, 456kink I, 570 transport property, 178kink II, 571 trapping potential, 46rest energy, 507 traveling-wave method, 335, 336

Solovev, 135, 146 traveling-wave solution, 335, 336specific heat, 577 Tsironis, 574-576spectral parameter, 104 tunneling, 209spinning soliton, 466, 467 tunnelling effect, 26spontaneous coherent state, 41 Turitsyn, 115squeezed state two-component solitary wave, 561

of double-Cooper-pairs, 42 two-soliton state, 147, 158squeezing coefficient, 296 type-I superconductor, 25squeezing effect, 218 type-II superconductor, 25


Uncertainty relationin linear quantum mechanics, 292

variable transformation, 332vector nonlinear Schrodinger equation,

304, 313Venzel, 508vortex, 25Vysloukh, 274

Wadati, 382Wang, 502, 595water, 577wave function, 85Webb, 546, 547Weinberg, 83Weinstein, 370Whitfield, 591Whitham, 247winding number, 192Wu, 267

Xiao, 541Xu, 612

Yamchuk, 74Young, 508

Zabusky, 153, 154Zakharov, 88, 134, 136, 150, 163, 216, 242,

256, 264, 313, 330, 340, 347, 417Zakharov-Shabat equation, 126, 212, 457Zolotaryuk, 559

Quantum Mechanics in NonLinear Systems

Documents

van der merwe

linear quantum mechanics

macroscopic quantum effects

original quantum mechanics

macroscopic quantum effect

magnetic field hz

frozen magnetic flux

quantum hall effect