Mathematics provides physical science with a language that connects cause and effect. The equations and functions of mathematics provide a grammar and.

Mathematics provides physical science with a language that connects cause and effect. The equations and functions of mathematics provide a grammar and a vocabulary to understand and interpret the physical world. Mathematical language helps uncover hidden connections between seemingly disparate physical phenomena, and helps us to think about nature in new ways.

In this lecture, we will illustrate the power of mathematical thinking by tracing the story of solitary waves, beginning with their discovery by John Scott Russell in 1834.

The Solitary Wave of Translation

(Edinburgh, 1834)The KdV Equation

(Amsterdam, 1895)

The Birth of Experimental Mathematics

(Los Alamos, 1955)

Solitary Waves and Quantum Waves

(Princeton and New York, 1964-68)

New Frontiers(1972-Present)

Making Waves with Mathematics

Åke Hultkrantz (1920-2006), Swedish Anthropologist and scholar of the Saami, Shosone, and Arapaho peoples

The Solitary Wave of Translation

John Scott Russell(1808-1882)

This is a most beautiful and extraordinary phenomenon. The first day I saw it was the happiest day of my life. Nobody had ever had the good fortune to see it before or, at all events, to know what it meant. It is now known as the solitary wave of translation.

John Scott Russell, 1865

“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed.”

Union Canal, Gyle, Edinburgh, Scotland

“I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.”

John Scott Russell, Report on Waves, Fourteenth Meeting of

the British Association for the Advancement of Science,

September 1844

Plaque at Hermiston House, Union Canal

Russell re-created the solitary waves and discovered four facts about them:

(1)The shape of the wave is described by the hyperbolic secant function

(2)A larger solitary wave travels faster than a smaller one

(3)Solitary waves can cross each other “without change of any kind”

(4) A large enough mass of water creates two or more independent solitary waves

The Hyperbolic Secant Function

The Korteweg-De Vries Equation

Diederik Korteweg(1848-1941)

Gustav de Vries(1866-1934)

We are not disposed to recognize this wave (discovered by Scott Russell) as deserving the epithets “great” or “primary,” and we conceive that ever since it was known that the theory of waves in shallow water was contained in the equation the theory of solitary waves has been perfectly well-known.

George Biddle Airy, “Tides and Waves,” Encyclopedia Metropolitana, 1845

tt xx

g uu

The Search for a Law of Motion

A travelling wave in a narrow channel is described by a height function where

is the distance along the channel

is the elapsed time

is the height of the wave above the undisplaced fluid surface

( , )u x t

x

t

( , )u x t

To understand Russell’s wave, one must find a law of motion

that describes how the shape function changes with

time, and correctly predicts the four properties of solitary waves

observed by Russell.

( , )u x t

Kortweg and de Vries revisited the equations of shallow

water waves in a channel, assuming that the depth of the

channel was small compared to the wave length. They

assumed a wave moving to the right and in place of Airy’s

linear equation for the shape function, they

obtained the nonlinear Korteweg-de Vries Equation

tt xx

g uu

3

36 0

u u uu

t xx

What is the physical mechanism that generates solitary waves?

The rate of change of the shape function is determined by two

different terms:

• A term that tends to make waves disperse

• A term that tends to make waves focus

3

306

uu

ut xx

u

If one looks for solutions to the KdV equation of the form

one immediately finds a solution that describes a single

solitary wave:

explaining Russell’s observations about the wave shape and

the relationship between wave amplitude and speed of

propagation

( , ) ( )u x t u x ct

2sech1

( , )2 2

cu x t x tcc

When both dispersive and nonlinear terms are present,

focussing can competes with dispersion to create stable,

nondispersive waves. In special circumstances, it may be

possible for the competition to “balance” and create a

solitary wave which neither focuses nor disperses.

But under what circumstances, and why?

The Birth of Experimental Mathematics

Enrico Fermi(1901-1954)

Stanlisaw Ulam

(1909-1984)

Mary Tsingou Menzel

(1927- )

John Pasta(1918-1984)

It is Fermi who had the genius to propose that computers could be used to study a problem or test a physical idea by simulation, instead of simply performing standard calculus.

Thierry Dauxois

• Fermi, Pasta and Ulam wanted to study how energy is distributed between normal modes of a one-dimensional crystal

• The crystal consists of a lattice of collinear atoms each of which vibrates due to interaction with its two nearest neighbors

• The system can be modeled as a system of masses coupled by springs

The spring mass system is linear if the force exerted on

a given mass by its neighbor to the left or right is

proportional to the compression or extension of the

spring.

A linear spring-mass system normal modes of vibration

which, once started, continue repeating the

same motion over and over. FPUT believed that this

would change if the force law for the springs was

changed to a nonlinear law.

The FPTU model consists of:

• balls on a line, connected by springs

• is the displacement of the nth mass from its rest position

• Newton’s law is applied to each mass to find the motion

nu

64N

Linear Versus Nonlinear Springs

• In the linear spring model, each mass moves according to the law

• In the nonlinear spring model, each mass moves according to the law

1 1n n n n nm uk u uu u

1 1 11

n nn n n n nu umu uk uuu

Fermi, Pasta, Tsingou, and Ulam computed the motion of the 64 nonlinear oscillators on a computer called theMANIAC I (Mathematical Analyzer, Numerator, Integrator, and Computer) based on ideas of John von Neumann

MANIAC I

• For the linear model, there are normal modes of oscillation which, once started, persist forever

• Fermi, Pasta, Tsingou, and Ulam expected that in the nonlinear model, the energy of the system would be distributed over all accessible modes of vibration

• Instead, they found that, if the system of nonlinear oscillators was “started” in a low mode, it would come back to that mode repeatedly, so that the nonlinear oscillators were behaving as if they were "really" linear oscillators

The FPTU Model and Solitary Waves

Kruskal and Zambusky (1965) sought to understand why the FPTU nonlinear oscillators exhibited periodic behavior.

They computed a continuum limit of the system studied by FPTU. The continuum limit means:

• Take the spacing h between oscillators to zero

• Take the number N of oscillators to infinity

• Consider to be a discrete "sampling" of a continuous function,

ju

( , ) ( )j

u jh t u t

Approaching The Continuum Limit

32N

64N

128N

• It was long known that the continuum limit for the linear model gives Airy’s wave equation

• Kruskal and Zambusky found that the continuum limit of the FPTU nonlinear model gives the Korteweg-de Vries equation

the same equation that describes Russell’s solitary wave!

• They computed numerical solutions to the KdV equation and reproduced the periodic behavior of FPTU’s numerical experiment

2 2

2 2 2

1u u

x c t

3

36 0

u u uu

t xx

Kruskal and Zabusky called the new solitary waves

“solitons.” The mechanism produced them remained

mysterious

Solitary Waves and

Quantum Waves

“Theories permit consciousness to `jump over its own shadow’, to leave behind the given, to represent the transcendent, yet, as is self-evident, only in symbols.”

Hermann Weyl

Peter Lax (1926- )

Martin Kruskal (1925-2006)

John Greene(1928-2007)

Robert Miura(1938- )

Clifford Gardner(1924- )

• Mathematicians suspected that some as yet undiscovered conservation laws were responsible for the existence of solitary waves

• Conservation laws for the total mass and energy were known, and more conservation laws were found by hard calculation

Conservation Laws

(Conservation of mass)

(Conservation of Momentum)

(Conservation of Energy)

udx

2u dx

23 12 x

u u dx

“The next surge of momentum came with the arrival of Robert Miura who was asked by Kruskal to get his feet wet by searching for a conservation law at level seven. He found one and then quickly filled in the missing sixth. Eight and nine fell quickly…

Miura was challenged to find the tenth. He did it during a two-week vacation in Canada (There is also a rumor that he was seen about this time in Mt. Sinai, carrying all ten).”

Alan C. Newell, Solitons in Mathematics and Physics

The conservation laws pointed to a remarkable connection between two very different problems:

(1) the initial value problem for the KdV equation

(2) Schrödinger’s Equation

for the “wave function” of a particle moving along a Straight line under the influence of a potential

3

3

0

6

(( ,0 )

0

) u

u

x

u uu

t xx

u x

02( )( ) ( ) ( )x x k xu x

( )x0( )u x

A quantum mechanical particle in a one-dimensional

potential well has two possible states of motion:

(1) “Bound state” motion where the particle stays

localized near the well

(2) “Free motion” where the particle moves away

from the well

The wave function is largest in amplitude

where the particle is most likely to be found

0( )u x

( )x

The rules of quantum mechanics imply that:

(1) Any potential well of finite depth can have

at most finitely many bound states

(2) The deeper the well, the more bound states

will occur

Using this connection, Gardner, Greene, Kruskal, and

Miura (GGKM) discovered a remarkable method for

solving the KdV equation using ideas from quantum

mechanics

GGKM’s solution method connects two completely

different problems:

(1) The motion of waves in a shallow channel, and (2) The motion of a quantum-mechanical particle in

one dimension

by a precisely defined mathematical transformation.

Their discovery explained Russell’s third and fourth

observations about solitary waves from the KdV

equation.

Recall those observations:

(3) Solitary waves can cross each other “without change of any kind”

(4) A sufficiently large mass of water creates two or more independent solitary waves

The set of all possible quantum states is called the spectrum of . There are at most finitely many bound states, described by the bound state energies. The following correspondence holds:

Quantum Problem Water Wave Problem is the potential is the initial wave shape

is the is the mass of thestrength of the potential wave

Bound states of Solitons for KdV

Moreover, the spectrum of is the same as the spectrum of

0u

0u

( , )u x t

0u 0

u

0( ,0) ( )u x u x

0( )u x dx 0

( )u x dx

We now exploit the following facts:

Fact 1: If evolves according to the KdV

equation, the number of bound states (and hence, the

number of solitons) stays fixed

Fact 2: The shape of each soliton is determined by the

corresponding bound state energy

Fact 3: So long as

there is at least one bound state in the quantum problem.

The larger the integral is, the larger the number of

bound states.

0( ) 0u dxx

( , )u x t

We can now explain Russell’s third and fourth observations using the connection with quantum Theory:

(3) Solitary waves can cross each other “without change of any kind”

The solitary waves correspond to quantum-mechanical bound states, which are determined by the spectrum andtherefore do not change over time

(4) A sufficiently large mass of water creates two or more independent solitary waves

The larger the mass of water , the larger the number of quantum-mechanical bound states. Each bound state will correspond to a soliton.

0( )u dx x

Quiz Question

Can the waveform shown below generate solitons?

Hint: Remember that the Schrodinger equation has bound states only when 0

0( )x du x

Peter Lax formulated the connection between the two problems in terms of a Lax Pair of operators, one that determines the scattering data and the other that determines how the scattering data evolve in time

Lax showed that the KdV equation can be described

by a Lax Pair of operators and :

(spectral problem)

(time evolution)

2

2

3

3

( , )

( , ) (

( )

( ) 4 6 3 , )

u x t

uu x t x

L tx

B t Cx xx

t

( )L t ( )B t

( ) ( ), ( )L t B t L t

together with a law of motion that is equivalent to the original

KdV equation:

Lax’s law of motion is equivalent to the original KdV

Equation, but expresses it in a different form.

Key Fact: If a pair of operators obey Lax’s law of motion,

then the spectrum of is automatically preserved, and

soliton solutions are possible. Systems obeying Lax’s law

of motion are called completely integrable.

( )L t

Lax’s framework enabled mathematicians to look for

other nonlinear wave equations that were completely

integrable. In 1972, Zakharov and Shabat showed

that the nonlinear Schrödinger equation

could be solved by the inverse scattering method and

had soliton solutions.

2 2

2( , ) ( , ) 2 ( , ) ( , ) 0

ui x t x t u x t u x

ut

t x

2 2

2( , ) ( , ) 2 ( , ) ( , ) 0

ui x t x t u x t u x

ut

t x

The nonlinear Schrödinger equation, in a slightly different

guise, governs the conduction of pulses in optical fibers:

where τ measures time from the pulse center and z measures

length along the fiber. This equation admits “bright soliton”

solutions.

2 2

2

1( , ) ( , ( ,

2) ) ( , ) 0

ui z z

uu zu z

z

2 2 2( , ) sechu x t t z

The one-soliton solution takes the form

The two-soliton solution takes the form

These “bright solitons” can be used to transmit data in

optical fibers at high speed, with one soliton equal to

one bit

New Frontiers

All of the examples discussed so far concern waves

in one dimension. What about two dimensions – e.g.

surface waves in shallow water?

The Kadomtsev-Petviashvili Equation

3 2

3 24 6 3 0

u u uu

x x x y

ut

The solution represents unidirectional

long waves propagating in shallow water. If

does not depend on then the KP

equation reduces to the KdV equation

( , , )u x y t

( , , )u x y t y

3

364 0

u uu

x x

ut

The KP equation admits line solitons

Line Soliton for the KP Equation

Morning Glory Cloud

Morning Glory clouds occur on a regular basis in the southern part of North Australia’s Gulf of Carpentaria. They can be up to 1000 km long and up to 1 km high.

KP Line Solitons

Video courtesy of Professor Yuji Kodama, Ohio State University

KP Line Solitons – Y Type

Movie courtesy of Professor Yuji Kodama, Ohio State University

Movie courtesy of Mark Ablowitz Via

Yuji Kodama

KP Line Solitons in the Gulf of Mexico

The KP equation can be solved, in principal at least,

by the method of inverse scattering. The “potential” for

three line solitons closely resembles the potential for

three quantum-mechanical particles moving in one

dimension.

A fundamental tool in the scattering theory of these

systems is the Weinberg-Van Winter Equation, co-discovered

by Stephen Weinberg and Clasine Van Winter of the

University of Kentucky. It allows one to analyze the 3-particle

system in terms of 2-particle subsystems. This is roughly

equivalent to analyzing multiple line-solitons in terms of

single line solitons.

A thorough analysis of KP line solitons will require

mathematical tools from the following disciplines

within `pure’ mathematics:

• Combinatorics and graph theory

• Harmonic analysis

• Operator theory

• Topology

as well as physical insight from quantum theory

In memory of Clasine Van Winter (1929-2000)

Special Thanks To:

• My parents, Edmund Franklin Perry and Lena Bowers Perry, who were always both necessary and sufficient

• My family for their love, support, and understanding

• The College of Arts and Sciences for support during sabbatical years 2004-2005 and 2010-2011

• The National Science Foundation for continuing support

• Professor Paul Umbanhower, Northwestern University, for permission to use images from his oscillon experiments to publicize this lecture

• Amy Hisel, Jennifer Allen and their colleagues for help with publicity, design, and arrangements • Professor Yuji Kodama, Ohio State University, for KP line soliton videos

• Professor David Royster for assistance with MathType, Mathematica, and Power Point