Multi-solitons of a (2+1)- dimensional vector soliton system Ken-ichi Maruno Department of Mathematics, University of Texas -- Pan American Joint work.

Multi-solitons of a (2+1)-Multi-solitons of a (2+1)-dimensional vector soliton dimensional vector soliton

systemsystemKen-ichi Maruno

Department of Mathematics,University of Texas -- Pan American

Joint work with Y. Ohta & M. Oikawa Kobe Univ.

Kyushu Univ.Mini-Meeting: "Nonlinear Waves and More" August, 15, 2007

University of Colorado, Boulder

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KP-II Line Soliton SolutionKP-II Line Soliton SolutionNonlinear wave in Plasma and water wave

Example: Hirota D-operator Hirota D-operator

Wronskian SolutionWronskian Solution

N-soliton solutionN-soliton solution

f is a solution of the dispersion relations

We can choose another kind of functions.

Wronskian

Web StructureWeb StructureNon-stationary complex patterns

These are made from Wronskian Solutions of KP (Biondini & Kodama 2003)

Classification of all soliton solutions of KP (Kodama; Biondini & Chakravarty)

Web structureWeb structure

web structure

ＫＰ： (2003) Biondini & Kodama

coupled KP: (2002) Isojima, Willox &Satsuma

2D-Toda: (2004) Maruno & Biondini

Theory of KP hierarchy (Jimbo-Miwa, 1983)Theory of KP hierarchy (Jimbo-Miwa, 1983)

AKP (= KP): WronskianBKP ： PfaffianCKP: WronskianDKP: Pfaffian

Semi simple Lie algebraSolution

Extention of determinant

Solutions are written by PfaffianSolutions are written by Pfaffian

Square root of determinant of

even antisymmetric

matrix

Hirota & Ohta; Kodama & KM

Four A-solitonTwo D-soliton Three D-soliton

Patterns of DKP equation are very complicated.

Patterns of DKP are classified using Pfaffian.

A-type soliton related to A-type Weyl groupA-type Weyl group

D-type soliton related to D-type Weyl groupD-type Weyl group.

(See Kodama & KM, 2006)

Patterns of DKP are made from A and D-type Weyl groupWeyl group !

N-soliton interactionN-soliton interaction

Equations having determinant type solutions KP, 2D-Toda, fully discrete 2D-Toda (Biondini, Kodama, Chakravarty, KM)

Equations having pfaffian type solutions DKP (coupled KP) (Kodama & KM)

QuestionQuestion• Analysis of N-soliton interaction

of equations having other types of solutions e.g. Multi-component determinant

Vector NLS-type solitons

Vector NLS (coupled NLS)Vector NLS (coupled NLS) equation equation

Vector soliton interaction (vector NLS equation)

– R Radhakrishnan, M Lakshmanan, J Hietarinta 1997

RemarkRemark

●Bright soliton solutions of NLS are written in the form of two-component Wronskian (Nimmo; Date,Jimbo,Miwa, Kashiwara)●Bright soliton solutions of two-component vector NLS are written in the form of 3-component Wronskian (Ohta)

Multi-component determinant Multi-component determinant solution of NLS type equationssolution of NLS type equations

Component 1 Component 2

Two component KP hierarchy (cf. Jimbo & Miwa)

2-component Wronskian

Bilinear forms

in 2-component

KP hierarchy

Reduction to NLSReduction to NLS

Gauge factor

NLS 2-component Wronskian

n-component NLS (n+1)-component Wronskian

Physical Difference betweenPhysical Difference between KdV and NLS KdV and NLS

KdV equation Long wave (e.g. Shallow water wave)

NLS equation Short wave(e.g. Deep water wave)

Is there any physical phenomenon having both long wave and short wave?

Long wave

Short wave

Resonance Interaction betweenResonance Interaction between long wave and short wave long wave and short wave

ResonanceInteraction

Example: Surface wave andExample: Surface wave and internal wave internal wave

(Oikawa & Funakoshi) (Oikawa & Funakoshi)

Yajima-Oikawa System (Long wave- short wave resonance interaction eq.)

S

L

S: Short wave

L: Long wave

Long wave-short waveLong wave-short wave resonance interaction: History resonance interaction: History

• N. Yajima & M. Oikawa(1976) Interaction of langumuir waves with ion-acoustic waves in plasma, Lax pair (3x3 matrix), Inverse Scattering Transform, Bright soliton

• D.J. Benney (1976) Water wave• Y.C. Ma & L.G. Redekopp (1979) Dark soliton• V. K. Melnikov (1983) Extension to multi-

component and 2-dimensional case using Lax pair

• M. Oikawa, M. Funakoshi & M. Okamura: 2-dimensional system in stratified flow, Bright and Dark soliton solutions

• T. Kikuchi, T. Ikeda and S. Kakei (2003) Painleve V equation

• Nistazakis, Frantzeskakis, Kevrekidis, Malomed, Carretero-Gonzakez (2007): Spinor BEC

2-dimensional 2-component Yajima-Oikawa system(2-dimensional 2-component long wave-short wave resonance interaction equations)Melnikov: On EQUATIONS FOR WAVE INTERACTION, Lett. Math. Phys. 1983 Lax form

2-dimensional vector 2-dimensional vector Yajima-Oikawa Yajima-Oikawa SystemSystem(2-component)(2-component)

Vector form

Bilinear EquationsBilinear Equations

c : a constant, c=0 Bright soliton

Solution of 2-dimensional Solution of 2-dimensional 2-component2-component

Yajima-Oikawa system Yajima-Oikawa system• Belongs to 3-component KP hierarchy• Theory of multi-component KP

hierarchy (T. Date, M. Jimbo, M. Kashiwara, T. Miwa 1981; V. Kac, J. W. van de Leur 2003)

• Bilinear identities of 3-component Wronskians

• 3-component Wronskian with constraints of reality and complex conjugacy of complex functions

3-component KP hierarchy

3-component Wronskian

Short wave

Long wave

Phase shift

- L S1

S2

- L S1

S2

- LS1

S2

Interaction of 2-line soliton andInteraction of 2-line soliton and periodic soliton periodic soliton

V-shape

- LS1

S2

2-dimensional vector 2-dimensional vector Yajima-Oikawa Yajima-Oikawa SystemSystem(n-component)(n-component)

tau-function: N-component Wronskian

2D Matrix Yajima-Oikawa system

Multi-soliton (Wronskian) solution?

BEC?

SummarySummary• We constructed Wronskian solutions

of 2-dimensional vector YO system• Soliton interaction of vector YO

system has some unusual properties.

Y. Ohta, KM, M. Oikawa: J. Phys. A 40 7659-7672 (2007)

KM, Y. Ohta, M. Oikawa, in preparation

• Analysis of multi-soliton interaction• Dark soliton? Dromion? Lump?• Soliton interaction of matrix generalization?

Future ProblemsFuture Problems