Multisolitons of a 2 1dimensional vector soliton system

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Multi-solitons of a (21)-dimensional vector
soliton system

• Ken-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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UTPA (Edinburg)
Population 55,297
McAllen-Edinburg-Mission Area
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KP-II Line Soliton Solution
Nonlinear wave in Plasma and water wave
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Example Hirota D-operator
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Wronskian Solution
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N-soliton solution
f is a solution of the dispersion relations
We can choose another kind of functions.
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Wronskian
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Web Structure
Non-stationary complex patterns
These are made from Wronskian Solutions
of KP (Biondini Kodama 2003) Classification
of all soliton solutions of KP (Kodama
Biondini Chakravarty)
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Web structure
web structure KP(2003) Biondini
Kodama coupled KP (2002) Isojima,
Willox Satsuma 2D-Toda (2004) Maruno
Biondini
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Theory of KP hierarchy (Jimbo-Miwa, 1983)
Solution
Semi simple Lie algebra

• AKP ( KP) Wronskian
• BKP Pfaffian
• CKP Wronskian
• DKP Pfaffian

Extention of determinant
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Solutions are written by Pfaffian
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Square root of determinant of even antisymmetric
matrix
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Hirota Ohta Kodama KM
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Patterns of DKP equation are very complicated.
Four A-soliton
Two D-soliton
Three D-soliton
Patterns of DKP are classified using Pfaffian.
A-type soliton related to A-type Weyl group
D-type soliton related to D-type Weyl
group. (See Kodama KM, 2006)
Patterns of DKP are made from A and D-type
Weyl group !
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N-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)

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Question

• Analysis of N-soliton interaction of equations
  having other types of solutions
  e.g. Multi-component
  determinant

Vector NLS-type solitons
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Vector NLS (coupled NLS) equation
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Vector soliton interaction (vector NLS equation)

• R Radhakrishnan, M Lakshmanan, J Hietarinta 1997

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Remark

• 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)

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Multi-component determinant solution of NLS type
equations
Component 1
Component 2
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Two component KP hierarchy (cf. Jimbo Miwa)
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Bilinear forms in 2-component KP hierarchy
2-component Wronskian
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Reduction to NLS
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Gauge factor
NLS 2-component Wronskian n-component NLS
(n1)-component Wronskian
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Physical Difference between KdV and NLS
Long wave (e.g. Shallow water wave)
KdV equation
Short wave (e.g. Deep water wave)
NLS equation
Long wave
Short wave
Is there any physical phenomenon having both
long wave and short wave?
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Resonance Interaction between long wave and
short wave
Resonance Interaction
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Example Surface wave and internal wave
(Oikawa Funakoshi)
S
L
Yajima-Oikawa System (Long wave- short wave
resonance interaction eq.)
S Short wave L Long wave
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Long wave-short wave 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

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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
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2-dimensional vector Yajima-Oikawa
System(2-component)
Vector form
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Bilinear Equations
c a constant, c0 Bright soliton
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Solution of 2-dimensional 2-component
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

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3-component Wronskian
3-component KP hierarchy
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Short wave
Long wave
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S1
- L
S1
- L
Phase shift
S2
S2
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- L
S1
S2
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Interaction of 2-line soliton and periodic
soliton
V-shape
- L
S1
S2
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2-dimensional vector Yajima-Oikawa
System(n-component)
tau-function N-component Wronskian
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2D Matrix Yajima-Oikawa system
Multi-soliton (Wronskian) solution? BEC?
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(DKP hierarchy Jimbo-Miwa Hirota-Ohta
Kodama-KM)
Complex variables
Hietarinta
Physical interpretation, Multi-soliton
solution Vector and matrix generalization???
Multi-component Pfaffian?
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Summary

• We constructed Wronskian solutions of
  2-dimensional vector YO system
• Soliton interaction of vector YO system has some
  unusual properties.

Future Problems

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

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