Multi-solitons of a (2+1)- dimensional vector soliton system Ken-ichi Maruno Department of...
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Transcript of Multi-solitons of a (2+1)- dimensional vector soliton system Ken-ichi Maruno Department of...
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
UTPA(Edinburg)
Population 55,297
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McAllen-Edinburg-Mission Area
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