Multi-solitons of a Multi-solitons of a (2+1)-dimensional vector (2+1)-dimensional vector soliton system 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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Multi-solitons of a (2+1)- dimensional vector soliton system Ken-ichi Maruno Department of Mathematics, University of Texas -- Pan American Joint work.
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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
Population 700,634
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)
– 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
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
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