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Kenichi Maruno, Univ. of Texas-Pan American Joint work with Yasuhiro Ohta , Kobe University, Japan Bao-Feng Feng , UT-Pan American. An integrable difference scheme for the Camassa-Holm equation and numerical computation . Nonlinear Physics V, Gallipoli, Italy June 12-21, 2008. - PowerPoint PPT Presentation
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An integrable difference scheme for the Camassa-Holm equation and
numerical computation Kenichi Maruno, Univ. of Texas-Pan American
Joint work with Yasuhiro Ohta, Kobe University, JapanBao-Feng Feng, UT-Pan American
Nonlinear Physics V, Gallipoli, Italy June 12-21, 2008
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Camassa-Holm Equation: History
Fuchssteiner & Fokas (1981) : Derivation from symmetry studyCamassa & Holm (1993) :Derivation from shallow water waveCamassa, Holm & Hyman(1994) : Peakon Schiff (1998) : Soliton solutions using Backlund transformConstantin(2001), Johnson(2004), Li & Zhang (2005) : Soliton
solutions using ISTParker(2004); Matsuno (2005) : N-soliton solution using bilinear
methodKraenkel & Zenchuk(1999);Dai & Li (2005) : Cuspon solutions
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Soliton and Cuspon
Ferreira, Kraenkel and Zenchuk JPA 1999
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Soliton-Cuspon Interaction
Dai & Li JPA 2005
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Numerical Studies of the Camassa-Holm equation
Kalisch & Lenells 2005: Pseudospectral scheme Camassa, Huang & Lee 2005: Particle method Holden, Raynaud 2006,Cohen, Owren & Raynaud
2008: Finite difference scheme, Multi-symplectic integration
Artebrant & Schroll 2006: Finite volume method Coclite, Karlsen & Risebro 2008: Finite difference
scheme
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Problem What is integrable discretization of Camassa-
Holm equation? Need a good numerical scheme to simulate the
Camassa-Holm equation because there exists singularity such as peakon and cuspon.
Simulation of interaction of soliton and cuspon.
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Discrete Integrable Systems Differential-difference equations: Toda lattice,
Ablowitz-Ladik lattice, etc. Method of Discretization of integrable systems:
Ablowitz-Ladik, Suris (Lax formulation), Hirota (Bilinear formulation), etc.
Full discrete integrable systems: discrete-time KdV, discrete-time Toda relationship with numerical ⇒algorithms (qd algorithm, LR alogrithm, etc.)
Discrete Painléve equations Discrete Geometry (Discrete-time 2d-Toda, etc.) Ultra-discrete integrable systems (Soliton Cellular
Automata)
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Discretization using bilinear form(Hirota 1977)
Soliton Equation
Bilinear Form Discrete Bilinear Form
Discrete Soliton Equation
Discretization
Dependent variable transform
Dependent variable transform
tau-function tau-functionKeep solution structure!
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Bilinear Form of CH Equation Parker, Matsuno didn’t use direct bilinear form
of the CH equation, they used bilinear form of AKNS shallow water wave equation which is related to the CH equation.
To discretize CH equation using bilinear form, we need direct bilinear form of the CH equation.
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Bilinear Form of CH Equation
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Determinant form of solutions
2-reduction of KP-Toda hierarchy
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Discretization of bilinear form
2-reduction of semi-discreteKP-Toda hierarchy
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Semi-discrete Camassa-Holm Equation
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Semi-discrete Camassa-Holm
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Numerical Method
Tridiagonalmatrix
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Simulation of cuspon# of grids100Mesh size0.04Time step0.0004
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Simulation of 2-cuspon interaction
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Simulation of soliton-cuspon interaction
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Simulation of soliton-cuspon interaction(Cont.)
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Simulation of soliton-cuspon interaction
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Simulation of soliton-cuspon interaction (Cont.)
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Non-exact initial data
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Conclusions We propose an integrable discretization of the
Camassa-Holm equation. The integrable difference scheme gives very
accurate numerical results. We found a determinant form of solutions of
the discrete Camassa-Holm equation.