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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.