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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, 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.

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An integrable difference scheme for the Camassa-Holm equation and numerical computation

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  1. 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

  2. Camassa-Holm Equation: History Fuchssteiner & Fokas (1981) : Derivation from symmetry study Camassa & Holm (1993) :Derivation from shallow water wave Camassa, Holm & Hyman(1994) : Peakon Schiff (1998) : Soliton solutions using Backlund transform Constantin(2001), Johnson(2004), Li & Zhang (2005) : Soliton solutions using IST Parker(2004); Matsuno (2005) : N-soliton solution using bilinear method Kraenkel & Zenchuk(1999);Dai & Li (2005) : Cuspon solutions

  3. Soliton and Cuspon Ferreira, Kraenkel and Zenchuk JPA 1999

  4. Soliton-Cuspon Interaction Dai & Li JPA 2005

  5. 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

  6. 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.

  7. 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)‏

  8. Discretization using bilinear form(Hirota 1977)‏ Discrete Soliton Equation Soliton Equation Dependent variable transform Dependent variable transform Discrete Bilinear Form Bilinear Form Discretization tau-function tau-function Keep solution structure!

  9. 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.

  10. Bilinear Form of CH Equation

  11. Determinant form of solutions 2-reduction of KP-Toda hierarchy

  12. Discretization of bilinear form 2-reduction of semi-discrete KP-Toda hierarchy

  13. Semi-discrete Camassa-Holm Equation

  14. Semi-discrete Camassa-Holm

  15. Numerical Method Tridiagonal matrix

  16. Simulation of cuspon # of grids 100 Mesh size 0.04 Time step 0.0004

  17. Simulation of 2-cuspon interaction

  18. Simulation of soliton-cuspon interaction

  19. Simulation of soliton-cuspon interaction(Cont.)‏

  20. Simulation of soliton-cuspon interaction

  21. Simulation of soliton-cuspon interaction (Cont.)‏

  22. Non-exact initial data

  23. 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.

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