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

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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

Nonlinear Physics V, Gallipoli, Italy June
12-21, 2008
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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

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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
Discrete Soliton Equation
Dependent variable transform
Dependent variable transform
Discrete Bilinear Form
Bilinear Form
Discretization
tau-function
tau-function
Keep solution structure!
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Bilinear Form of CH Equation

• Parker, Matsuno didnt 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-discrete KP-Toda hierarchy
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(No Transcript)
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Semi-discrete Camassa-Holm Equation
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Semi-discrete Camassa-Holm
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Numerical Method
Tridiagonal matrix
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Simulation of cuspon
of grids 100 Mesh size 0.04 Time step 0.0004
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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.