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A convergent numerical scheme for the Camassa-Holm equation based on multipeakons. (English) Zbl 1111.35061
Summary: The Camassa-Holm equation \[ u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0 \] enjoys special solutions of the form \(u(x,t)= \sum^n_{i=1}p_i(t)e^{-|x-q_i(t)|}\), denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data \(u |_{t=0}=u_0\) in \(H^1(\mathbb{R})\) such that \(u-u_{xx}\) is a positive Radon measure, one can construct a sequence of multipeakons that converges in \(L^\infty_{\text{loc}}(\mathbb{R},H^1_{\text{loc}}(\mathbb{R}))\) to the unique global solution of the Camassa-Holm equation. The approach also provides a convergent, energy preserving nondissipative numerical method which is illustrated on several examples.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q51 Soliton equations
35B10 Periodic solutions to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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