Preprint 2005-004

A convergent numerical scheme for the Camassa–Holm equation based on multipeakons

Helge Holden and Xavier Raynaud

Abstract: The Camassa–Holm equation utuxxt+3uux−2uxuxx−uuxxx=0 enjoys special solutions of the form u(x,t)=∑ipi(t)exp(-|x-qi(t|), denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data u|t=0=u0 in H1(R) such that uuxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc(R,H1loc(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.

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Helge Holden, <>
Xavier Raynaud, <>
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<> January 13 2005.

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