American Institute of Mathematical Sciences

July  2006, 14(3): 505-523. doi: 10.3934/dcds.2006.14.505

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

 1 Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim 2 Department of Mathematical Sciences, Norwegian University of Science and Technology, NO--7491 Trondheim, Norway

Received  February 2005 Revised  May 2005 Published  December 2005

The Camassa--Holm equation $u_t$$-$uxxt+3u$u_x-2u_x$uxx-uuxxx=0 enjoys special solutions of the form $u(x,t)=$Σi=1n$p_i(t)e^{-|x-q_i(t)|}$, denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data $u|_{t=0}=u_0$ in $H^1$(R) such that u-uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc(R, Hloc1(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.
Citation: Helge Holden, Xavier Raynaud. A convergent numerical scheme for the Camassa--Holm equation based on multipeakons. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 505-523. doi: 10.3934/dcds.2006.14.505
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