Preprint 2005-004

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

Abstract: 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)=∑_ip_i(t)exp(-|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=u₀ in H¹(R) such that u−u_xx is a positive Radon measure, one can construct a sequence of multipeakons that converges in L^∞_loc(R,H¹_loc(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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Available as PDF (296 Kbytes), Postscript (1040 Kbytes) or gzipped PostScript (352 Kbytes; uncompress using gunzip).

Author(s):

Helge Holden, <holden@math.ntnu.no>

Xavier Raynaud, <raynaud@math.ntnu.no>

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<raynaud@math.ntnu.no> January 13 2005.