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

Abstract

The Camassa--Holm equation$u_t$$-$uxxt+3u$u_x-2u_x$uxx-uuxxx=0 enjoys special solutionsof the form $u(x,t)=$Σi=1n$p_i(t)e^{-|x-q_i(t)|}$, denotedmultipeakons, 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 canconstruct a sequence of multipeakons that converges inLloc∞(R, Hloc1(R)) to the unique global solution of theCamassa--Holm equation. The approach also provides a convergent,energy preserving nondissipative numerical method which isillustrated on several examples.