A Convergent Numerical Scheme for the Camassa–holm Equation Based on Multipeakons

The Camassa–Holm equation ut−uxxt+3uux−2uxuxx−uuxxx = 0 enjoys special solutions of the form u(x, t) = Pn i=1 pi(t)e −|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 u − uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc(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.