Asymptotics for peakon-antipeakon collision in a general Camassa-Holm model

Abstract

We analyze a peakon - antipeakon collision for essentially non-integrable versions of the Camassa-Holm equation. Using the matching and weak asymptotics methods, we construct an asymptotic solution that satisfies both a one-parameter family of equations (with a parameter r), and two energy laws. It is shown that the original waves with amplitudes $A_1>0$ and $A_2<0$ are reflected upon collision and form new waves with amplitudes $B_1<0$ and $B_2>0$ such that, depending on the parameter r, either $B_1=A_2$ and $B_2=A_1$, or $B_i$ are arbitrary numbers provided $B_1+B_2=A_1+A_2$.