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 A1>0 and A2<0 are reflected upon collision and form new waves with amplitudes B1<0 and B2>0 such that, depending on the parameter r, either B1=A2 and B2=A1, or Bi are arbitrary numbers provided B1+B2=A1+A2.