The general peakon-antipeakon solution for the Camassa-Holm equation

TL;DR

This paper explicitly derives the peakon-antipeakon solution for the Camassa-Holm equation, revealing wave breaking phenomena and the effects of dissipation in a non-symmetric setting.

Contribution

It provides an explicit solution for the peakon-antipeakon case of the Camassa-Holm equation, including non-symmetric and dissipative scenarios, enhancing understanding of wave breaking.

Findings

Solution experiences finite-time wave breaking

Explicit form clarifies variable interactions

Dissipative effects analyzed in non-symmetric case

Abstract

We compute explicitly the peakon-antipeakon solution of the Camassa-Holm equation $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ in the non-symmetric and $\alpha$-dissipative case. The solution experiences wave breaking in finite time, and the explicit solution illuminates the interplay between the various variables.