On solutions for the b-family of peakon equations

TL;DR

This paper studies a family of peakon equations characterized by parameters, unifying solutions for well-known equations like Camassa-Holm and Degasperis-Procesi, and explores their dynamic behaviors including collision types.

Contribution

It introduces a unified framework for peakon solutions across the b-family of equations and analyzes their solution forms and collision dynamics.

Findings

Peakon solutions are unified in a difference form and weak sense.

Different collision behaviors are observed depending on the parameter b.

Camassa-Holm and Degasperis-Procesi equations are special cases with distinct wave interactions.

Abstract

We investigate a family of peakon equations, labelled by two parameters $b$ and $\kappa$, all of which admit one-peakon solutions in a unified form. The well known Camassa-Holm equation and Degasperis-Procesi equation are derived from the b-family peakon equations by choosing $b=2$ and 3, respectively. For all values of $b$, their two peakon-type solutions are shown in difference form and in weak sense. At the same time, the dynamic behaviors are shown in difference phenomenons, such as the peaked waves collide elastically in the case $b=2$, while the peaked waves collide inelastically in other cases.