Dynamics of conservative peakons in a system of Popowicz

TL;DR

This paper analyzes the dynamics of peakons in a non-integrable two-component Hamiltonian system coupling the Camassa-Holm and Degasperis-Procesi equations, revealing explicit solutions and novel interaction features.

Contribution

It provides explicit solutions for two-peakon dynamics and explores the unique interaction behaviors in the non-integrable Popowicz system.

Findings

Two-peakon system is Liouville integrable.

Explicit two-peakon solutions are derived.

Novel peakon interaction features are described.

Abstract

We consider a two-component Hamiltonian system of partial differential equations with quadratic nonlinearities introduced by Popowicz, which has the form of a coupling between the Camassa-Holm and Degasperis-Procesi equations. Despite having reductions to these two integrable partial differential equations, the Popowicz system itself is not integrable. Nevertheless, as one of the authors showed with Irle, it admits distributional solutions of peaked soliton (peakon) type, with the dynamics of $N$ peakons being determined by a Hamiltonian system on a phase space of dimension $3N$. As well as the trivial case of a single peakon ($N=1$), the case $N=2$ is Liouville integrable. We present the explicit solution for the two-peakon dynamics, and describe some of the novel features of the interaction of peakons in the Popowicz system.