On the non-integrability of the Popowicz peakon system

TL;DR

The paper investigates the integrability of the Popowicz peakon system, a coupled PDE system inspired by integrable equations, and finds evidence of non-integrability while constructing explicit solutions and describing peakon dynamics.

Contribution

It provides strong evidence that the Popowicz system is non-integrable and constructs explicit travelling wave solutions, including peakons, along with analyzing N-peakon dynamics.

Findings

Evidence suggests the Popowicz system is non-integrable.

Explicit travelling wave solutions are constructed.

N-peakon dynamics are described by a Hamiltonian system.

Abstract

We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev\'e analysis provides strong evidence that the Popowicz system is non-integrable. Nevertheless, we are able to construct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N.