Dynamics of conservative peakons in a system of Popowicz

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.