Dynamics of conservative peakons in a system of Popowicz

• Published in: Physics letters. A 2019-01, Vol.383 (5), p.406-413
• Main Authors: Barnes, L.E., Hone, A.N.W.
• Format: Article
• Language: English
• Subjects: Distributions; Hamiltonian dynamics; Peakons; Popowicz system
• ISSN: 0375-9601; 1873-2429
• DOI: 10.1016/j.physleta.2018.11.015

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. •The Popowicz system is not integrable, but it has peaked soliton (peakon) solutions which are conservative.•A proof of the reduction of the Hamiltonian structure of the Popowicz system to the submanifold of N-peakon solutions is given.•For N=2 we explicitly integrate the equations of motion and describe the interaction of two peakons.