On the non-integrability of the Popowicz peakon system

Hone, Andrew N.W., Irle, Michael V. (2009) On the non-integrability of the Popowicz peakon system. Dynamical Systems and Differential Equations, 2009 (2009). pp. 359-366. ISSN 1078-0947. (doi:10.3934/proc.2009.2009.359)

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.

Item Type: Article
DOI/Identification number: 10.3934/proc.2009.2009.359
Uncontrolled keywords: Camassa-Holm equation, Degasperis-Procesi equation, peakons
Subjects: Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations
Q Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations
Q Science > QA Mathematics (inc Computing science) > QA801 Analytic mechanics
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Andrew N W Hone
Date Deposited: 21 Jun 2014 00:57 UTC
Last Modified: 05 Jun 2019 14:57 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/41490 (The current URI for this page, for reference purposes)
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