# American Institute of Mathematical Sciences

2009, 2009(Special): 359-366. doi: 10.3934/proc.2009.2009.359

## On the non-integrability of the Popowicz peakon system

 1 Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom 2 Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom

Received  July 2008 Revised  April 2009 Published  September 2009

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 Painlevé 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 $3N$.
Citation: Andrew N. W. Hone, Michael V. Irle. On the non-integrability of the Popowicz peakon system. Conference Publications, 2009, 2009 (Special) : 359-366. doi: 10.3934/proc.2009.2009.359
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