American Institute of Mathematical Sciences

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

On the non-integrability of the Popowicz peakon system

Andrew N. W. Hone 1, and  Michael V. Irle 2,

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 $3N$.
Keywords: peakons, Degasperis-Procesi equation, Camassa-Holm equation.
Mathematics Subject Classification: Primary: 37K05, 37K10; Secondary: 37J99.
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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