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On the non-integrability of the Popowicz peakon system

Pages: 359 - 366, Issue Special, September 2009

 Abstract        Full Text (146.1K)              

Andrew N. W. Hone - Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom (email)
Michael V. Irle - Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury CT2 7NF, United Kingdom (email)

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 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:  Camassa-Holm equation, Degasperis-Procesi equation, peakons
Mathematics Subject Classification:  Primary: 37K05, 37K10; Secondary: 37J99

Received: July 2008;      Revised: April 2009;      Published: September 2009.