1. On the non-integrability of the Popowicz peakon system
- Author
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Hone, Andrew N. W. and Irle, Michael V.
- Subjects
QA801 ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,QA372 ,FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,Exactly Solvable and Integrable Systems (nlin.SI) ,QA377 ,Nonlinear Sciences - Pattern Formation and Solitons ,Nonlinear Sciences::Pattern Formation and Solitons - 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., Comment: 8 pages, AIMS class file. Proceedings of AIMS conference on Dynamical Systems, Differential Equations and Applications, Arlington, Texas, 2008
- Published
- 2009