On the non-integrability of a fifth order equation with integrable two-body dynamics

Holm, Darryl D. and Hone, Andrew N.W. (2003) On the non-integrability of a fifth order equation with integrable two-body dynamics. Theoretical and Mathematical Physics, 137 (1). pp. 1459-1471. ISSN 0040-5779. (doi:https://doi.org/10.1023/A:1026060924520) (Full text available)

Abstract

We consider the fifth order partial differential equation (PDE) u4x,t−5uxxt+4ut+uu5x+2uxu4x−5uu3x−10uxuxx+12uux=0, which is a generalization of the integrable Camassa-Holm equation. The fifth order PDE has exact solutions in terms of an arbitrary number of superposed pulsons, with geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N=2. Numerical simulations show that the pulsons are stable, dominate the initial value problem and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the non-existence of a suitable Lagrangian or bi-Hamiltonian structure, and obtaining negative results from Painlev\'{e} analysis and the Wahlquist-Estabrook method, we assert that the fifth order PDE is not integrable.

Item Type: Article
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 22:29 UTC
Last Modified: 26 Jun 2017 15:03 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/41496 (The current URI for this page, for reference purposes)
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