Abstract
The theory of exterior differential systems is appl ied to study integrability of a set of related partial differential equations expressed in the ter ms of differential forms using Cartan's method. The Camassa-Holm equation and the Degasperis-Procesi equations are special cases that are described by these exterior differential systems. Some conservat ion laws are obtained in the cases of the more relevant equations. A closed differential ideal is constructed for each case studied.
Highlights
It has been known for a while that the simplest nonlinear evolution equations which have solitary wave solutions or solitons are known to have an infinite number of conservation laws
This in turn is related to the concept of integrability of a particular partial differential equation or system of such equations (Estabrook and Wahlquist, 1975)
A Lax pair can usually be determined and a Backlund or auto-Backlund transformation might be written down. It was shown by Wahlquist and Estabrook (1975; 1976) that by applying results from the theory of exterior differential systems, as well as prolongation techniques, it is possible to determine integrability of a related partial differential system
Summary
It has been known for a while that the simplest nonlinear evolution equations which have solitary wave solutions or solitons are known to have an infinite number of conservation laws. It was shown by Wahlquist and Estabrook (1975; 1976) that by applying results from the theory of exterior differential systems, as well as prolongation techniques, it is possible to determine integrability of a related partial differential system. An exterior differential system which reproduces the given equation on the transverse manifold is developed for each case.
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