Exterior Differential Systems, Prolongations and the Integrability of Two Nonlinear Partial Differential Equations

TL;DR

This paper formulates a generalized KdV equation as an exterior differential system, analyzes its prolongation structure, extends the analysis to the Camassa-Holm equation, and discusses conservation laws and Bäcklund transformations.

Contribution

It introduces a novel exterior differential system approach to analyze the integrability and prolongation structures of nonlinear PDEs like KdV and Camassa-Holm.

Findings

Prolongation structures are determined for various variable powers.

Nontrivial algebraic structures are identified in the prolongations.

A Bäcklund transformation is constructed for the equations.

Abstract

A generalized KdV equation is formulated as an exterior differential system, which is used to determine the prolongation structure of the equation. The prolongation structure is obtained for several cases of the variable powers, and nontrivial algebras are determined. The analysis is extended to a differential system which gives the Camassa-Holm equation as a particular case. The subject of conservation laws is briefly discussed for each of the equations. A Backlund transformation is determined using one of the prolongations.