Self-adjointness of a generalized Camassa-Holm equation

TL;DR

This paper proves that the Camassa-Holm equation is self-adjoint, similar to KdV equations, and uses this property to construct conservation laws based on its symmetries.

Contribution

It establishes the self-adjointness of a generalized Camassa-Holm equation, a property previously known for KdV-type equations, and derives related conservation laws.

Findings

Camassa-Holm equation is self-adjoint.

Conservation laws are constructed from symmetries.

Extends properties of KdV equations to Camassa-Holm.

Abstract

It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, Ibragimov has shown that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization of the Camassa-Holm equation, we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.