Self-adjointness of a generalized Camassa–Holm equation

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, it is shown in [1,2] that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization [3] of the Camassa–Holm equation [4], 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.