The energy method for high-order invariants in shallow water wave equations

Third-order dispersive evolution equations are widely adopted to model one-dimensional long waves and have extensive applications in fluid mechanics, plasma physics and nonlinear optics. The typical representatives are the KdV equation, the Camassa-Holm equation and the Degasperis-Procesi equation. They share many common features such as complete integrability, Lax pairs and bi-Hamiltonian structure. In this paper we revisit high-order invariants for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint operator (1 - partial differential xx)-1. Several applications to seek high-order invariants of the Benjamin-Bona-Mahony equation, the regularized long-wave equation and the Rosenau equation are also presented.(c) 2023 Elsevier Ltd. All rights reserved.