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−∂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.