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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. Among them 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 $(1-\partial_{xx})^{-1}$. 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.