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.

翻译：第三顺序分解进化方程式被广泛采用,用于模拟单维长波,在流体力学、等离子物理和非线性光学方面广泛应用,其中包括KdV方程式、Camassa-Holm方程式和Degasperis-Procesi方程式,它们有许多共同特征,如完全融合、Lax对子和双-Hamiltonian结构。在本文中,我们用能源方法结合一个Skew-adjoint接线操作员$(1-\party\ ⁇ xx}} ⁇ -1美元)对这三类浅水波方程式的高度差异性进行了重新审视,还介绍了寻求Benjamin-Bona-Mahony方程式、正规化长波方程方程式和罗索方程的高顺序变量。