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

TL;DR

This paper develops an energy method using a skew-adjoint operator to analyze high-order invariants in third order dispersive equations like KdV, Camassa-Holm, and Degasperis-Procesi, with applications to related wave equations.

Contribution

It introduces a novel energy method for deriving high-order invariants in shallow water wave equations, expanding understanding of their conserved quantities.

Findings

Derived high-order invariants for KdV, Camassa-Holm, and Degasperis-Procesi equations.

Applied the method to Benjamin-Bona-Mahony, regularized long wave, and Rosenau equations.

Enhanced the analytical tools for studying integrability and conservation laws in dispersive wave models.

Abstract

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.