On the Galilean invariance of some dispersive wave equations

TL;DR

This paper addresses the lack of Galilean invariance in classical water wave models by proposing modifications to derive new models that preserve this symmetry, improving the accuracy of solitary wave solutions compared to full Euler equations.

Contribution

The authors introduce a mechanism to modify classical dispersive wave equations to ensure Galilean invariance, a property lost in traditional models like BBM and Peregrine systems.

Findings

New Galilean invariant models derived from classical water wave equations.

Enhanced solitary wave solution interactions aligning better with full Euler solutions.

Preservation of Galilean symmetry improves the physical accuracy of water wave modeling.

Abstract

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.