Conservation Laws and Web-Solutions for the Benney--Luke Equation

TL;DR

This paper develops a perturbation method based on conservation laws to analyze web-type solutions of the Benney-Luke equation, supported by numerical simulations that explore their relationship with the Kadomtsev-Petviashvili equation.

Contribution

It introduces a novel perturbation approach utilizing conservation laws for the Benney-Luke equation and provides new numerical simulations for non-decaying web solutions.

Findings

Conservation laws determine the evolution of web solutions.

Numerical simulations confirm analytical predictions.

Web solutions relate to Kadomtsev-Petviashvili equation.

Abstract

A long wave multi-dimensional approximation of shallow water waves is the bi-directional Benney-Luke equation. It yields the well-known Kadomtsev-Petviashvili equation in a quasi one-directional limit. A direct perturbation method is developed; it uses the underlying conservation laws to determine the slow evolution of parameters of two space dimensional, non-decaying web-type solutions to the Benney-Luke equation. New numerical simulations, based on windowing methods which are effective for non-decaying data, are presented. These simulations support the analytical results and elucidate the relationship between the Kadomtsev-Petviashvilli and the Benney-Luke equations and are also used to obtain amplitude information regarding particular web solutions. Additional dissipative perturbations to the Benney-Luke equation are also studied.