Extended shallow water wave equations

TL;DR

This paper derives extended shallow water wave equations using asymptotic expansions, capturing higher-order nonlinear and dispersive effects beyond traditional models, applicable in various dimensions and geometries.

Contribution

It introduces higher-order extended models for shallow water waves, including KdV, Benjamin-Bona-Mahony, Camassa-Holm, and Green-Naghdi equations, advancing the accuracy of wave modeling.

Findings

Derived extended KdV, BBM, and Camassa-Holm equations in 1D.

Extended cylindrical and KP equations in 2D.

Formulated extended Green-Naghdi equations.

Abstract

Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation, incorporating all appropriate dispersive and nonlinear terms. Specifically, first we derive the extended Korteweg-de Vries (KdV) equation, and then proceed with the extended Benjamin-Bona-Mahony and the extended Camassa-Holm equations in (1+1)-dimensions, the extended cylindrical KdV equation in the quasi-one dimensional setting, as well as the extended Kadomtsev-Petviashvili and its cylindrical counterpart in (2+1)-dimensions. We conclude with the case of the extended Green-Naghdi equations.