On the Korteweg-de Vries approximation for uneven bottoms

TL;DR

This paper examines the validity of the Korteweg-de Vries approximation for water waves over uneven bottoms, identifying cases where it diverges from Boussinesq solutions and proposing a modified model.

Contribution

It extends the KdV approximation to more general bathymetries and introduces a topographically modified KdV model for better accuracy.

Findings

The standard KdV approximation diverges for certain bathymetries.

A modified KdV model improves approximation accuracy for uneven bottoms.

Numerical comparisons validate the proposed model's effectiveness.

Abstract

In this paper we focus on the water waves problem for uneven bottoms on a two-dimensionnal domain. Starting from the symmetric Boussinesq systems derived in [Chazel, Influence of topography on long water waves, 2007], we recover the uncoupled Korteweg-de Vries (KdV) approximation justified by Schneider and Wayne for flat bottoms, and by Iguchi in the context of bottoms tending to zero at infinity at a substantial rate. The goal of this paper is to investigate the validity of this approximation for more general bathymetries. We exhibit two kinds of topography for which this approximation diverges from the Boussinesq solutions. A topographically modified KdV approximation is then proposed to deal with such bathymetries. Finally, all the models involved are numerically computed and compared.