Comparisons between the BBM equation and a Boussinesq system

Abstract

This project aims to cast light on a Boussinesq system of equations modelling two-way propagation of surface waves. Included in the study are existence results, comparisons between the Boussinesq equations and other wave models, and several numerical simulations. The existence theory is in fact a local well-posedness result that becomes global when the solution satisfies a practically reasonable constraint. The comparison result is concerned with initial velocities and wave profiles that correspond to unidirectional propagation. In this circumstance, it is shown that the solution of the Boussinesq system is very well approximated by an associated solution of the KdV or BBM equation over a long time scale of order $\frac{1}{\epsilon}$, where $\epsilon$ is the ratio of the maximum wave amplitude to the undisturbed depth of the liquid. This result confirms earlier numerical simulations and suggests further numerical experiments, some of which are reported here. Our results are related to recent results of Bona, Colin and Lannes [11] comparing Boussinesq systems of equations to the full two-dimensional Euler equations (see also the recent work of Schneider and Wayne [26] and Wright [30].