A comparison of solutions of a Boussinesq system and the Benjamin-Bona-Mahony equation.

Abstract

We compare solutions of a regularized Boussinesq model system for water waves to solutions of the Benjamin-Bona-Mahony equation. Both the system and the equation are obtained as formal approximations to the Euler equations of motion in which terms of order epsilon2 are neglected, where epsilon is a small parameter related to the wave amplitude and the inverse wavelength. For each choice of initial free-surface profile in an appropriate Sobolev space, we show that the Boussinesq system has a unidirectional solution which exists on a time interval 0 ≤ t ≤ T of length T = O( 1/e ), and that on this time interval the solution agrees with the corresponding solution of the Benjamin-Bona-Mahony equation to within Cepsilon2t, where C is independent of epsilon and t. Therefore the solution of the system agrees with the solution of the equation to within the accuracy of either model.