Singular Solutions of a Boussinesq System for Water Waves

Abstract

Studied here is the Boussinesq system $$ η_t+u_x+(ηu)_x+au_{xxx}−bη_{xxt} =0,\\ u_t+η_x+uu_x+cη_{xxx}−du_{xxt} =0,$$ of partial differential equations. This system has been used in theory and practice as a model for small-amplitude, long-crested water waves. The issue addressed is whether or not the initial-value problem for this system of equations is globally well posed. The investigation proceeds by way of numerical simulations using a computer code based on a a semi-implicit, pseudo-spectral code. It turns out that larger amplitudes or velocities do seem to lead to singularity formation in finite time, indicating that the problem is not globally well posed.