Singular Solutions of a Boussinesq System for Water Waves

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.
Min Chen (