  The aim of this paper is to present a survey and a detailed numerical study
on a remarkable Boussinesq system describing weakly nonlinear, long surface
water waves. In the one-dimensional case, this system can be viewed as a
dispersive perturbation of the hyperbolic Saint-Venant (shallow water) system.
The asymptotic stability of the solitary waves is numerically established.
Blow-up of solutions for initial data not satisfying the non-cavitation
condition as well as the appearence of dispersive shock waves are studied.
