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 noncavitation condition as well as the appearance of dispersive shock waves are studied.