On the ill-posedness of a weakly dispersive one-dimensional Boussinesq system

We study the Cauchy problem for a one-dimensional dispersive system of Boussinesq type which models weakly nonlinear long wave surface waves. We prove that such a system is ill-posed in H ^s (?) for s < 0 in the sense that the solution does not depend continuously on the initial data. We also provide criteria for the formation of singularities.