Periodic Travelling Waves and its Inter-relation with Solitons for the 2D abc-Boussinesq System

Via a variational approach involving Concentration-Compactness principle, we show the existence of $x$-periodic travelling wave solutions for a general 2D-Boussinesq system that arises in the study of the evolution of long water waves with small amplitude in the presence of surface tension. We also establish that $x$-periodic travelling waves have almost the same shape of solitons as the period tends to infinity, by showing that a special sequence of $x$-periodic travelling wave solutions parameterized by the period converges to a solitary wave in a appropriate sense.