LAUSR

Abstract In this paper, we study exact solutions of the classical Boussinesq (CB) system, which describes propagations of shallow water waves. By using the bilinear form, with exponential expansions, we obtain solitary wave solutions of the CB system. Based on asymptotic analysis method, we study the elastic and elastic–inelastic–coupled interactions of the obtained solitary wave solutions. With extended three–wave method, we obtain the periodic solitary solution of the CB system. And with polynomial expansions, we get the rational solutions of the CB system. These interesting exact solutions may be useful in the study of some phenomena appeared in shallow water waves.