Solitons, Backlund transformation and Lax pair for a variable-coefficient generalized Boussinesq system in the shallow water

Under investigation in this paper is a variable-coefficient generalized Boussinesq system, which describes the propagation of long weakly non-linear and weakly dispersive surface waves in the shallow water. Bilinear form, Backlund transformation and Lax pair are derived by virtue of the Bell polynomials. One- and two-soliton solutions are constructed via the Hirota method. Propagation characteristics and collision behaviors of the solitons are discussed through the graphical analysis. Elastic collisions of the two bell-and periodic-shaped solitons are obtained. Head-on collision between the two solitons occurs when the sign of the two wave numbers are different; Otherwise, the overtaking collision happens.