Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq–Burgers equations from shallow water waves

Under investigation in this paper is the set of the Boussinesq–Burgers (BB) equations, which can be used to describe the propagation of shallow water waves. Based on the binary Bell polynomials, Hirota method and symbolic computation, the bilinear form and soliton solutions for the BB equations are derived. Bäcklund transformations (BTs) in both the binary-Bell-polynomial and bilinear forms are obtained. Through the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair for the BB equations are presented as well. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Shock wave and bell-shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface. In both the head-on and overtaking collisions, the shock waves for the u profile change their shapes, which denotes that the collisions for the u profile are inelastic. However, the collisions for the v profile are proved to be elastic through the asymptotic analysis. Our results might have some potential applications for the harbor and coastal design.