Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional Broer-Kaup-Kupershmidt system in the shallow water of uniform depth

Abstract

Under investigation in this paper is a (2+1)-dimensional Broer-Kaup-Kupershmidt system for the nonlinear and dispersive long gravity waves on two horizontal directions in the shallow water of uniform depth. Bilinear forms, Bäcklund transformation and Lax pair are derived based on the Bell polynomials and symbolic computation. One- and two-soliton solutions with a real function ϕ(y) are constructed via the Hirota method, where y is the scaled space coordinate. Propagation and interaction of the solitons are illustrated graphically: (i) ϕ(y) affects the shape of the solitons. (ii) Interaction of the solitons including the elastic and inelastic interactions are discussed. When the solitons’ interaction is elastic, the amplitude, velocity and shape of the soliton remain invariant after the interaction except for a phase shift, and the smaller-amplitude soliton has a larger phase shift. (iii) Height of the water surface above a horizontal bottom can be a bell-shaped soliton or an upside-down bell-shaped soliton under certain conditions, while horizontal velocity of the water wave always keeps bell-shaped.