Gramian solutions and solitonic interactions of a (2+1)-dimensional Broer–Kaup–Kupershmidt system for the shallow water

Abstract

Purpose This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth. Design/methodology/approach Pfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties. Findings N-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied. Originality/value Shallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.