New dynamical behavior of two waves for (2+1)-dimensional Broer-Kaup equation

Abstract

New exact solutions including periodic breather wave, kink breather wave and doubly breather wave solutions are obtained for (2+1)D BK equation by using Painleve analysis, variable separation approach, the homoclinic test method and generalized CK method via the linearization of equation, variable separation and equivalent transformation, respectively. The dynamical behavior and interaction between different waves are investigated. These results enrich the dynamic features of higher dimensional nonlinear system.

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

“…Nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth have been said to be governed by a (2 + 1)-dimensional BKK system as follows (Ying and Lou, 2001; Ruan and Chen, 1998; Tang et al , 2020; Rizvi et al , 2020; Wen, 2011; Kassem and Rashed, 2019; Yamgoué et al , 2019; Jiang et al , 2017; Lan et al , 2017): where u = u ( x , y , t ) denotes the height of the water surface above a horizontal bottom, v = v ( x , y , t ) is the horizontal velocity of the water wave, and the subscripts represent the partial derivatives with respect to the scaled space variables x , y and time variable t . When y = x , System (1) has been reduced to a (1 + 1)-dimensional BKK system, which is used to describe the propagation of the long waves in shallow water (Ying and Lou, 2001).…”

“…Binary Bell polynomial approach (Tang et al , 2020), Hirota bilinear method (Tang et al , 2020; Rizvi et al , 2020; Wen, 2011), hidden symmetries of the Lie optimal system (Kassem and Rashed, 2019) and rational sine-Gordon expansion method (Yamgoué et al , 2019) have been applied to System (1). Types of the solutions of System (1), such as the N -soliton solutions ( N is a positive integer) (Wen, 2011; Kassem and Rashed, 2019), cuspon (Kassem and Rashed, 2019), breather (Jiang et al , 2017), lump and rogue wave solutions (Rizvi et al , 2020), have been obtained. Bäcklund transformation, Lax pair, one- and two-soliton solutions of System (1) have been given (Lan et al , 2017).…”

“…Nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth have been said to be governed by a (2 þ 1)-dimensional BKK system as follows (Ying and Lou, 2001;Ruan and Chen, 1998;Tang et al, 2020;Rizvi et al, 2020;Wen, 2011;Kassem and Rashed, 2019;Yamgoué et al, 2019;Jiang et al, 2017;Lan et al, 2017):…”