On multiple soliton similariton-pair solutions, conservation laws via multiplier and stability analysis for the Whitham-Broer-Kaup equations in weakly dispersive media

In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham-Broer-Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one-, two- and three-solitonic similaritons on the behavior of propagation of waves, in quasi-periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti-soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three-soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients.