Exact Solutions and Conservation Laws of the Generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt System via Lie Symmetry and Riccati Equation Methods

The generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt (GWBKBK) system plays a crucial role in modeling nonlinear wave phenomena, particularly long-wave dispersion in shallow water, which is essential for understanding wave propagation in oceans and other fluid media. In this study, we conduct a Lie symmetry analysis of the GWBKBK system, reducing it to a system of ordinary differential equations (ODEs) in five distinct cases, each yielding exact solutions. For one of these reduced systems, we combine Lie symmetry reduction with two Riccati equation methods: the generalized Riccati equation mapping approach and the Weierstrass-type Riccati equation expansion technique. This allows us to derive new exact solutions, including kink solitary waves, dark solitons, bright solitons, and several singular solitary wave solutions. The diverse wave profiles of these solutions are visualized through three-dimensional and contour plots.