Application of the generalized double reduction method to the (1+1)-dimensional Kaup–Boussinesq (K–B) system: Exploiting Lie symmetries and conservation laws

This paper explores the application of the generalized double reduction method to the (1+1)-dimensional Kaup–Boussinesq system, which models nonlinear wave propagation. Generalized double reduction method is a structured and systematic approach in the analysis of partial differential equations. We first identify the Lie point symmetries of the Kaup–Boussinesq system and construct four non-trivial conservation laws using the multiplier method. The association between the Lie point symmetries and the conservation laws is established, and the generalized double reduction method is then applied to transform the Kaup–Boussinesq system into second-order differential equations or algebraic equations. The reduction process allowed us to derive two exact solutions for the Kaup–Boussinesq system, illustrating the method’s effectiveness in handling nonlinear systems. This work highlights the effectiveness of the generalized double reduction method in simplifying and solving nonlinear systems of partial differential equations, contributing to a deeper understanding of the systems.