Analytical and Numerical Solutions of the Whitham-Broer-Kaup System for Long Waves in Small-amplitude Shallow Water

The principal focus of this work is to explore numerical solutions of the classical Whitham-Broer-Kaup equations using the local discontinuous Galerkin method. The numerical methodology employs an explicit strong-stability-preserving higher-order Runge-Kutta method to estimate the temporal derivatives and the local discontinuous Galerkin method to estimate the spatial derivatives, generating a large coupled ordinary differential equations system. The analytical improved \( (G'/G) \)-expansion technique is also employed in this article to obtain new exact travelling wave solutions of the governing coupled Whitham-Broer-Kaup equations. Finally, some numerical experiments are performed, and all the simulation results are displayed in several tables and various two-dimensional and three-dimensional plots, which validate the satisfactory accuracy and efficacy of the implemented method.