On the traveling wave solutions of the (1 + 1)-dimensional Broer–Kaup system for modeling the bi-directional propagation of long waves in shallow waters

Abstract

The development of an effective and efficient local discontinuous Galerkin algorithm for the solution of the coupled ( 1 + 1 ) -dimensional Broer–Kaup system is the primary focus of this research. In this algorithm, spatial discretization is attained using the local discontinuous Galerkin method, while temporal discretization is handled using the explicit total variation diminishing higher-order Runge-Kutta method. However, the ( G ′ G ) -expansion method is also implemented to produce the exact solutions for the coupled ( 1 + 1 ) -dimensional Broer–Kaup system. The obtained solutions are the traveling wave solutions. The efficiency and reliability of the proposed method are analyzed by comparing the generated numerical simulations to various traveling wave solutions using several tables and figures. The exact solutions and the numerical simulations of the coupled ( 1 + 1 ) -dimensional Broer–Kaup system are shown to correspond exceptionally well.