Traveling Wave Solution of the Kaup–Boussinesq System with Beta Derivative Arising from Water Waves

The main purpose of this paper is to construct the traveling wave solution of the Kaup–Boussinesq system with beta derivative arising from water waves. By using the complete discriminant system method of polynomial, the rational function solution, the trigonometric function solution, the exponential function solution, and the Jacobian function solution of the Kaup–Boussinesq system with beta derivative are obtained. In order to further explain the propagation of the Kaup–Boussinesq system with beta derivative in water waves, we draw its three-dimensional diagram, two-dimensional diagram, density plot, and contour plot by using Maple software.