A Class of Exact Solution of (3+1)-Dimensional Generalized Shallow Water Equation System

Shallow water wave equation has increasing use in many applications for its success in eliminating spurious oscillation, and has been widely studied. In this paper, we investigate (3+1)-dimensional generalized shallow water equation system. Based on the ( G ′ / G ) $(G'/G)$ -expansion method and the variable separation method, we choose ξ ( x , y , z , t ) = f ( y + c z ) + a x + h ( t ) $\xi (x,y,z,t) = f(y + cz) + ax + h(t)$ and suppose that a i ( i = 1 , 2 , … , m ) ${a_i}(i = 1,2, \ldots,m)$ is an undetermined function about x , y , z , t $x,y,z,t$ instead of a constant in eq. (3), which are different from those in previous literatures. With the aid of symbolic computation, we obtain a family of exact solutions of the (3+1)-dimensional generalized shallow water equation system in forms of the hyperbolic functions and the trigonometric functions. When the parameters take special values, in addition to traveling wave solutions, we also get the nontraveling wave solutions by using our method; these obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves. The ( G ′ / G ) $(G'/G)$ -expansion method is a very general and powerful tool that will lead to further insights and improvements of the nonlinear models.