Symmetry Reductions, Dynamical Behavior and Exact Explicit Solutions to a Class of Nonlinear Shallow Water Wave Equation

By using Lie symmetry analysis and dynamical systems method for a class of nonlinear shallow water wave equation, the exact solutions based on the Lie group method are provided. Especially, the bifurcations and exact explicit parametric representations of the traveling solutions are given, and the possible solitary wave solutions and many uncountable infinite periodic wave solutions to the nonlinear equation are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, we give some exact analytic solutions by using the power series method. This result enriches the types of solutions of nonlinear shallow water wave equation and has important physical significance for further study of this kind of equation.