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.

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一类非线性浅水波方程的对称约简，动力学行为和精确显式解

通过对一类非线性浅水波方程使用李对称性分析和动力学系统方法，给出了基于李群方法的精确解。特别是给出了行进解的分支和精确的显式参数表示，并获得了非线性方程的可能的孤波解和许多不可数的无限周期波解。为了保证上述解决方案的存在，确定了所有参数条件。此外，我们使用幂级数方法给出了一些精确的解析解。该结果丰富了非线性浅水波方程解的类型，对进一步研究这类方程具有重要的物理意义。