Approximate Symmetry Of The Nonlinear Wave Equation With Dissipation Reduction And Infinite Series Solutions

In recent years, nonlinear science has been widely used in mathematics and physics, chemistry, economy and other fields. In the research process, we inevitably encounter a wide variety of nonlinear equations, which leads us to con-sider how to solve the nonlinear system of nonlinear partial differential equations (PDES), and how to explore the features of solutions of nonlinear systems. It is well known that symmetry group theory is one of the effective ways in studying exact solutions of nonlinear PDES.As studying the nonlinear systems, there are some nonlinear PDES with perturbation, we need to seek their approximate solutions. In order to study perturbations of PDES, some symmetry perturbation methods based on the Lie theory have been established. This paper is devoted to using the approximate symmetry method recently proposed to the nonlinear wave equation with dissi-pation. The structure of this paper is as follows, we:(1) Introduce the related preliminaries including the approximate symmetry method. Actually, the approximate symmetry reduction method is the combi-nation of both Lie symmetry and perturbation theory.(2) Obtain the approximate symmetry reduction and infinite series solution for the nonlinear wave equation with damping.(3) Obtain the approximate symmetry reductions and infinite series solution for the nonlinear wave equation with dissipation.(4) Draw conclusions of this paper and give some topics to be considered later.