Lie Symmetries,1-dimensional Optimal System And Optimal Reductions Of (2+1)-dimensional Nonlinear Schroedinger Equation~*

With the development of nonlinear, a large number of nonlinear partial differential equations arrising. In particular, the mathematical models of the problems involved continuous time variable are established by evolution equation, and the solution to the problems depend on solving corresponding various problems related to these evolution equation. At present, there is no effective and unified way to solve this kinds of equations in mathematics. Hence, looking for solving method for the equations is one of the important research topics in the field of partial differential equations. Since, the quantitiese or the qualitative relations between physics variables can be determined by solving the nonlinear evolution equations. Especially, the relations among physics variables can be inspected by figures of the exact solutions of the nonlinear evolution equations. Hence, the study of solving exact solutions and solitary wave solution to nonlinear evolution equations has become a hot topic in the field of nonlinear science.In recent years, the development of the computer, especially the emergence of symbolic computation software promotes the study of solving the nonlinear partial differential equation. Although there is no systematic and unified way to solving the nonlinear evolution equations, many efficient methods have been established and developed to solve some integrable nonlinear evolution equations. For instance, Darboux transformation, Backlund transformation, Funetional variable separation approach, Classical and Non-Classical Lie group approaehes, Function expansion method and so on. Meantime, many new methods have been developing.Due to there is no efficient method to solve the nonlinear evolution equations, developing a method for specific evolution equation is necessary and it is important in theoretical and practical points. In this paper, we study the Lie algebra properties and solving methods of (2+1)-dimensional Nonlinear Schrodinger Equation, using the Lie algebra admitted by the equation. Concequently, we have the following academic results:1. For understanding and grasping in the Lie method, we review thesymmetry method of partial differential equation,; 2. Determine the9-dimensional classical Lie algebra of the (2+1) dimensional schrodinger equation;3. Calculate1-dimensional optimal system of the Lie algebra of the equation;4. Give the reductions of the equation;5. Some exact solutions of the equation are given.The structure of the article is elucidated as follows. In chapter1, as the introduction parts, we give the research background, present situation of the topics on symmetry, including main work of this article and its research significance. In chapter2, we give some theories of lie symmetry method. In chapter3, we give the specific calculation process of Lie symmetries,1-dimensional optimal system and reductions of (2+1)-dimensional Nonlinear Schroedinger Equation according to the obtained Lie algebra. Concluding remarks are given in chapter4.