The Application Of Symmetry And Perturbation Method For Solving Some Nonlinear Differential Equations

With the continuous progress of science and technology, nonlinear science has been devel-oped rapidly. Recently, nonlinear science is substantially studied and widely applied in biology,chemistry, mathematics, communication, economics and so on, in which a large number of non-linear systems appeared. In the past, many methods have been established and developed tosolve the nonlinear systems, for instance, the inverse scattering method, Darboux transforma-tion, Ba¨cklund transformation, varible separation approach and so on. To obtain the symmetryand symmetry reduction for nonlinear systems, there are two methods: CK direct method,symmetry group direct method and classical (or non-classical) Lie group approach. The for-mer solves nonlinear di?erential equations algebraically and the latter is based on group theory.Whether it is integrable or non-integrable nonlinear systems, we all hope to get their exact solu-tions, so as to explain the systems embedded in the practical problem better. But sometimes itis very di?cult to obtain their exact solutions. Many mathematicians and physicists proposedseveral methods to get approximate solutions of nonlinear systems, such as Adomian decom-position method, perturbation method, homotopy analysis method and so on. In this thesis,based on symmetry and homotopy analysis method, we investigate several nonlinear di?erentialequations.The thesis is arranged as follows:Chapter 1 Introducing the history and progress of soliton brie?y and several methods forsolving exact solutions and approximate solutions of nonlinear mathematical physics equations.Chapter 2 Based on symmetry method and symbolic computation, symmetry reductionsand symmetry transformation groups of nonlinear system are studied. First, nearly concentricKdV equation is reduced to low-dimensional partial di?erential equation by traditional CKmethod. Then, on the basis of modified CK direct method by Professor Sen-Yue Lou et al, thefinite symmetry transformation group of coupled Kadomtsev-Petviashvili equation is obtained.Chapter 3 Extending the applied range of extended symmetry method, some symme-try reductions and exact solutions of general (2+1)-dimensional variable coe?cients nonlinearSchro¨dinger equation and Kadomtsev-Petviashvili equation are obtained. Then with numericalsimultation and figures analysis, some physical explanation for these exact solutions are given.Chapter 4 Some approximate solutions of Klein-Gordon-Schro¨dinger equation are obtainedby homotopy analysis method. Then with numerical simulation and error analysis, the validityof the approximate solutions is verified.