Study On Nonlocal Symmetry And Symmetry-Preserving Discrete Scheme

Symmetry theory plays a more and more important role in the field of mathematical physics and so on, has become one of powerful tools in studying integrable properties and exact solutions of nonlinear system. Based on symmetry theory and symbolic com-putation, the main work is carried out in three aspects:nonlocal symmetries of nonlinear differential equations are investigated in this dissertation, some new exact solutions of these equations are obtained; Symmetry-preserving discrete, algorithm of nonlinear evo-lution equations is studied, and the method is extended to higher dimensional equation-s; Furthermore, based on symbolic computation platform Maple, a software package of nonlocal symmetry automatic deduction is developed. The main work is carried out as follows:In chapter1, an introduction of the research background and the current situation re-view related to this dissertation, which including symmetry theory, symmetry-preserving discrete and symbolic computation is devoted. The main works of this dissertation are also illustrated.In chapter2, nonlocal symmetries of several differential equations are studied. First-1y, based on the auxiliary systems, a method of constructing nonlocal symmetry is pro-posed, and the method is applied to several classical equations, such as Boussinesq equa-tion, MKdV equation, AKNS system, etc. The nonlocal symmetries of these equations are obtained, and localized to corresponding closed systems. Secondly, we study PIB equation by using Lie group method and construct nonlocal symmetries of reduced e-quations. Some new exact solutions.of the PIB equation are obtained by using symmetry reduction method. Finally, nonlinear diffusion-convection equation is studied by using symmetry-based method, nonlocal-related potential systems and nonlocal symmetries are constructed.In chapter3, exact solutions of several classical equation are constructed. The sym-metry groups of the closed systems are obtained through classical Lie group method, op-timal systems of the symmetry groups are constructed. Some new exact solutions of these equations are obtained by using the symmetry reduction method, among which the inter-actions of elliptic periodic wave and soliton as well as periodic wave and kink soliton are found. In order to study the properties of the solutions, we select appropriate parameters and draw the corresponding images of above solutions.In chapter4, symmetry-preserving discrete format of nonlinear evolution equations is discussed. Symmetry-preserving discrete formats of MKdV equation, Boussinesq e- quation are constructed by using symmetry-preserving discrete method. We put forward a method which can construct symmetry-preserving discrete formats of high dimension-al nonlinear evolution equations, and apply this method to (2+1)-dimensional Burgers equation. Corresponding symmetry-preserving discrete formats which inherit Lie point symmetries of continuous equations are obtained.In chapter5, a software package of constructing nonlocal symmetries of PDEs is developed. Based on some auxiliary systems of PDEs, we put forward a mechaniza-tion method of constructing nonlocal symmetries and develop the corresponding software package NonSymI. Multiple instances are used to prove the effectiveness of this package, thus the efficiency of constructing nonlocal symmetries is improved greatly.In chapter6, the summary and discussion of this dissertation are given, and the outlook of future works are discussed.