Exact Solutions Of Some Nonlinear Partial Differential Equations And Applications Of Moving Frames

Nonlinear partial differential equations are important mathematical models in various fields of sciences such as mathematics and physics. In this dissertation, based on mathematical mecha-nization idea, AC=BD model and moving frames theory, some topics are studied including exact solutions of nonlinear partial differential equations, complete system of differential invariants of differential equations and the method of equivariant moving frames to solve partial differential equations. There are six chapters in this dissertation:In Chapter1, the development and research achievements of soliton theory, mathematical mechanization, moving frames theory, differential invariant theory, exact solutions of mathe-matical physics equations and the brief of research work in this dissertation are introduced.In Chapter2, the basic knowledge associated with this dissertation is introduced, including some fundamental knowledge about Lie groups and it’s applications to differential equations.In Chapter3, AC=BD theory and its applications to differential equations are introduced.In Chapter4, based on the theory in Chapter3, the extended coupled sub-equation ex-pansion method is improved to solve nonlinear partial differential equations. The generalized Schrodinger-Boussinesq equation and coupled nonlinear Klein-Gordon-Schrodinger equations are given to prove the validity and advantages of this method.In Chapter5, the moving frames theory and it’s application to find the complete system of differential invariants of differential equations is introduced. The constructive computational al-gorithms to determine the complete systems of differential invariants of the mathematical physics equations:Hirota-Ramani equation and Drinfel’d-Sokolov-Wilson system are presented.In Chapter6, with the theoretical results in Chapter5, the application of moving frames theory to solve nonlinear partial differential equations is explored. A new method, the equivariant method of moving frames, to solve some nonlinear partial differential equations is given and enriches AC=BD model.