Some Mechanical Methods In Soliton Theory And Differential Geometry

In this dissertation, we discuss some problems of soliton theory, the integrable systems and some theorems of differential geometry with the aid of Wu method (including Wu algebraic elimination method and Wu differential elimination theory). Some mechanical methods are presented to obtain exact solutions(including soliton solutions, periodic solutions, doubly-periodic solutions and rational solutions) of nonlinear evolution equations. Wu-Ritt differential characteristic set theory together with Reid method is applied to linear partial differential equations which possess physical significance to obtain the size of solutions. It is also applied to differential geometry to prove some theorems mechanically.In Charter 1, we introduce some related definitions, the origin and development of soliton theory and the relations between soliton theory and differential geometry. The main works and achievements which have been obtained are presented.Charter 2 is devoted to AC=BD model and its applications in partial differential equations and differential geometry. First, we give basic notations, basic theory of C-D pair and C-D integrable systems with the algorithm to construct C-D pair. How to seek transformation u = Cv is an important aspect in Charter 2. Then, AC=BD model is applied to differential geometry. C-D pair and general C-D integrable system are defined.In Chapter 3, we study the applications of the improved homogenous balance method. We apply the method together with Wu algebraic elimination method to Boussinesq equation, and many new soliton and doubly-periodic solutions are obtained. Applying the method to SK, KK, KP, DLW equations and KdV equation with variable coefficients, we obtain not only their Backlund transformations but also new exact solutions.Chapter 4 deals with some mechanical methods to obtain soliton solutions of nonlinear evolution equations including new extended-tanh function method, extended Ric-cati equation method, project Riccati equation method and generalized Riccati equation method. The multiple soliton solutions of the generalized Riccati equation are obtained. With these solutions, we obtain more exact solutions(including solitary wave solutions, periodic wave solutions and rational solutions) of a kind of nonlinear evolution equations, such as Burgers equation, general Burgers-Fisher equation and Kuramoto-Sivashinsky equation. More soliton-like solutions of the Burgers equation with variable coefficients are obtained by use of our method. Wu method is the most important basic tool during the course of solving these equations.In Chapter 5, we present the mechanical methods to obtain doubly-periodic solutionsof nonlinear partial differential equations. First, we give the improved Jacobi elliptic function expansion method. It is more effective than the sine-cosine method, the sn-cn method and tanh-method. We apply the method to the combined KdV and mKdV equations to obtain Jacobi elliptic function solutions and other exact solutions. Then, we also present the first kind and the second kind of elliptic equation method, with which more Jacobi elliptic function solutions of these two kinds of equations are obtained. With these solutions, we obtain more doubly-periodic solutions of a class of nonlinear evolution equations, such as the coupled KdV and mKdV equations. These solutions are degenerated to soliton solutions under degenerated conditions. Finally, our method is applied to higher dimensional KP equation with variable coefficients to obtain more doubly-periodic solutions.In the last chapter, we study basic theory of Wu-Ritt method and its applications. We apply Wu-Ritt method and Reid theory to linear partial differential equations to obtain the size of solutions. We also apply Wu-Ritt differential characteristic set to differential geometry to prove some theorems mechanically.