Constructive Methods In Solving Nonlinear Differential Equations And Symbolic Computation

In this dissertation, under the instruction of mathematical mechanization and AC=BD model, and by means of computer algebraic system sotfware, some topics are studied, in-cluding some problems of solving some significant soliton equations in the theory of soliton, symmetry of differential-difference equations and algebraic geometric solutions for some soliton equations.In Chapter 1, we introduce the history and development of mathematical mecha-nization and computer algebra, soliton theory, solutions of the soliton equations, soltion integrable system and algebraic geometry. Some works and achievements on these subjects involved in this dissertation are presented at home and abroad. as well as the main work of this dissertation.In Chapter 2, the main content and idea of AC= BD model theory and the con-struction of the exact solutions of differential equations under the guidance of this theory are presented.In Chapter 3, based on the idea of algebraic method, algorithm reality and mecha-nization for solving nonlinear evolution equations, the new project equations method are presented and many Jacobi elliptic functions solutions are obtained, as well as the first integrable method is generalized to the nonlinear partial differential equations.In Chapter 4, hyperelliptic functions solutions with genus 2 of some soliton equations are obtained by using direct method, and some new solutions are presented by combining with the symmetry transformation group, as well as some new symmetry of some (2+ 1)-dimensional differential-difference equations are obtained by direct method.In Chapter 5, pluri-genus Riemann theta functions solutions of (2+1)-dimensional lattice, (3+1)-dimensional JM equation and generalized variable-coefficient fKdV equa-tion are obtained by means of the Hirota bilinear method and the Riemann theta function. As well as starting from a matrix spectral problem with rank 2, we construct the quasi-periodic solutions of the generalized coupled mKdV equation and Schodinger equation.