Researches On Several Problems About Solving Nonlinear Evolution Equations

Under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, researches on several problems about solving nonlinear evolution equations, especially about integrable systems, has been considered in this dissertation, including C - D variable separation approach based on the idea of AC = BD introduced by Prof. H. Q. Zhang, a uniform and mechanical model of C - D pair construction based on the pseudo-differential division with remainder and the mechanical calculation of exact solutions to nonlinear evolution equations.As three kinds of well-known nonlinear phenomena in the whole of nature, Solitons, Chaos and Fractals, which can generate nonlinear evolution equations, are introduced in chapter one. The origin and development of some subjects about mathematics mechanization, soliton theory and integrable systems are involved. The work and achievements of the domestic and foreign scholars are presented also. Main works of this thesis is summarized at last.Differential Algebra related to the idea of AC = BD are presented firstly in chapter two. The second and the third parts is devoted to AC = BD and its application to nonlinear partial differential equations with basic notations, basic theory of C - D pair and C - D integrable systems. Based on the idea of AC = BD and the pseudo-differential division with remainder, a uniform and mechanical model of constructing C- D pair of formal variable separation approach from one variable to two variables is put forward, which will accelerate the application of C - D pair theory to other well-known and effective methods. The definition of C - D variable separation approach is proposed and give novel explanation of formal variable separation approach at last.Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, two mechanical algorithms-variable coefficient Riccati equation method and Jacobi elliptic functions method for constructing the exact solutions of nonlinear evolution equations are introduced firstly. The effectiveness of the algorithm is proved by means of application to some higher dimensional nonlinear differential equations, such as (2+1)-dimensional cubic nonlinear Schrhdinger equation, the averaged dispersion-managed fiber system equation and the (2+1)=dimensi0nai dispersive long wave equation based on Wu elimination theory and symbolic computation software Maple.As is well known, there are three fundamental methods-classical Lie group method, nonclassical Lie group approach and CK's direct method to solve nonlinear systems especially to obtain reduction solutions, which always reduce higher dimensional equations to lower dimensional ones. A PDE approach is proposed in chapter four based on reduction idea and some group invariant solutions of (2+1)-dimension shallow water wave equation and (2+1)-dimension PKP equation are listed.