Some Problems In Theory Of Soliton And Their Mechanization Realization

This dissertation considers the following problems in the theories of soliton, Hamiltonian systems and fractional differential form:1. the realization of mechanization for constructing exact solutions of nonlinear evolution equations;2. Backhand transformation;3. the realization of mechanization for Painleve Test;4. the inverse problem in infinite dimensional Hamiltonian systems;5. fractional differential forms.Chapter 1 of this dissertation is devoted to reviewing the history and development of the soliton theory, mathematics mechanization, Hamiltonian systems (inverse problem) and fractional differential form, with an emphasis on the achievements on the subjects involved in this dissertation.Chapter 2 considers the construction of exact solutions of partial differential equations (PDEs) under the guidance of the theory of AC = BD. The basic theory of AC =BD and the algorithm to construct the C-D pair, introduced by Prof. H. Zhang, are illustrated through some concrete transformations. In particular, based on the division with remainder AC = BD+R, some concrete algorithms for constructing the exact solutions are also presenteted. The computing programs are given in terms of the symbolic computation software Maple. Finally, a method for constructing A, the nonlinear evolution equaitons with nonlinear terms of any order, is discussed.Based on the ideas of algebraic method, algorithm realization, and mechanization for solving nonlinear evolution equations, Chapter 3 deals with the construction of exact solutions for nonlinear evolution equations by use of Wu-method and symbolic computation. Its mam contents are as follows:1) The generalized Hirota-Satsuma KdV system and a coupled MKdV equation are solved by using an unproved extended-tanh function method, and some new exact solutions are found.2) The improved extended-tanh function method is further improved in order to solve the nonlinear evolution equations with nonlinear terms of any order, and the solutions in more general forms are thus obtained.3) A generalized extended-tanh function method, which includes the above extended-tanh methods as special cases, is presented and applied to construct the exact solutions of SLA equation.4) The Jacobi elliptic function method is extended to solve (2+1)-dimensional dispersive long wave equations.5) The complex-tan function method and its application are presented.6) The projective Riccati equation method is generalized to obtain some new exact solutions for general Zakharov-Kuzentsov equation.7) A generalized Riccati equation expansion method is presented to obtain some soliton-like solutions.In Chapter 4, by use of the symbolic computation softwares Maple and Mathematica, the generalized HBM method is improved to seek the Backhand transformation of nonlinear evolution equations. The Backlund transformation and exact solutions of two classes of nonlinear evolution equations as follows are also discussed:1) Backlund transformation for nonlinear evolution equaiton with nonlinear terms of any order;2) Backlund transformation for variable coefficient nonlinear evolution equations. Chapter 5 deals with the Painleve property of nonlinear partial differential equation basedupon the Wu-Ritt differential elimination theory. Firstly, the basic theory and algorithm of Wu-Ritt differential elimination are presented. Secondly, the general theory of Painleve analysis is discussed, and a new algorithm to test the P-property is proposed which has been realized on computer. Through the new algorithm, the resonance point can be found out without seeking the recursion relations, and moreover the Painleve property of the nonlinear partial differential algebraic equation is finally judged by the Wu-Ritt differential elimination theory automatically on Maple.Chapter 6 mainly discusses the following inverse problems of infinite dimensional Hamilto-nian systems:1) The canonical representation of some Hamiltonian systems in mathematical physical problems;2) The ordered analytic representatio...