Soliton Theory, Several Exact Solution Method Research And Application

In this disscrtation, under the guidance of mathematics mechanization and by means of computer algebraic system software, some problems of solving some significant nonlinear evolution equations in the theory of soliton arc discussed and some methods for constructing the exact solutions of nonlinear evolution equations arc presented and improved. The methods presented arc realized on symbolic computation system Mathcmatica or Maple. The main work is as follows:Chapter 1 is devoted to reviewing the history and development of the soliton theory, mathematics mechanization, with an emphasis on some achievements on the subjects involved in this dissertation.Chapter 2 considers the construction of exact solutions of partial differential equations (PDEs) under the instruction of the AC = BD theory introduced by Prof. H.Q. Zhang. Some examples arc chosen to illustrate to how to use the principle and the application range. At last, the method for constructing A: the nonlinear evolution equations with nonlinear terms of any order arc discussed.In chapter 3, the homogeneous balance method is improved to seek the Backhund transformation for some nonlinear soliton equations with nonlinear terms of any order. Based on the Backlund transformation, some exact solutions arc obtained. By use of the truncated Painleve expansion method, the Backlund transformation and exact solutions of a generalized nonlinear Schrodingcr equation arc investigated.In chapter 4, the extended- tanh function method is improved and the generalized Ric-cati equation expansion method is presented. Some general exact solutions for some nonlinear soliton equations with nonlinear terms of any order arc derived by use of the improved extended-tanh function method. Rich exact solutions (including solitary solutions, soliton-like solutions, periodic solutions, period-form solutions, rational solutions) arc constructed for the (2+1)-dimcnsional breaking soliton equation, (3 +l)-dimcnsional Jumbo-Miwa equation by means of the generalized Riccati equation expansion method.In chapter 5, first, the projective Riccati equation expansion method is extended to obtain some new exact travelling wave solutions for two nonlinear evolution equations with high order nonlinear terms and a nonlinear evolution equation. Second, the generalized projective Riccati equation method is presented to construct some exact solutions for the generalized (2+l)-dimcnsion Schrodingcr equation.In chapter 6, based on the Q-dcformed hyperbolic functions, the generalized projective Riccati equation method and the complex envelope ansatz method, the generalized Q-dcformcd hy-pcrbolic functions method is developed. Then by use of the method, three nonlinear Schrodingcr cquations with physical significance are investigated. 1) We study an averaged fiber system equation, which governs the dynamics of the core of the dispersion-managed soliton, and simulate the soliton propagation and solitons interaction scenario by computer; 2) We study the higher order nonlinear Schrodingcr (HNLS) equation to seek for some general exact analytical solutions. In particular, the novel soliton solutions described W—shaped arc found. Then W-shaped soli-ton propagation and W-shaped solitons interaction arc discussed and simulated; 3) We study the nonlinear Schroodingcr equation model (NLSE) with varying dispersion, nonlincarity, and gain or absorption. Rich soliton-like solutions including two arbitrary functions arc obtained. By choosing the arbitrary constants and arbitrary functions suitably, soliton propagation and solitons interaction arc discussed and simulated, such as, snake-solitons' propagation and interaction, boomcrange-like solitons propagation and interaction, periodic dispersion bright and dark solitons' propagation and interaction, and so on.