Symmetry Groups And Exact Analytical Solutions For Several Nonlinear Partial Differential Equations

Soliton theory is one of several prolonged prosperous important hot topics in nearly half a century, some nonlinear partial differential equations with soliton solution is constantly dis-coverd in physics and even the whole of natural science field, here sparks of wisdom collision when the rigorous mathematics and the inspired pratical physics. Solution of nonlinear partial differential equation has been always one of the modern science problems in the forefront, of which numerical solutions and exact solutions are convenient access to the analysis and study of natural phenomena. While the numerical solution can not express the global features infinitely of the original equation. In many cases, people not need to know the specific number of in-dividual solutions, but wonder the general qualitative features of the original equation, which is often more profound description of the problem. And there are still non-linear instability of reconciliation and a series of reliability problems, therefore, the analytic solution of the non-linear partial differential equation research are of great significance in understanding the inner structure and the nature of the mathematical model and the movement law in the nonlinear action. In the investigation of solving nonlinear partial differential equations, many physicists and mathematicians proposed and developed a number of methods, such as:inverse scattering method, Backlund and Darboux transformation, hyperbolic function expansion method, vari-able separation method, Riccati equation expansion method, sub-equation expansion method and so on. Among them, two most effective major categories are the symmetry method and the direct constructed method in searching the exact analytical solutions of nonlinear partial differ-ential equations. Recently, with the thought of mathematics mechanization, the development of computer science, and the motivation of physics, symbolic computation system has developed as a powerful tool of variety of scientific and engineering problems. Based on computer sym-bolic system Maple, the generalized symmetry group method and the generalized sub-equation method are investigated in several nonlinear partial differential equations in this thesis.The thesis is arranged as follows:Chapter I Make a berif introduction about soliton and several methods for searching ana-lytical solutions of nonlinear partial differential equations in this chapter.Chapter 2 Introducing the Lie group method and then studying two examples;based on the generalized symmetry group method and symbolic computation, firstly, both the Lie point groups and the non-Lie symmetry groups of a (3+1)-dimensional nonlinear evolution equation and Maccari's system are obtained. Furthermore, some exact solutions of the two equations are derived from some simple solutions by the generalized symmetry groups.Chapter 3 Based on the generalized sub-equation expansion method and symbolic compu-tation, rich exact analytical solutions of the (3+1)-dimensional Gross-Pitaevskii equation with time-and space-dependent potential, time-dependent nonlinearity, and gain or loss are obtained, which include bright and dark solitons, Jacobi elliptic function solutions, and Weierstrass ellip-tic function solutions. With computer simulation, the main evolution features of some of these solutions are shown by some figures. Nonlinear dynamics of a soliton pulse is also investigated under the different regimes of soliton management.