Bifurcation And Exact Solutions Of A Class Of Nonlinear Schr (?) Dinger Equations

Exact solutions of nonlinear partial differential equation and its solutions are a frontier research topic and heat questions in the study of nonlinear science. A series of exact solutions of nonlinear partial differential equations has been put forward. But the uni?ed method of solving all the equations has not been found, the solutions about partial differential equation are always studied. This thesis mainly solved the Schr¨odinger equations in physical background of nonlinear by two different ways.In chapter one, the introduction of the historical background and the research status about partial differential equation are introduced. Then illustrated the research signi?-cance and research contents of this thesis.The second chapter introduced the basic theory knowledge needed in this thesis, including the soliton theory, the dynamical systems and bifurcation theories, common type of nonlinear partial differential equation etc.The third chapter narrated the relevant solutions of the nonlinear partial differential equation. Describing two methods used to solve equation in this thesis: the modi?ed generalized Tanh-function method, the method of dynamical system and the theory of bifurcations.In the fourth chapter, we used the modi?ed generalized Tanh function method, the method of dynamical system and the theory of bifurcations to solved three nonlinear Schr¨odinger equations. At the same time using Maple software and Matlab software obtained phase portraits of Hamilton system and the 3D graphics of solutions. Through comparing the two methods, we can get that the method of dynamical system and the theory of bifurcations are more effective and get the better solutions.