Study On Branches And Exact Solutions Of Schr (?) Dinger Equation Equations

Nonlinear science is a basic science to study the commonness of nonlinear phenomena. It is an important part of nonlinear science to study the traveling wave solutions of nonlinear partial differential equations and their solutions. Since there is not still a uniform and universal method for solving partial differential equations, it is very important and valuable to seek some effective methods to solve the partial differential equations.In this thesis, the exact traveling wave solutions of the nonlinear Schr?dinger equation of power law exponential order, Schr?dinger-Improved-Bousinesq system and Schr?dinger-Benjamin-Ono system are solved by using the method of dynamical system. Firstly, by means of a suitable traveling wave transformation, the partial differential equation is transformed into an ordinary differential equation, and then it’s transformed into a planar dynamical system. Secondly, using Maple software, combining with the qualitative theories of ordinary differential equation, the phase portraits of the system are given and the nonlinear dynamics of the system are discussed.The exact solutions that we obtain, including soliton solutions, kink wave solutions, periodic solutions and breaking wave solutions. They are represented by Jacobian elliptic functions and hyperbolic functions. From the results, we can see that the results by the dynamic system method are rich and beautiful; from the solution process, the dynamic system method is simple, direct and effective, and the result enrich the diversity of solution structures of the equations.