Exact Solution Of Nonlinear Schr?dinger Equation

Schr ¨odinger equation is the most basic equation in quantum mechanics,and the nonlinear Schr ¨odinger equation is a nonlinear equation with solitary wavelet solution.It is closely related to many nonlinear problems in theoretical physics,such as nonlinear optics,ion acoustic wave of plasma and so on.It is widely used in the basic model of nonlinear wave,and has a very wide range of applications in many fields of physics.It comes from the study of nonlinear wave propagation in dispersive and inhomogeneous media,such as plasma and inhomogeneous media.It is a general equation to describe the amplitude evolution of nonlinear wave train in weak nonlinearity,strong dispersion and hyperbolic.In recent years,many physicists and mathematicians pay attention to the nonlinear Schr ¨odinger equation,and have many important results in the study of its solution.In this paper,three different methods are used to solve the nonlinear Schr ¨odinger equation,which is transformed from Landau Lifshitz equation.The structure of this paper is as follows:The first chapter is the introduction,which mainly introduces the physical background,source,introduction of the equation,the research status at home and abroad and the main results of this paper.The second chapter is the preparatory knowledge,mainly introduces the solution method used in this paper.The third chapter is the first part of the paper,which is to modify the ferromagnetic chain equation by defining a new function,to get the form of nonlinear Schr ¨odinger equation,and then to find out the exact solutions of its soliton solution,periodic solution by the method of separating variables.We first find its soliton solution by the method of separating variables,and then extend this method to the one-dimensional higher-order equation which is extended by the nonlinear Schr ¨odinger equation and added with self consistent field.The applicability of the method of separating variables is explained,and the exact and uniform solutions of the two equations are obtained.The fourth chapter is the second part of the paper.In this chapter,two methods are used to study several solutions of the modified nonlinear Schr ¨odinger equation.Not only the soliton solution but also its periodic solution are obtained.At last,the research contents of this paper are simply summarized,and the prospect and direction of the application of these methods in high-dimensional equations in the future are put forward.