Numerical Solution To A Kind Of The Nonlinear Schr(?)dinger Equation

Nonlinear Schr(?)dinger equation is an important branch of mathematics.It is widely used in high-energy physics,quantum mechanics,biomedicine,nonlinear optics and many other fields.If we want to understand the principles of these physical phenomena,we must study the solutions of nonlinear Schr(?)dinger equation,We can't find the exact solution of the nonlinear Schr(?)dinger equation,so it's very important to find the numerical solution of the nonlinear Schr(?)dinger equation.In this paper,we study the numerical solution to a initial periodic boundary value problem for a kind of nonlinear Schr(?)dinger equation.We propose a new implicit difference scheme preserving conservations of the mass and energy.We prove that the solution of difference solution exists and converges to the exact solution of original problem with O(?~2+h~2)in L~? norm,where ? is the time-step and h is the space-step.The dissertation consists of fiver chapters.In chapter 1,we introduce the physical background,research significance and research status of the nonlinear Schr(?)dinger equa-tion,including the well posed analytical solution and numerical solution,and summarize the main research content of this paper.In chapter 2,we will introduce some common notations and some important lemmas.In chapter 3,we construct a finite difference scheme.The main method is to use cen-ter difference quotient in space,forward difference quotient in time and center difference quotient in interior.We prove this difference scheme satisfies two discrete conservation laws.This step is very important for the prior estimation of subsequent difference de-compositionIn chapter 4,we prove some properties of the difference scheme.First,we prove the existence of the difference solution of the difference scheme by using the Brouwer fixed point theorem,and we prove the boundedness of the difference solution under the L~? norm by using two discrete conservation laws.At last,We prove that the solution of difference solution converges to the exact solution of original problem with O(?~2+h~2) in L~? norm.In chapter 5,we make a conclusion on the whole work and state future work.