The Difference Scheme Of A Class Of Nonlinear Schr(?)dinger Equations

As it is well known,the Schr(?)dinger equation is an important mathematical model,which is widely used in quantum mechanics,high energy physics,optics,optical soliton communication and many other fields.This paper discusses the periodic initial-boundary value problem of a class of nonlinear Schr(?)dinger equations with third-order derivatives and quintic nonlinear terms.This model is derived from optical soliton communication.This paper proposes a nonlinear difference scheme for the equation,which proves that the difference scheme satisfies the law of conservation of energy and the law of conservation of mass,and proves the existence of the solution of the difference scheme and the convergence and stability of the difference scheme.First,we constructed a finite difference scheme.The main method is to use the central difference quotient in space,forward difference quotient at the boundary in time,and central difference quotient internally.It is proved that the difference format satisfies the discrete mass and energy conservation laws.Then we proved the relevant properties of the difference decomposition of the constructed difference scheme.Using the Brouwer fixed point theorem,we proved the existence of the difference scheme when the time-space step ratio satisfies certain conditions,and applied two discrete conservation laws Proof of the boundedness of difference decomposition.Finally,the a priori estimates of the difference decomposition is used to discuss the stability and convergence of the difference scheme and give an error estimate.It is proved that the difference decomposition converges to its analytical solution according to the maximum norm,and its convergence order isO(?~2+h~2).