Numerical Solutions Of Nonlinear Schr(?)dinger Equation With Wave Operator And Its Application

In this paper,the nonlinear Schr(?)dinger equation with wave operator are studied in four aspects,the first focuses on the stability of the equilibrium solution of the equation;in the second part,we mainly study the solution of the exact solution of the equation,and obtain the periodic solution of Jacobi elliptic function,the multistage envelope solution of Lam?e function and some explicit solitary wave solutions;in the third part,we construct four different finite difference scheme,prove its convergence and stability,and verify its accuracy by numerical example;in the fourth part,we study the linear stability of the solution of the equation in the case of perturbation.The first chapter gives the research background and research status of nonlinear Schr(?)dinger equation and nonlinear Schr(?)dinger equation with wave operator,and lists the structure and the main content.The second chapter uses the Jacobi elliptic function method combined with Jacobi and try to pre elliptic function and Lam?e function of nonlinear Schr(?)dinger equations with wave operator are solved.The equation of Jacobi elliptic function periodic solutions and periodic solutions of Lam?e function and gives the multi order envelope solitary wave solutions and other solutions in the limit of the display.The third chapter,we first give the conservation laws of nonlinear Schr?inger equation with wave operator.The numerical solutions method of the three layer linear conservative schemes,the three layer nonlinear implicit schemes,the four layer explicit condition conservative difference scheme and three layer linear conditional non conservative difference scheme of the nonlinear Schr(?)dinger equation with wave operator are constructed based on finite difference method,meanwhile,we give two conservation laws,and the convergence and stability of the schemes are given,through the numerical simulation to verify the numerical solutions accuracy of O(?~2+ h~2),the conservation and four kinds of quality difference scheme.In the fourth chapter,the existence,stability and dispersion of travelling wave solutions of the nonlinear Schr(?)dinger equation with wave operator is studied,the traveling wave solutions of oscillation and stability and unstable conditions and the dispersion relation are given.The fifth chapter sums up the paper and outlook for the follow-up study.