The Numerical Methods For A Kind Of Nonlinear Schr?dinger Equations

Nonlinear Schr?dinger equation is one of the most important equations with universal significance in the modern science. It's widely applied in many areas of physics, such as the fluid dynamics, solid state physics, nonlinear optics, electromagnetism etc.. In the 1870 s, when people studied the dingero Schr equation with the inverse scattering methods, they found that the solution of the Schr?dinger equation was composed of a series of soliton waves and dispersive waves. Then, the soliton solution is widely studied and applied because of the well properties. Especially, the optical soliton generated by the optical communication system, is applied widely. And it's described by the cubic NLSE.Nonlinear Schr?dinger equation is a kind of nonlinear parabolic partial differential equation. There is no universal and effective method to solve it. So numerical methods are often used to study its properties.In this paper, some efficient numerical methods are proposed to compute the cubic nonlinear Schr?dingerequation and analyze the error of the local truncation. In addition, the stability of the finite difference scheme is also analyzed.In the part of the numerical experiments, different numerical experiments are designed to show the effectiveness of the difference schemes. Results show that the finite difference schemes are effective.Finally, the contents of this paper are summarized and the further work is also discussed.