The Numerical Methods For The Nonlinear Schr(?)dinger Equation And Related Coupled Equations

The nonlinear partial differential equation has important application in nonlinear optic, condensed matter physics, Bose-Einstein Condensate, plasma physics and fluid mechanics. Numerical method is an important tool to get the characteristic of the nonlinear partial differential equation and to obtain it's potential physical application. Taking the nonlinear Schr(o|¨)dinger (NLS) equation and the related coupled equations as examples, the new numerical methods are studied. Although many teoretical and numerical methods have been used to investigate the NLS equation, as far there are little numerical methods for accurately simulating the generalized nonlinear Schr(o|¨)dinger (GNLS) equation. The main idea of the split-step method is splitting the nonlinear Schr(o|¨)dinger equation into the linear one and the nonlinear one at one temporal interval. Then the nonlinear can be solved exactly and the linear one will be solved by the finite difference method, or finite element method, and so on. The time relaxation approach is also aim to deal with the nonlinear term, which not only can avoid the complicated interation computation efficiently but also can obtain a high accuracy. In this paper, a new numerical method is applied to this GNLS equation. It combined the split-step method and the time relaxation approach to overcome the complaxation of the GNLS equation. The numerical solutions can simulate accuratly the propagation of solitons of various cases in which different parameters are choosed.Additionaly, the system of a wave equation coupled with the Schr(o|¨)dinger have extensivly application among various branch of physica and engineering. The study for the soliton solutions of these coupled equations are hot topics during long times. The finite difference method, the finite element method and the spectral method are often been applied to the coupled equations. The coupled Schr(o|¨)dinger-KdV equations and the coupled Schr(o|¨)dinger-Boussinesq equations are a spectial type among the various coupled equations. They are often described the interaction of Langmuir waves and the ion-acustic waves. We study the coupled Schr(o|¨)dinger-KdV equations by the finite element method (FEM). The quadratic Lagrange function and B-spline function are choosed as the base function, respectively. The semi-discrete and the full-discrete scheme are conducted to simulate the exact solutions numerically. These schemes can preserve the conservations of the equations. Numeical experiments verified the accuracy and efficiency of our schemes. In order to investigate extensionally, the finite difference method is applied to this coupled equation too. Three types of difference scheme are conducted, such as the time-spliting finite difference scheme, the Crank-Nicolson scheme and the three-level finite difference scheme, and so on. The numerical comparisions are made among these schemes. Simultaneously, the couled Schr(o|¨)dinger-Boussinesq equations are studied by the FEM. The B-spline function is choosed as the base function and a semi-discrete scheme is conducted. The existence, uniqueness and the error estimates are proved. The numerical experiments are conducted to verify the accuracy of our scheme.