Numerical Studies On Nonlinear Schrodinger Equations And Related Issues

The study on numerical methods for partial differential equations is not only an important branch of computational mathematics, but also has extensive applications in many other fields, such as computational physics, chemistry, biology, et al. Most models describing various phenomena in the world are nonlinear differential equations. One of them is the nonlinear Schrodinger (NLS) equation which plays a key role in many fields of physics. Therefore, some nonlinear Schrodinger equations and their related issues are numerically resolved and studied in this thesis. The main numerical methods applied are the orthogonal spline collocation (OSC) method, the finite difference method, and the split step technique.Firstly, discrete-time OSC schemes are proposed for the NLS equation with wave operator and the coupled nonlinear Klein-Gordon-Schrodinger equation. The conservation, convergence and stability of the schemes are strictly proved in theory. Numerical experiments are carried out to verify the theorical results.The split step method is efficiently combined with the finite difference method, the spectral method, or the finite element method and so on. Surprisingly, the combination of the split step method and the OSC method is missed. Therefore, the combination is tried in this thesis for the coupled NLS equation in one dimension. The new method is named as split-step OSC (SSOSC) method by the author. The SSOSC method is compared with three other discrete-time OSC schemes, and the new method is superior. Thus, the first step is successful.As the SSOSC method is reliable and efficient, it is extended to multidimensional NLS equations. And this extension is also successful, which is examed by many numerical tests. Since no research on three-dimensional problems by the OSC method has been found, this step is really inspiring. Moreover, sometimes the time-dependent potential in the NLS equation cannot be integrated exactly which weakens the advantage of the split step method, so numerical integration is utilized by the author. The approximate strategy is efficient and does not reduce the original accuracy. It is worth noting that this approximation is efficient not only for the SSOSC method, but also for the split-step finite difference method and the split-step Fourier spectral method.Ginzburg-Landau (GL) equations are more complicated than the NLS ones. Finite difference schemes are presented for the Kuramoto-Tsuzuki equation which is a one-dimensional cubic GL equation in form. The relation between the nonlinear schemes and the linearized ones is discussed in theory, and also verified by numerical examples. For the two-dimensional cubic GL equation, a split-step finite difference (SSFD) scheme and a split-step alternating direction implicit (ADI) scheme are proposed. The connection between the two schemes is discussed, and the plane wave solution of the equation is used to analyze the schemes. Linearized analysis is applied to discuss the stability, and numerical tests are also carried out to examine the schemes. Split-step compact finite difference schemes are constructed for the multi-dimensional cubic-quintic GL (CQ GL) equations. Firstly, the split step method is applied, and the CQ GL equations are separated into two nonlinear subproblems and one or several linear ones. The linear subproblems are resolved by the compact finite difference schemes. As the nonlinear subproblems cannot be integrated exactly as usual, the normal split-step compact finite difference (SSCFD) method is failed. Theorefore, the author utilizes the Runge-Kutta method, and the accuracy order is not reduced. Extensive numerical experiments are carried out to show that the proposed numerical approximation is successful and efficient. And even for the situations that the nonlinear subproblems can be integrated exactly, the present method is still as good as the usual SSCFD one. It implies that the requirement of integrating the nonlinear subproblems exactly may be not always necessary. This is an extension and improvement of the split step method.