橢圓與光學晶格位能井之非線性薛丁格方程的數值研究

Abstract

本論文主要運用數值方法來探討非線性薛丁格方程的解之穩定性分析，其中這邊考慮的非線性薛丁格方程式是含有位能井（trap potential）。首先，我們先介紹非線性薛丁格方程式以及兩種不同的位能井。接著，簡述我們所運用的數值方法。最後，我們呈現一些數值模擬結果。經由我們的數值模擬，發現在一些解的形式與位能井的選擇是可以觀察到解的穩定性變化。

In this thesis, we would like to find out the stability of the solution which linearized around the special solution of the nonlinear Schrodinger equation. Here, we consider nonlinear Schrodinger equations with the potential part, optical lattices form and elliptic form. In the first part of the thesis, we introduce roughly the nonlinear Schrodinger equation and two kinds of trap potentials. Then, we show the mathematical analysis of the numerical method that we used. At the end, we give some numerical results of our numerical experience.