以保結構算則求解非線性薛丁格方程

摘要瘋狗浪較普通海浪高出一倍以上。在沒有電子儀器的航海時代，瘋狗浪被斥為無稽之談——能在此滔天巨浪中倖存之人不多，加之當時學界普遍認為海浪高不過30英呎（9.1m）。1995年元旦，在大西洋北海的鑽油台上，靈敏的電子儀器記錄到了以時速72.4公里撲打平台的巨浪，其浪高25.9m，足足較當時週遭的海平面有效波高高出13.9m。瘋狗浪這一現象也會在海波之外的介質中出現。2007年，科學家們發現了光學中的瘋狗浪，此一成果發表在《自然》期刊上。摘要在本論文的第一部份，我們嘗試從線性與非線性兩種機制來瞭解瘋狗浪這一現象。從非線性機制中，可推導出非線性薛丁格方程式。同樣地，在光纖的雙折射現象中，我們可以推導出耦合的非線性薛丁格方程式。摘要在論文的第二部份，我們將討論能解此非線性薛丁格方程式的數值方法。其中，非線性薛丁格方程式被分為線性與非線性兩部份，在數值求解時交替解之。另外，此數值方法也可解耦合的非線性薛丁格方程式。摘要論文的第三部份則給出了數值結果。首先是帶有實解的例子，我們能由此得知程式是否無誤。接著我們解了半古典（semi-classical）的非線性薛丁格方程式。最後我們嘗試模擬瘋狗浪以及旋轉玻色－爱因斯坦凝聚中的兩個量子化渦格。

關鍵字

瘋狗浪；非線性薛丁格方程式；耦合非線性薛丁格方程式

並列摘要

Rogue wave has been captivating to researchers. People have not fully understood rogue wave yet. The first strong scientific evidence was presented in 1995, just nearly a quarter-century ago. Since then, people gradually noticed that this phenomenon could occur in other media as well. The first optical rogue wave was reported in 2007. In the first part of this thesis, we try to understand this special phenomenon. The definition of rogue wave is introduced and then the physical mechanisms of rogue wave are given according to linear mechanism and nonlinear mechanism. From the nonlinear mechanism, nonlinear Schrödinger equation (NLS) can be obtained analytically. We would like to know more about rogue wave by investigating NLS equation. Similarly, the coupled NLS equations are obtained by considering the linear and nonlinear response in the birefringent optical fibers. The coupled NLS equations can describe the pulse propagation in those fibers. In the second part of this thesis, we review a numerical method that can solve NLS equation for which the equation is separated into linear and nonlinear part. The latter can be solved iteratively. The method has been modified in order to solve coupled NLS equations. In the third part of this thesis, we first consider examples that admit exact solutions in order to know the proposed algorithms work well in both NLS and coupled NLS cases. We then move to solve semi-classical NLS equation by using this method. Last but not least, we simulate rogue wave and two quantized vortex lattices in a rotating trapped Bose-Einstein condensate (BEC). We hope to know more about these phenomena through simulations.