非線性薛丁格方程基本理論的探討

Study of the Underlying Theory of the Nonlinear Schrodinger Equations

管俊奎(Jiun-Kuei Guan)

指導教授 ： 李榮耀

國立交通大學/理學院/數學建模與科學計算研究所/碩士(2014年)

https://doi.org/10.6842/NCTU.2014.00275

摘要

在此篇論文中, 我們研習用橢圓函數 dn(u; k)去表述Nonlinear Schrὂdinger equation (NLS)的某些特殊週期解 q

iq_t+q_xx+2〖|q|〗^2 q = 0,

此 dn 函數是定義在某個黎曼空間上的, 所以首先我們介紹黎曼空間的理論, 其次再介紹橢圓函數, 最後再利用黎曼空間和橢圓函數的理論去分析及解NLS的特殊解及其退化解。

關鍵字

黎曼空間 ； 橢圓函數 ； 非線性薛丁格方程

並列摘要

In this paper, we want to use the elliptic function dn(u; k) to express analytically some special solutions of the Nonlinear Schrὂdinger equation (NLS),

iq_t+q_xx+2〖|q|〗^2 q=0

The function dn(u; k) is defined on some Riemann surface, so we first introduce the theory of Riemann surfaces, and then we introduce elliptic functions. Finally, we use theory of Riemann surfaces and elliptic functions to analyze and solve some special solutions of the NLS and discuss the degenerates of those solutions.

並列關鍵字

Riemann surface ； elliptic function ； nonlinear Schrodinger equation

參考文獻

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