標題: 黎曼曲面與橢圓函數的理論及其對正弦高登方程的應用
The Theories of Riemann Surfaces and Elliptic Functions with Application to the sine-Gordon Equation
作者: 陳建澤
Chen, Jian-Ze
李榮耀
Lee, Jong-Eao
應用數學系所
關鍵字: 黎曼曲面;橢圓函數;正弦高登方程;單擺;Riemann surface;elliptic function;sine-Gordon equation;pendulum
公開日期: 2012
摘要: 我們有興趣的是,研究正弦高登方程的一些特殊解,正弦高登方程如下: u_tt - u_xx + sin[u(x,t)] = 0 其中 -∞ < x < ∞ ,而且 t > 0 。 經由變數變換,我們可以將原本的方程式變成以下的形式: u_ss + sin[u(s)] = 0 這是一個對於時間 s 的單擺運動方程式,而且我們可以繼續推導變成 u_s = √2[E + cos(u)],其中 E 是一個常數。 可是 √2[E + cos(u)] 是一個複數上的雙值函數,所以我們介紹黎曼曲面 R 的理論,使得這一個函數在這個曲面上變成了一個可以分析的單值函數。 接下來,我們介紹橢圓函數的古典理論,並且利用它去對 u_ss + sin[u(s)] = 0 求解,並分析相關的性質。
The Goal of this paper is to solve the sine-Gordon equation, u_tt - u_xx + sin[u(x,t)] = 0, where -∞ < x < ∞ and t > 0. By using the method of substitution, we get u_ss + sin[u(s)] = 0, which is a simple pendulum motion at time s with the angular displacement u, and it implies u_s = √2[E + cos(u)], where E is constant. But √2[E + cos(u)] is a two-valued function on C, so we introduce the theory of the Riemann surface R such that it comes to a single-valued analytic function on this surface. Next, we introduce the classical theory of the elliptic functions, to solve u_ss + sin[u(s)] = 0, and analyze the associated properties.
URI: http://140.113.39.130/cdrfb3/record/nctu/#GT079922501
http://hdl.handle.net/11536/72170
Appears in Collections:Thesis