Sine-Gordon方程基本理論的探討

Abstract

Sine-Gordon 方程是一個二階的偏微分方程。方程式經由轉換變成一單擺運動，並且我們得到在複數平面上是一個雙值函數，所以我們引進一個新的空間，即黎曼面，使得 在這個空間上為可解析的單值函數，並且我們研究在黎曼面上的積分以及數值解。 再來我們運用古典橢圓函數的理論來尋求單擺方程的特殊解及其他性質。

Sine-Gordon equation is a second order partial differential equation. We consider the traveling wave solution. Then it comes to the pendulum motion. But it is a two-valued function on . So we need a new space, which is known as the Riemann surface, such that it becomes analytic single-valued function on the Riemann surface. And then we study the classical elliptic function theory to solve some special solutions of it , and discuss their properties.