在B型代數結構下之N相黎曼空間的單擺運動之確切理論與數值計算

Abstract

Sine-Gordon 方程 u_xx-u_yy+sin⁡u=0是被廣泛應用的偏微分方程式，而其某些特殊解滿足非線性二階微分方程 (d^2 u)/(dt^2 )+sin⁡u=0，此為單擺運動方程式。當求解 (d^2 u)/(dt^2 )+sin⁡u=0 我們首先利用 sin⁡u 的Maclaurin級數來替代 sin⁡u 使得原微分方程變為 (d^2 u)/(dt^2 )+P(u)=0，其中 P(u) 為多項式。此方程的解存在於N相黎曼空間。我們利用正確的代數結構來建構這些黎曼空間，使我們可以在黎曼空間中執行路徑積分進而得到方程數值解。之後我們研究古典橢圓函數，利用 Jacobian 橢圓函數來分析單擺運動方程。最後，我們利用 Jacobian 橢圓函數來導出此方程式的確切解與週期。

The Sine-Gordon equation u_xx-u_yy+sin⁡u=0 is a well-known Partial differential equation, and there are some special solutions satisfy the nonlinear second-order differential equation (d^2 u)/(dt^2 )+sin⁡u=0 which is the Pendulum motion. As we solving the differential equation (d^2 u)/(dt^2 )+sin⁡u=0. We first replace sin⁡u by the Maclaurin Series of sin⁡u to get the differential equation of the form (d^2 u)/(dt^2 )+P(u)=0 , where P(u) is a polynomial. Solutions of such equations reside in Riemann Surfaces of genus N. We construct these Riemann Surfaces with the correct algebraic structures. So we can perform path integrals on the Riemann Surfaces to get the numerical solution of the equation. Next, we investigate the classical Elliptic functions, and use the Jacobian Elliptic function to analyze this nonlinear pendulum motion. Finally, we derive the exact solutions and the periods of those solutions by the Jacobian Elliptic functions.