非線性偏微分方程Sine-Gordon方程的理論與特殊解

Abstract

在此篇論文中，我們的目標是找到Sine-Gordon方程的行進波解。我們想運用黎曼空間的理論和橢圓函數的理論來解Sine-Gordon方程。Sine-Gordon方程是一個二階偏微分方程，其定義如下: 𝑢𝑡𝑡−𝑢𝑥𝑥+𝑠𝑖𝑛[𝑢(𝑥,𝑡)]=0,𝑤ℎ𝑒𝑟𝑒−∞<𝑥<∞,𝑡>0。 令 𝑢(𝑥,𝑡)=𝑢( 𝜃(𝑥,𝑡) ),𝜃=κ𝑥−𝜔𝑡 and (𝜔2−κ2)=1。我們得到 𝑢𝜃=√2(𝐸+𝑐𝑜𝑠𝑢),𝐸 是一個積分常數。 因為 √2(𝐸+𝑐𝑜𝑠𝑢) 在複數平面ℂ是一個雙值函數，我們建立了黎曼面ℜ，使得√2(𝐸+𝑐𝑜𝑠𝑢)在黎曼面ℜ上是一個單值函數。最後我們藉由泰勒級數逼近𝑐𝑜𝑠𝑢來解決這個問題。此外，我們也運用MATHEMATICA來幫助我們解出Sine-Gordon方程的數值解。

In this paper, our goal is to find the travelling wave solution of the Sine-Gordon equation. We want to apply the theory of Riemann Surface and the theory of Elliptic Functions to solve the Sine-Gordon equation. The Sine-Gordon equation is a second-order partial differential equation which is defined by 𝑢𝑡𝑡−𝑢𝑥𝑥+𝑠𝑖𝑛[𝑢(𝑥,𝑡)]=0,𝑤ℎ𝑒𝑟𝑒−∞<𝑥<∞,𝑡>0. Let 𝑢(𝑥,𝑡)=𝑢( 𝜃(𝑥,𝑡) ),𝜃=κ𝑥−𝜔𝑡 and (𝜔2−κ2)=1. Then we obtain 𝑢𝜃=√2(𝐸+𝑐𝑜𝑠𝑢),where 𝐸 is an integration constant. SinceSince √2(𝐸+𝑐𝑜𝑠𝑢) is a two-valued function in complex plane ℂ, we construct the Riemann surface ℜ such that √2(𝐸+𝑐𝑜𝑠𝑢) becomes a single-valued function in ℜ. Finally, we solve the problem in ℜ by approximation of 𝑐𝑜𝑠𝑢 by the Taylor series at 0. Moreover, we also use MATHEMATICA to help us solve the Sine-Gordon equation numerically.