標題: 黎曼曲面與Weierstrassian橢圓函數的理論及其對Korteweg-deVries方程的應用
The Theory of Riemann Surfaces and the Weierstrass Elliptic Functions with Application to the Korteweg-deVries Equation
作者: 黃建順
李榮耀
Huang, Jian-Shun
Lee, Jong-Eao
應用數學系所
關鍵字: 黎曼曲面;偏微分方程;橢圓函數;Weierstrassian Elliptic Functions;Riemann Surfaces;Partial Differential Equations;Korteweg-deVries Equation
公開日期: 2016
摘要: Korteweg-deVries方程式是個非線性偏微分方程,而KdV方程式如下: u_t(x,t)-6*u(x,t)u_x(x,t)+u_xxx(x,t)=0,t>0,-∞≤x≤∞ 對於特殊解,我們可以把偏微分方程式轉變成微分方程式,再利用變數變換的方法,我們可以將原本的方程式變成以下的形式: u_θ^2(θ)=2*u^3(θ)+cu^2(θ)+2*A*u(θ)+B 如果我們要求解出函數u(θ),基本上這是一個要解一個積分函數,而此積分函數是具有平方根的形式,以及根號內是一個三次多項式。 對於平方根在複數平面上它是一個多值函數,我們在複數平面上建立黎曼曲面,並且藉由適當的代數建構,使得平方根在黎曼曲面上是一個單值函數。 而對於根號內的三次多項式,我們介紹Weierstrassian橢圓函數的古典理論,並且利用它去求u_θ(θ)=√(2*u^3(θ)+cu^2(θ)+2*A*u(θ)+B) 的解,並分析相關的性質。
The Korteweg-deVries equation is nonlinear partial differential equations, and the KdV equation is as follows: u_t(x,t)-6*u(x,t)u_x(x,t)+u_xxx(x,t)=0,t>0,-∞≤x≤∞ For traveling solutions, we can transform partial differential equations into differential equation, and the KdV equation becomes the following form: u_θ^2(θ)=2*u^3(θ)+cu^2(θ)+2*A*u(θ)+B To solve u(θ) we transfer this ode into integral equation namely, The inverse problem where the integral involves square root(a multi-valued function). We develop Riemann surfaces with proper algebraic structure to make the function √ to be single-valued. Then we introduce the classical theory of Weierstrassian elliptic functions, to solve the solution of u_θ(θ)=√(2*u^3(θ)+cu^2(θ)+2*A*u(θ)+B) .
URI: http://etd.lib.nctu.edu.tw/cdrfb3/record/nctu/#GT070152208
http://hdl.handle.net/11536/143407
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