黎曼空間的理論和其在微分方程上的應用

Abstract

此篇文章主要在探討擁有 d^2u/dt^2+PN(u)=0 形式的非線性二階微分方程，其解的函數理論，其中PN(u)是2N 或2N-1次多項式。此方程的解存在於N-1相黎曼空間上。我們要利用正確的代數結構來建構這些黎曼空間。以此為基準，無論是理論或數值上我們可以在黎曼空間執行路徑的積分，並在此原則上獲得其解。其中PN(u)的根扮演了重要的角色，而複數分析是我們主要的工具。

In this paper, we study the function theory of the solutions of the nonlinear second-order equations which have the following forms, d^2u/dt^2+PN(u)=0 where PN(u) is a polynomial of degree 2N-1 or 2N. Solutions of such equations reside on Riemann surfaces of genus N-1. We construct those Riemann surfaces with the correct algebraic structures. From which, we are able to perform path integrals on the Riemann surfaces theoretically and numerically, and, in principle, solutions can be derived. The roots of PN(u) play the essential roles in every aspects, and complex analysis is our main tool.