非線性薛丁格方程基本理論的探討

Abstract

在此篇論文中, 我們研習用橢圓函數 dn(u; k)去表述Nonlinear Schrὂdinger equation (NLS)的某些特殊週期解 q

iq_t+q_xx+2〖|q|〗^2 q = 0,

此 dn 函數是定義在某個黎曼空間上的, 所以首先我們介紹黎曼空間的理論, 其次再介紹橢圓函數, 最後再利用黎曼空間和橢圓函數的理論去分析及解NLS的特殊解及其退化解。

In this paper, we want to use the elliptic function dn(u; k) to express analytically some special solutions of the Nonlinear Schrὂdinger equation (NLS),

iq_t+q_xx+2〖|q|〗^2 q=0

The function dn(u; k) is defined on some Riemann surface, so we first introduce the theory of Riemann surfaces, and then we introduce elliptic functions. Finally, we use theory of Riemann surfaces and elliptic functions to analyze and solve some special solutions of the NLS and discuss the degenerates of those solutions.