非線性薛丁格方程的基本理論及特殊解

Abstract

在此論文中，我們利用橢圓函數dn(u,k)表示 Nonlinear Schrodinger equation(NLS)的某些特殊解q iq_t+q_xx+2|q|^2q=0 橢圓函數dn(u,k)定義在黎曼空間上，因此我們先介紹黎曼空間的理論，接著再介紹橢圓函數，最後利用黎曼空間和橢圓函數的理論去解 NLS 的特殊解並分析其退化。

In this paper, we express some special solutions q of the Nonlinear Schrodinger equation(NLS) iq_t+q_xx+2|q|^2q=0 by elliptic function dn(u,k). Since the function dn(u,k) is defined on the Riemann surface, we introduce the theory of Riemann surfaces at first, and then we introduce classical elliptic functions. Finally, we use the theory of Riemann surfaces and elliptic function to solve some special solution of NLS and analyze the degeneracies of the NLS solutions.