冪次非線性薛丁格方程式之孤波解的研究

Abstract

非線性薛丁格方程式是與相當多的物理模型有關，包括非線性光學、水波與量子世界的描述等。本計畫將針對具有冪次非線性項的薛丁格方程式進行探討，欲瞭解不同型態的孤波解在此冪次非線性薛丁格方程式中的穩定狀況。在藉由孤波解附近的線性化展開後，得到一線性算子，此一線性算子將扮演重要角色。對此線性算子求其譜分佈，即是本計畫所要關心的課題，並且要對其所對應的不同型態的孤波解做一一的分類與研究。

Nonlinear Schrodinger equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of nonlinear Schrodinger equations. In this project we hope to study these spectra in detail, both analytically and numerically. Such nonlinear Schrodinger equations arise in many physical settings, including nonlinear optics, water waves, and quantum physics. Mathematically, nonlinear Schrodinger equations with various nonlinearities are studied as basic models of nonlinear dispersive phenomena. In this project, we will stick to the case of a pure power nonlinearity for the sake of simplicity.