Travelling Waves for the Gross-Pitaevskii Equation

本文的目標是探討Fabrice Bethuel,Philippe Gravejat,和 Jean-Claude Saut 在Gross- Pitaevskii方程式中關於二,三維度的行波解 c∂1u + ∆u + u(1 − |u|2) = 0 前四章節我們透過最小化能量在動量固定下來探討解之存在性,並提供一些構造 這些定理的動機。 最後一章節我們討論Gross-Pitaevskii equation未來的研究方向。In this thesis, Fabrice Bethuel, Philippe Gravejat, and Jean-Claude Saut discussed the existence of travelling wave solutions to the Gross-Pitaevskii equation in RN , where N = 2, 3. c∂1u + ∆u + u(1 − |u|2) = 0 In the first four sections, we survey the theorems based on minimizing energy under momentum constraint. Also, we give some motivations about how the theorems are constructed. In the final section, we discuss the future works of Gross-Pitaevskii equation