Proceedings of the International Congress of Mathematicians Seoul 2014
Proceedings of the International Congress of Mathematicians Seoul 2014
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发表时间：
2015
影响因子：
1.3
通讯作者：
S. Jang;Young Rock Kim;Dae-Woong Lee;Ikkwon Yie
中科院分区：
文献类型：
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作者：
S. Jang;Young Rock Kim;Dae-Woong Lee;Ikkwon Yie
碱原子的超冷蒸汽中玻色-爱因斯坦凝聚（BEC）的实现，对磁阱和光晶格内凝聚态的稀原子气体的理论和实验研究产生了巨大的推动作用。本文简要介绍了基于平均场理论的BEC数学模型、理论和数值方法。我们从三维（3D）的Gross-Pitaevskii方程（GPE）开始，用于模拟弱相互作用玻色子的单组分BEC，将其缩放为三参数模型，并展示如何在某些极限区域将其简化为二维（2D）和一维（1D） GPE。给出了BEC基态和动力学的数学理论和数值方法。讨论了旋转BEC中具有角动量旋转项的GPE扩展、偶极BEC中具有长程各向异性偶极-偶极相互作用的GPE扩展以及自旋轨道耦合BEC中耦合GPE的扩展。最后，本文得出了一些结论，并对未来的研究方向进行了展望。数学学科分类（2010）。主要35 q55;二次70 f10。
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling onecomponent BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with longrange anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed. Mathematics Subject Classification (2010). Primary 35Q55; Secondary 70F10.