Collaborative Research: New Directions in Atomic Bose-Einstein Condensates

这一提议的目的是在原子玻色-爱因斯坦凝聚体(BEC)的背景下探索一些新的、新兴的方向。该项目将扩大私人投资机构最近就被困凝析油中的涡旋动力学主题开展的合作工作。在那里，我们将探索与阿默斯特学院一个合作实验小组获得的实验结果以及与数学和物理的其他领域的实验结果之间的联系。其中最值得注意的包括这种非线性相干态的谱理论及其结构相变，具有有趣的非线性分叉现象的基于粒子的动力学模型的约化，以及监测涡团的存在、稳定性和动力学的计算工具。此外，将在最近建立的自旋-轨道耦合的BEC方向上考虑这些主题的多组分概括。这些项在非线性问题中包含更复杂的算符结构，线性部分同时具有色散(拉普拉斯)和类狄拉克部分，这些项的相互作用及其对非线性状态(如暗或亮孤子和涡旋)的影响将被研究。这方面的结果将与华盛顿州立大学一个合作实验小组正在进行的实验进行比较。这项研究有望为研究非线性相干结构提供新一代的理论和计算工具，特别是为了研究玻色-爱因斯坦凝聚体的原始原子物理结构。此外，这里开发的理论/数学工具将对谱理论、非线性常微分方程和偏微分方程组等领域产生更广泛的影响。还可以预见，我们的发现将与许多物理学领域建立联系，例如流体动力学、非线性光学和统计力学(相变)。另一方面，开发的计算工具将探索数值分叉理论、动力系统甚至蒙特卡罗/分子动力学技术之间的接口，以及它们在现有物理系统中的潜在用途。重要的是，这项研究将是真正应用数学的真正协同，不仅涉及理论方法和计算技术的发展，而且还涉及它们与物理实验的直接联系。最后，该项目将着重于维持一支充满活力的多学科团队，拥有强大而多样的研究生核心，因此将始终如一地致力于产生一系列重大的更广泛的影响。相关研究将通过高质量的数学和物理期刊出版物传播，并将在美国和海外的非线性科学和数学物理会议上发表。

The aim of this proposal is to explore a number of novel, emerging directions in the context of atomic Bose-Einstein condensates (BECs). The project will extend the recent collaborative work of the PIs on the theme of vortex dynamics in trapped condensates. There, we will explore both connections with experimental results obtained by a collaborating experimental group at Amherst College and ones with other areas of mathematics and physics. These include most notably the spectral theory of such nonlinear coherent states, structural phase transitions thereof, reductions to particle-based dynamical models with interesting nonlinear bifurcation phenomena, and computational tools to monitor the existence, stability and dynamics of vortex clusters. Also, a multi-component generalization of these themes will be considered in the recently established direction of spin-orbit coupled BECs. These contain a more complex operator structure within the nonlinear problem, with a linear part featuring both a dispersive (Laplacian) and a Dirac-like part and the interplay of these terms and their impact on nonlinear states such as dark or bright solitons and vortices will be studied. The results in this direction will be compared to ongoing experiments by a collaborating experimental group at Washington State University. This research is expected to provide a new generation of both theoretical and computational tools for studying nonlinear coherent structures with a particular view towards the pristine atomic physics setup of Bose-Einstein condensates. Furthermore, the theoretical/mathematical tools developed here will bear broader impacts towards areas such as spectral theory and nonlinear ordinary and partial differential equations, among others. It is also envisioned that our findings will create connections with a number of areas of Physics such as Fluid Dynamics, Nonlinear Optics and Statistical Mechanics (of Phase Transitions). On the other hand, the developed computational tools will explore the interface between numerical bifurcation theory, dynamical system and even Monte-Carlo/Molecular Dynamics techniques and their potential use for the physical system at hand. Importantly, the research will be a genuine synergy in truly Applied Mathematics, involving not only the development of theoretical methods and computational techniques, but also their direct connection with physical experiments. Finally, the project will be critically focused on sustaining a dynamic and multi-disciplinary team with a strong and diverse core of graduate students and hence will be consistently geared towards having a significant set of broader impacts. The relevant research will be disseminated via high quality journal publications both in Mathematics and in Physics and will be presented at Nonlinear Science and Mathematical Physics conferences both in the US and abroad.