CAREER: Solitons in Bose-Einstein Condensates: Generation, Manipulation and Pattern Formation

panayotis Kevrekidis，马萨诸塞大学阿默斯分校，题目：职业：玻色-爱因斯坦凝聚中的孤子：产生、操纵和模式形成本项目的目的是研究原子物理（玻色-爱因斯坦凝聚）背景下孤子波结构的行为。由于其结构的鲁棒性和准弹性相互作用，孤子是过去在光学环境中用于信息传输的天然构建块，并且可以自然地扩展为这种新的物质波设置中的信息载体。此外，这些结构可以适当地操纵，波导或用于构建微观（原子）水平的各种图案。研究将在三个不同的层面上进行：(1)利用不稳定性和/或实验上可用的机制（如费什巴赫共振）产生这些孤立波的层面；(2)操纵波的方法（用光镊拖动波或引导波通过结）；(3)最后，在组合它们的层次上创建模式，并识别它们的稳定状态和结构转变。这些步骤将用于两种主要类型的相互作用：a)原子之间的吸引相互作用（例如，聚焦非线性和负散射长度，如锂的情况）；b)对于排斥相互作用（例如，对于离焦非线性和正散射长度，如铷和钠的情况）。将要研究的模型将是具有外部势的偏微分方程的连续体模型（线性或二次，或其组合），以及准离散模型（与所谓的光学晶格中的周期性外部势相关）。将使用的技术将涉及规则和奇异摄动方法，线性和调制稳定性分析，规则和可能的指数渐近，数值分岔理论以及直接数值模拟和分子动力学技术（研究模式及其结构转变）。本研究项目的主要重点是对玻色-爱因斯坦凝聚体（BECs）在非常可控、超低温的原子物理环境下产生的孤立波进行详细研究。自从它们最近的实验实现（为此获得了2001年诺贝尔物理学奖）以来，bec一直是数学界和物理界不断增长的密集实验和理论努力的中心。对bec的研究也与对超导和超流体这两个激动人心的重要领域（2003年诺贝尔物理学奖的主题）的深入理解有着密切的联系。从数学物理的角度来看，研究bec最有趣和最吸引人的方面之一是它们丰富的非线性波现象学，各种可能的设置（一维到三维）和详细的实验控制，这些实验控制允许对这些波动态演变的外部条件进行精确的工程/操纵。这项研究工作的主要目的是扩展和加深我们对基本结构和波及其在bec中的作用和重要性的理解，而且更广泛地（由于类似的数学描述）在非线性光学（光纤和波导）以及波物理中。顺便说一句，应该指出的是，这项工作将严重依赖于计算资源和伴随使用的数值代码来模拟这些现象；还应该指出的是，这种物质波活动的长期前景之一是构思和构建新的设备，这些设备将引导和更广泛地控制物质波的运动，并有可能用于纳米尺度的量子信息处理。这些方面使我们期望在联邦战略利益领域（如高性能计算、材料和制造）实施该项目可能会产生显著的好处。

Abstract: CAREER Award DMS-0349023Panayotis Kevrekidis, University of Massachusetts at AmherstTitle: CAREER: Solitons in Bose-Einstein Condensates: Generation, Manipulation and Pattern FormationThe aim of this project is to examine the behavior of solitary wavestructures in the setting of atomic physics (Bose-Einstein Condensates). Solitons as per their structural robustness and quasi-elastic interactions are natural building blocks that have been used for information transmission in optical settings in the past and could naturally be extended as information carriers in this new matter wave setup. Furthermore, these structures can be appropriately manipulated, waveguided or used to construct various patterns at this microscopic (atomic) level. The study will be undertaken at three different levels: (1) The level of creating these solitary waves by taking advantage of instabilities and/or experimentally available mechanisms (such as the Feshbach resonance); (2) The one of manipulating the waves (either dragging them by means of an optical tweezers or waveguiding them through junctions); (3) And, finally, at the level of combining them to create patterns and to identify their steady states and structural transitions. These steps will be carried through for the two principal types of interactions: a) For attractive interactions between the atoms (e.g., for focusing nonlinearities and negative scattering lengths as in the case of lithium); b) For repulsive interactions (e.g., for defocusing nonlinearities and positive scattering lengths as in the case of rubidium and sodium). The models that will be examined will be both continuum models of partial differential equations with external potentials (linear or quadratic, or combination thereof), as well as quasi-discrete ones (relevant for periodic external potentialsas in the case of the so-called optical lattice). The techniques that will be used will involve regular and singular perturbationmethods, linear and modulational stability analysis, regular and possibly exponential asymptotics, numerical bifurcation theory as well as direct numerical simulations and also molecular dynamics techniques (to study patterns and their structural transitions). The main focus of this research project is a detailed study of solitary waves generated in the very controllable, ultra-low temperature, atomic physics context of Bose-Einstein condensates (BECs). Since their recent experimental realization (for which the 2001 Physics Nobel prize was awarded), BECs have been the center of an intensive and ever growing experimental and theoretical effort in the Mathematics and Physics communities. The examination of BECs has also strong ties with a deeper understanding of the exciting and important fields of superconductivity and superfluidity (which were the theme of the Physics Nobel prize in 2003). From a Mathematical Physics perspective, one of the most interesting and appealing aspects of studying BECs is their rich nonlinear wave phenomenology, the wide variety of possible settings (one to three dimensions) and the detailed experimental control that permits a precise engineering/manipulation of the external conditions under which these waves dynamically evolve.The main purpose of this research effort is to extend and deepen our understanding of the fundamental structures and waves and their roleand importance in BECs, but also more generally (due to the similarmathematical description) in nonlinear optics (optical fibers andwaveguides) as well as wave physics. As an aside, it should be noted that this effort will heavily rely on computational resources and the concomitant use of numerical codes that model these phenomena; it should also be remarked that one of the longer term perspectives of this activity on matter waves is to conceive and construct novel devices that would guide and more generally control the motion of the matter waves and could potentially be used for quantum information processing at the nanoscale. These aspects lead us to expect that significant benefits may result from the implementation of this project in areas of strategic federal interest such as high performance computing and materials and manufacturing.