RUI: Waves in Hamiltonian Systems with Applications to Bose-Einstein Condensates

RUI：哈密顿系统中的波及其在玻色-爱因斯坦凝聚中的应用

• 批准号: 0806636
• 负责人: Todd Kapitula
• 金额: $ 14.3万
• 依托单位: Calvin University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2011-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806636&HistoricalAwards=false
• 关键词: RUI; Waves; Hamiltonian; Systems; Applications

KapitulaDMS-0806636 The governing equations for the wave function of aBose-Einstein condensate (BEC) are Hamiltonian; in particular,they are a variant of the nonlinear Schrodinger equation known asthe Gross-Pitaevskii equation. The analysis of this physicalsystem leads to several intriguing fundamental mathematicalproblems for Hamiltonian systems which deal with the existence,spectral stability, and nonlinear stability of waves. The majormathematical problems to be addressed in this proposal include(a) the construction and spectral stability of fullythree-dimensional waves in the presence of a cigar magnetic trapvia the technique of spatial dynamics, (b) the construction andspectral stability of two-dimensional periodic waves in thepresence of an optical lattice, and (c) a careful study ofeigenvalues with negative Krein signature. Regarding (c), thereare two avenues that are to be explored: (1) the determination ofembedded eigenvalues of negative sign in the case that thesolitary wave is realized as a limit of a family of periodicwaves, and (2) the determination of the exact number ofeigenvalues with negative sign. Especially since the 1997 Nobel Prize winning work of S.Chu, C. Cohen-Tannoudji, and W. Phillips on the exotic quantumphenomenon known as BECs, there has been a great deal of excitingexperimental and theoretical work in the study of BECs. In orderto form a BEC, it is necessary that the matter is cooled tobillionths of a degree above absolute zero; hence, BECs areextremely fragile, and it is likely to be some time before anypractical applications are developed. Nevertheless, they haveproved to be useful in exploring a wide range of questions infundamental physics. Examples include experiments that havedemonstrated interference between condensates due towave-particle duality, the study of superfluidity and quantizedvortices, and the slowing of light pulses to very low speeds. Vortices in BECs are also currently the subject ofanalogue-gravity research, studying the possibility of modelingblack holes and their related phenomena in the lab. Themathematical results of this proposal will help boththeoreticians and experimentalists better understand the dynamicsof not only patterns such as vortices, necklaces, and solitons inBECs, but also the dynamics of waves and patterns in Hamiltoniansystems which are used to model interesting phenomena such aswaves in fluids and light propagation in optical fibers. Theinclusion and training of undergraduate students is an integralpart of this proposal. Introducing students to the interplaybetween applications and experiments, numerics, and formal andrigorous analysis, will lead them to being excited about seeingthe connections between the physical world and the mathematicalworld. This will further lead the participating students towardsa deeper appreciation for the usefulness of mathematics in otherdisciplines, and perhaps then allow them to be open to the ideaof doing more serious applied mathematics at the graduate level. Calvin College has a history of producing successful Ph.D.students in mathematics and statistics. Approximately one-thirdof these students have been women, who are significantlyunderrepresented in the mathematical sciences. Furthermore,Calvin has been very successful in the education and training ofsecondary education teachers. As a consequence of this award,the investigator will be able to fruitfully support thecollege-wide effort in training future researchers and educatorsin the mathematical sciences.

KapitulaDMS-0806636 玻色-爱因斯坦凝聚体（BEC）的波函数的控制方程是哈密顿的;特别地，它们是被称为Gross-Pitaevskii方程的非线性薛定谔方程的变体。 对这个物理系统的分析引出了几个有趣的哈密顿系统的基本物理问题，这些问题涉及波的存在性、谱稳定性和非线性稳定性。 本方案主要解决的数学问题包括：（a）通过空间动力学技术在雪茄磁阱中构造全三维波及其谱稳定性;（B）在光晶格中构造二维周期波及其谱稳定性;（c）对具有负Krein特征的特征值的细致研究。 关于（c），有两条途径有待探索：（1）在孤立波被实现为一个行波族的极限的情况下，确定负符号的嵌入特征值;（2）确定负符号特征值的确切数目。 特别是1997年诺贝尔奖得主朱棣文、C. Cohen-Tannoudji和W.自从菲利普斯提出奇异量子现象BEC以来，人们在BEC的研究方面做了大量令人兴奋的实验和理论工作。 为了形成BEC，物质必须冷却到绝对零度以上十亿分之一度;因此，BEC非常脆弱，在任何实际应用开发之前可能还需要一段时间。尽管如此，它们在探索基础物理学中的广泛问题时被证明是有用的。 例子包括已经证明了由于波粒二象性而导致的凝聚体之间的干涉的实验，超流性和量子化涡旋的研究，以及光脉冲减慢到非常低的速度。BEC中的涡旋也是目前类比重力研究的主题，研究在实验室中模拟黑洞及其相关现象的可能性。 该提议的数学结果将帮助理论家和实验家更好地理解不仅是BEC中的涡旋、项链和孤立子等图案的动力学，而且是Hamilton系统中的波和图案的动力学，这些系统用于模拟有趣的现象，例如流体中的波和光纤中的光传播。 对本科生的包容和培训是这一建议的一个组成部分。向学生介绍应用和实验、数学以及形式和华丽分析之间的相互作用，将使他们对看到物理世界和数学世界之间的联系感到兴奋。 这将进一步引导参与的学生对数学在其他学科中的有用性有更深的认识，也许还能让他们接受在研究生阶段做更严肃的应用数学的想法。卡尔文学院在培养数学和统计学博士生方面有着悠久的历史。这些学生中大约有三分之一是女性，而女性在数学科学领域的代表性明显不足。此外，卡尔文在中学教师的教育和培训方面也非常成功。 作为这个奖项的结果，调查员将能够卓有成效地支持整个学院在培养未来的研究人员和教育工作者在数学科学的努力。