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

玻色-爱因斯坦凝聚（BEC）的波函数控制方程为哈密顿方程；特别是，它们是非线性薛定谔方程的一种变体，即格罗斯-皮塔耶夫斯基方程。这个物理系统的分析导致了几个有趣的哈密顿系统的基本数学问题，这些问题涉及波的存在性、谱稳定性和非线性稳定性。在这个提议中要解决的主要数学问题包括：(a)通过空间动力学技术在雪茄磁阱存在下的全三维波的构造和光谱稳定性，(b)在光学晶格存在下的二维周期波的构造和光谱稳定性，以及(c)具有负Krein签名的特征值的仔细研究。关于(c)，有两种途径可以探索：(1)在孤立波作为一组周期波的极限实现的情况下确定负号的嵌入特征值，以及(2)确定带有负号的特征值的确切数量。特别是自1997年朱棣文、科恩-坦努吉和菲利普斯获得诺贝尔奖以来，人们对被称为bec的奇异量子现象进行了大量令人兴奋的实验和理论研究。为了形成BEC，物质必须冷却到绝对零度以上的十亿分之一度；因此，BECs非常脆弱，可能还需要一段时间才能开发出任何实际应用。尽管如此，它们已被证明在探索广泛的基础物理问题方面是有用的。例如，由于拖粒二象性，冷凝物之间的干涉实验，超流体和量子化漩涡的研究，以及光脉冲减慢到非常低的速度。bec中的涡旋目前也是模拟重力研究的主题，研究在实验室中模拟黑洞及其相关现象的可能性。这一建议的数学结果将帮助理论家和实验家更好地理解动力学，不仅是模式，如漩涡，项链，和孤子在bec中，而且波的动力学和模式的哈密顿系统，用于模拟有趣的现象，如流体中的波和光在光纤中的传播。对本科生的包容和培训是这一提议的一个组成部分。向学生介绍应用和实验、数值、正式和严格的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，也许这样他们就能在研究生阶段接受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方面的成功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，她们在数学科学领域的代表性明显不足。此外，卡尔文在中学教育教师的教育和培训方面非常成功。作为这个奖项的结果，研究者将能够富有成效地支持学院范围内的努力，培养未来的研究人员和教育工作者在数学科学。

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.