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

KapitulaDMS-0806636玻色-爱因斯坦凝聚体(BEC)波函数的控制方程是哈密顿的；具体地说，它们是被称为Gross-Pitaevskii方程的非线性薛定谔方程的变体。对这一物理系统的分析引出了哈密顿系统的几个有趣的基本数学问题，这些问题涉及波的存在性、谱稳定性和非线性稳定性。这一建议涉及的主要数学问题包括：(A)通过空间动力学技术研究雪茄磁陷阱中全三维波的构造和光谱稳定性；(B)光学晶格中二维周期波的构造和光谱稳定性；以及(C)具有负Krein特征的特征值的仔细研究。关于(C)，有两个途径需要探索：(1)在孤立波被实现为周期波族的极限的情况下确定负号嵌入的特征值；(2)确定负号特征值的确切个数。特别是自从1997年朱夏莲、C.Cohen-Tannoudji和W.Phillips获得诺贝尔奖以来，人们对BECs这一奇异的量子现象进行了大量激动人心的实验和理论研究。为了形成BEC，有必要将物质冷却到绝对零度以上十亿分之一度；因此，BEC非常脆弱，很可能需要一段时间才能开发出任何实际应用。然而，事实证明，它们在探索基础物理中的广泛问题时是有用的。例如，实验证明了凝聚体之间由于波粒二元性而产生的干涉，超流和量子化涡旋的研究，以及将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者不仅更好地理解BEC中涡旋、项链和孤子等图案的动力学，而且还可以理解哈密顿系统中的波和图案的动力学，这些图案被用来模拟有趣的现象，如流体中的波和光纤中的光传输。本科生的纳入和培训是这项提议的一个组成部分。向学生介绍应用和实验、数值、形式和严格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作用有更深的理解，并可能使他们对在研究生水平上进行更严肃的应用数学的想法持开放态度。卡尔文学院有培养成功的数学和统计学博士生的历史。这些学生中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，加尔文在中等教育教师的教育和培训方面取得了非常成功的成就。作为这一奖项的结果，研究人员将能够卓有成效地支持全校范围内培训未来数学科学研究人员和教育工作者的努力。

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.