RUI: Nonlinear spectral problems in Hamiltonian systems

RUI：哈密顿系统中的非线性谱问题

• 批准号: 1108783
• 负责人: Todd Kapitula
• 金额: $ 13.83万
• 依托单位: Calvin University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-05-15 至 2015-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108783&HistoricalAwards=false
• 关键词: RUI; Nonlinear; spectral; problems; Hamiltonian

Since the late 1980s, there has been a great deal of both theoretical and applied work in the study of orbital stability of waves for Hamiltonian systems, and in particular the relation of the energy spectrum to that of the linearized spectrum. Some of the results have led to instability criteria, whereas other results have led to index theorems relating the energy spectra to the (potentially) unstable linearized spectra. A common feature in all of this work is that the eigenvalue problems have been linear in the spectral parameter. This project is devoted to a study of a novel class of problems, the self-adjoint polynomial pencils, which are (a) a natural generalization of the self-adjoint linear pencils previously studied, and (b) arise quite naturally in the study of partial differential equations for which there are second-order or higher temporal derivatives. In this research, index type theorems for these nonlinear eigenvalue problems, as well as instability criteria will be developed. The mathematical tools that the Principal Investigator will use and refine are the analytic Evans function (e.g., the transmission coefficient in mathematical physics) and the recently developed meromorphic Krein matrix. Both of these tools have the property that eigenvalues for the linearized problem are realized as zeros (for the Krein matrix it is the zeros of the determinant). The successful conclusion of this project will shed light on how these tools relate to each other, and show ways that they can be jointly used to solve problems of interest to mathematicians, physicists, and engineers. Results of the research will be disseminated broadly through journal publications, and conference and seminar presentations. The results of this funded research will help both theoreticians and experimentalists better understand the dynamics of nonlinear waves in Hamiltonian systems, i.e., systems which conserve energy. Particular physical problems which are modeled by Hamiltonian systems include (a) the dynamics of matter waves in Bose-Einstein condensates, (b) wave propagation in fluids, and (c) light propagation in optical fibers. Much of the work funded by this grant will be collaborative, and colleagues, e.g., at Michigan State University, the University of Illinois, and the University of Kansas, will play an active role in the research. The inclusion and training of undergraduate students is an integral part of this project. With the financial support provided via this grant more students will be introduced to the benefits and excitement of collaborative research. The interplay of applications, numerics, and formal and rigorous analysis at a level not seen in their class work leads the participating students to becoming intrigued and excited about the connections between the physical world and the mathematical world. These students will be better prepared for graduate work in the mathematical sciences, and will also have a better appreciation and understanding of the usefulness of mathematics in the physical sciences. The projects are of such a nature that the participating students can be expected to produce results which are publishable in an appropriate journal. Calvin College, which is an Undergraduate Institution, has a history of producing successful Ph.D. students in mathematics and statistics. Approximately one-third of these students have been women, who are significantly underrepresented in the field of mathematics. Furthermore, Calvin has been very successful in the education and training of secondary education teachers.

自 20 世纪 80 年代末以来，在哈密顿系统波的轨道稳定性研究，特别是能谱与线性化谱的关系方面，已经进行了大量的理论和应用工作。一些结果导致了不稳定标准，而其他结果则导致了将能谱与（潜在）不稳定线性化谱相关联的指数定理。所有这些工作的一个共同特征是特征值问题在谱参数中是线性的。该项目致力于研究一类新问题，即自伴多项式铅笔，它是（a）先前研究的自伴线性铅笔的自然推广，以及（b）在存在二阶或更高阶时间导数的偏微分方程的研究中很自然地出现。 在本研究中，将开发这些非线性特征值问题的指数型定理以及不稳定性准则。 首席研究员将使用和完善的数学工具是解析埃文斯函数（例如数学物理中的传输系数）和最近开发的亚纯 Kerin 矩阵。这两种工具都具有以下特性：线性化问题的特征值被实现为零（对于 Kerin 矩阵，它是行列式的零）。该项目的成功结束将揭示这些工具如何相互关联，并展示如何联合使用它们来解决数学家、物理学家和工程师感兴趣的问题。研究结果将通过期刊出版物、会议和研讨会演讲广泛传播。这项资助的研究结果将帮助理论家和实验家更好地理解哈密顿系统（即能量守恒系统）中非线性波的动力学。由哈密顿系统建模的特定物理问题包括（a）玻色-爱因斯坦凝聚体中物质波的动力学，（b）流体中的波传播，以及（c）光纤中的光传播。这笔赠款资助的大部分工作将是协作性的，密歇根州立大学、伊利诺伊大学和堪萨斯大学等机构的同事将在研究中发挥积极作用。本科生的包容和培训是该项目的一个组成部分。通过这笔赠款提供的财政支持，更多的学生将了解到合作研究的好处和兴奋。应用、数字以及正式和严格的分析之间的相互作用达到了课堂作业中未曾见过的水平，这使得参与的学生对物理世界和数学世界之间的联系变得好奇和兴奋。这些学生将为数学科学的研究生工作做好更好的准备，并且也会更好地欣赏和理解数学在物理科学中的用处。这些项目的性质使得参与的学生有望产生可在适当的期刊上发表的成果。加尔文学院是一所本科院校，拥有培养成功博士学位的历史。数学和统计学的学生。这些学生中大约有三分之一是女性，她们在数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也非常成功。