RUI: Nonlinear spectral problems in Hamiltonian systems

自20世纪80年代后期以来，人们在哈密顿系统的轨道稳定性，特别是能谱与线性化能谱的关系方面做了大量的理论和应用研究。一些结果导致不稳定的标准，而其他结果导致指数定理有关的能量谱的（潜在的）不稳定的线性化谱。所有这些工作的一个共同特点是，本征值问题的谱参数是线性的。该项目致力于研究一类新颖的问题，即自伴多项式束，它们是（a）之前研究的自伴线性束的自然推广，以及（B）在研究中非常自然地出现的偏微分方程，其中存在二阶或更高的时间导数。 在这项研究中，这些非线性特征值问题的指标型定理，以及不稳定性准则将开发。 主要研究者将使用和完善的数学工具是分析埃文斯函数（例如，数学物理中的透射系数）和最近发展的亚纯Krein矩阵。这两种工具都具有线性化问题的特征值被实现为零的特性（对于Krein矩阵，它是行列式的零）。该项目的成功完成将揭示这些工具如何相互关联，并展示它们可以联合用于解决数学家，物理学家和工程师感兴趣的问题的方法。研究结果将通过期刊出版物、会议和研讨会报告广泛传播。这项资助研究的结果将有助于理论家和实验学家更好地理解哈密顿系统中非线性波的动力学，即，节约能源的系统。由哈密顿系统模拟的特殊物理问题包括：（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。这项赠款资助的大部分工作将是协作性的，同事们，例如，密歇根州立大学、伊利诺斯大学和堪萨斯大学的研究人员将在这项研究中发挥积极作用。对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，更多的学生将被介绍给合作研究的好处和兴奋。应用程序，数值，以及正式和严格的分析在他们的课堂作业中没有看到的水平的相互作用导致参与的学生变得好奇和兴奋的物理世界和数学世界之间的联系。这些学生将更好地为数学科学的研究生工作做好准备，也将更好地欣赏和理解数学在物理科学中的用途。这些项目的性质是，参与的学生可以预期产生的结果是在适当的期刊上发表。卡尔文学院是一所本科院校，有着培养成功博士的历史。学生在数学和统计。这些学生中约有三分之一是妇女，她们在数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。

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.