Global Dynamics of Nonlinear Dispersive Evolution Equations and Spectral Theory

非线性色散演化方程的全局动力学和谱理论

基本信息

批准号：
1902691
负责人：
Wilhelm Schlag
金额：
$ 27万
依托单位：
Yale University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2021-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902691&HistoricalAwards=false
关键词：
Global Dynamics Nonlinear Dispersive Evolution

项目摘要

This project aims at understanding the propagation of waves in a wide sense. On the one hand, the PI will investigate the behavior of waves in space, as they interact nonlinearly with themselves and matter over large space-time scales. The ultimate goal is to explain how the energy, which is stored in a wave undergoing a nonlinear dynamical evolution, ultimately splits into quantized pieces and a wave "at the horizon". The latter refers to energy, possibly of large size, which is infinitely spread out and does not interact with anything in a noticeable fashion. In contrast with this macroscopic behavior, the project also aims at understanding the behavior of waves on the microscopic scale, such as in crystals or quasi-crystals. The goal is to explain transitions from an insulating state to that of a conductor, which these materials may exhibit as they undergo changes on the molecular level. Such changes may occur through the insertion of impurities, or changes in the environment. Both the macroscopic as well as the microscopic behavior of waves is of crucial importance to science and engineering, and profoundly affects our daily modern lives. Modern communication relies on waves transmitted over large distances both in space but also along glass fiber cables. For the latter the properties of the material are crucial and both nonlinear effects as well as aforementioned microscopic phenomena decide the suitability of the underlying medium. More technically speaking, the PI intends to further investigate the rigorous mathematical theory of focusing dispersive semilinear evolution equations. A major open problem is to analyze the resolution of any solution into moving solitons and radiation. Some success has been achieved in recent years on this important problem, but for nonintegrable equations we are far from a satisfactory understanding. The PI is currently involved in the study of this problem in the dissipative setting in which some damping is added to the equation. The Hamiltonian setting appears to be very difficult at the moment, especially in the subcritical regime. The methods involved derive from dynamical systems, invariant manifold theory, and dispersive PDEs. The quantum mechanical problems alluded in the previous paragraph belong to the area of Anderson localization. Together with his long-standing collaborator Michael Goldstein at Toronto, but also with young collaborators which are joining the field, the PI intends to bring the body of techniques which were developed over the past 20 years based on large deviation estimates, the avalanche principle, semi-algebraic sets, and harmonic analysis such as (pluri)subharmonic functions and the Cartan estimate, to bear on both linear and nonlinear problems in dynamical systems and spectral theory. Ultimately, the goal here is also to better describe the behavior of wave propagation in disordered media.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目旨在从广义上理解波的传播。一方面，PI 将研究空间波的行为，因为它们在大时空尺度上与自身和物质进行非线性相互作用。最终目标是解释存储在经历非线性动力学演化的波中的能量如何最终分裂成量子化的部分和“地平线”的波。后者指的是能量，可能是大尺寸的，它无限分散并且不以明显的方式与任何事物相互作用。与这种宏观行为相反，该项目还旨在了解微观尺度上的波行为，例如晶体或准晶体中的波行为。目标是解释从绝缘状态到导体状态的转变，这些材料在分子水平上发生变化时可能会表现出这种转变。这种变化可能是通过杂质的插入或环境的变化而发生的。波的宏观和微观行为对于科学和工程都至关重要，并深刻影响着我们的日常生活。现代通信依赖于在太空和玻璃纤维电缆上长距离传输的波。对于后者，材料的特性至关重要，非线性效应以及上述微观现象决定了底层介质的适用性。从技术上讲，PI打算进一步研究聚焦色散半线性演化方程的严格数学理论。一个主要的开放问题是分析任何解决方案对移动孤子和辐射的分辨率。近年来在这个重要问题上已经取得了一些成功，但对于不可积方程我们还远远没有令人满意的理解。 PI 目前正在耗散环境中研究这个问题，其中在方程中添加了一些阻尼。目前，汉密尔顿设置似乎非常困难，尤其是在亚临界状态下。所涉及的方法源自动力系统、不变流形理论和色散偏微分方程。上一段提到的量子力学问题属于安德森局域化领域。 PI 打算与他在多伦多的长期合作者 Michael Goldstein 以及正在加入该领域的年轻合作者一起，将过去 20 年来基于大偏差估计、雪崩原理、半代数集，以及调和分析（例如（复）次调和函数和嘉当估计），以解决动力系统和谱理论中的线性和非线性问题。最终，这里的目标也是更好地描述无序介质中波传播的行为。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。