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.

该项目旨在在广泛的意义上理解波浪的传播。一方面，PI将研究空间中的波的行为，因为它们与自身的非线性相互作用并在大时空尺度上进行物质。最终目标是解释如何存储在非线性动力学进化的波浪中的能量如何将最终分为量化的碎片和“地平线”的波浪。后者是指能量，可能是大尺寸的能量，它无限分散，并且不会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的行为，例如在晶体或准晶体中。目的是解释从绝缘状态到导体的过渡，这些材料在分子水平发生变化时可能会显示出来。这种变化可能通过插入杂质或环境变化而发生。波浪的宏观和微观行为对科学和工程至关重要，并且对我们的日常现代生活产生了深远的影响。现代通信依赖于在太空中且沿玻璃纤维电缆的大距离传输的波浪。对于后者，材料的性质至关重要，非线性效应以及上述显微镜现象决定了基础培养基的适用性。从技术上讲，PI旨在进一步研究集中分散半线性进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中的任何解决方案中的孤子和辐射。近年来，在这个重要问题上已经取得了一些成功，但是对于不可整合的方程式，我们远非令人满意的理解。 PI目前参与了该问题的研究，在耗散性环境中，将一些阻尼添加到方程式中。目前，哈密顿的环境似乎非常困难，尤其是在亚临界体制中。涉及的方法源自动态系统，不变歧管理论和分散PDE。上一段中提到的量子机械问题属于安德森本地化区域。 PI与他在多伦多的长期合作者迈克尔·戈德斯坦（Michael Goldstein）一起，以及与年轻的合作者一起，PI旨在根据大型偏差估计值，雪崩原理，半代数集合，半分析和诸如（pluri）的（plururi）子括号估算，在过去20年中开发的技术和诸如cartan toctimation and cartan toctimation and Cartan eversive and Cartan toctimations cartan eversime和光谱理论。最终，这里的目标也是更好地描述无序媒体中波浪传播的行为。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。