Global Dynamics of Nonlinear Dispersive Evolution Equations and Spectral Theory

这个项目的目的是从广义上理解波的传播。一方面，PI将研究波在空间中的行为，因为它们在大的时空尺度上与自己和物质进行非线性相互作用。最终目标是解释储存在经历非线性动力学演化的波中的能量最终是如何分裂成量子化的碎片和“在地平线上”的波的。后者指的是能量，可能是很大的能量，它无限地扩散，不会以明显的方式与任何东西相互作用。与这一宏观行为形成对比的是，该项目还旨在了解微观尺度上的波的行为，如晶体或准晶体中的波。其目的是解释从绝缘态到导体的转变，这些材料在分子水平上经历变化时可能会表现出这种转变。这种变化可以通过杂质的插入或环境的变化来发生。波浪的宏观和微观行为对科学和工程都具有至关重要的意义，并深刻地影响着我们的日常现代生活。现代通信依赖于长距离传输的波，不仅是在太空中，也是沿着玻璃纤维电缆传输。对于后者，材料的性质是至关重要的，非线性效应和上述微观现象都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发展方程的严格数学理论。一个主要的开放问题是分析任何解分解为运动的孤子和辐射的问题。近年来，在这一重要问题上已经取得了一些成果，但对于不可积方程，我们的理解还远远不够。PI目前主要是在方程中加入一定的阻尼力的耗散环境下对这一问题的研究。哈密顿的设定目前似乎非常困难，特别是在亚临界状态下。所涉及的方法来源于动力系统、不变流形理论和色散偏微分方程组。上一段提到的量子力学问题属于安德森局部化领域。与他在多伦多的长期合作者Michael Goldstein以及加入该领域的年轻合作者一起，PI打算将过去20年发展起来的基于大偏差估计、雪崩原理、半代数集和调和分析(如(多)次调和函数和Cartan估计)的技术体系应用于动力系统和频谱理论中的线性和非线性问题。最终，这里的目标也是为了更好地描述波在无序介质中的传播行为。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.