Global Dynamics of Nonlinear Dispersive Evolution Equations and Spectral Theory

这个项目旨在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，因为它们在大时空尺度上与自身和物质非线性相互作用。最终的目标是解释能量是如何储存在经历非线性动态演化的波中，最终分裂成量子化的碎片和“在视界”的波。后者指的是能量，可能是大尺寸的，它无限地扩散，不以一种明显的方式与任何东西相互作用。与这种宏观行为相反，该项目还旨在了解波在微观尺度上的行为，例如晶体或准晶体。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经历变化时可能表现出这种转变。这种变化可能通过杂质的插入或环境的变化而发生。波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。现代通信依赖于在太空和玻璃纤维电缆上远距离传输的波。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的解析度到运动孤子和辐射。近年来在这一重要问题上取得了一些成功，但对于不可积方程，我们还远远没有令人满意的认识。PI目前正在研究在方程中加入一些阻尼的耗散设置下的这个问题。目前看来，哈密顿的设定是非常困难的，尤其是在亚临界状态下。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。前文提到的量子力学问题属于安德森局域化的范畴。与他在多伦多的长期合作伙伴Michael Goldstein一起，以及正在加入该领域的年轻合作者，PI打算将过去20年来基于大偏差估计，雪崩原理，半代数集和谐波分析（多倍）次谐波函数和Cartan估计等发展起来的技术体，用于动力系统和谱理论中的线性和非线性问题。最终，这里的目标也是为了更好地描述波在无序介质中的传播行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.