Spectral Theory and Nonlinear Waves

本项目致力于研究波在结构化和无序介质中的传播。现代社会广泛依赖涉及波传输的工程应用。手机通信、卫星数据传输和互联网上的信息——所有这些都是通过数千英里的玻璃光纤电缆发送的——都是由描述波运动的数学控制的。波传播的一个显著特征，也是本项目的主题，是它的普遍性。事实上，在天文尺度上传播的波，如LIGO探测到的引力波，以及在微观尺度上传播的波，如激光中以电磁辐射形式由原子发射的波，都是由完全相同的数学理论控制的。因此，培养PI研究领域的年轻专家对于整个社会的福祉和安全具有最重要的战略意义；这也是这个项目的目标之一。经验表明，这种更广泛的影响只能由积极在知识前沿工作的科学家来实现。PI最近完成了对2球的等变临界波映射的微扰分析，没有对微扰进行任何对称假设。这种非线性分析是基于贝塞尔型势的波函数的半经典表示，这是PI在十多年前在广义相对论中的Price定律的背景下发展起来的。这一新技术计划应用于其他非线性发展方程，这些方程允许分离变量并简化为耦合半经典波动方程的无限系统。半经典参数h的作用通常由角动量的倒数来发挥。谱理论正在崭露头角的另一个领域是拓扑孤子的渐近稳定性，特别是一维标量场的扭结解。PI最近与他人一起演示了如何处理具有非一般势和长期非线性的Klein-Gordon方程。每个基本标量场方程都表现出具有阈值共振的非一般势，因此属于本项目要考虑的类型。PI关于准周期局部化及其分支的工作将继续进行。非均匀双曲动力系统将提供一个特别具有挑战性的新视角，其中有强烈的迹象表明，可能会带来安德森定位技术。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 is devoted to the study of wave propagation in both structured and disordered media. Modern society relies extensively on engineering applications involving transmission of waves. Cell phone communications, satellite data transmissions and information on the internet - all sent along thousands of miles of glass fiber cables - are controlled by mathematics describing the motion of waves. A striking feature of the wave propagation, and the topic of this project, is its universality. In fact, waves propagating on astronomical scales such as the gravitational waves detected by LIGO, as well as waves on a microscopic scale such as those emitted by atoms in the form of electromagnetic radiation in a laser, are governed by exactly the same mathematical theories. It is therefore of the utmost strategic importance for the well-being and safety of the society at large to train young specialists in the PI’s area of research; this is also one of the goals of this project. Experience shows that this broader impact can only be achieved by scientists who are actively working at the frontier of knowledge. The PI recently completed a perturbative analysis of equivariant critical wave maps into the 2-sphere, without any symmetry assumptions on the perturbations. This nonlinear analysis is based on the semi-classical representations of wave functions for Bessel-type potentials which the PI developed more than a decade ago in the context of the Price law in general relativity. Application of this new technique is planned to other nonlinear evolution equations, which admit separation of variables and a reduction to an infinite system of coupled semi-classical wave equations. The role of the semi-classical parameter h is typically played by the reciprocal of the angular momentum. Another area in which spectral theory is now coming to the fore is the rapidly developing field of asymptotic stability of topological solitons, specifically of kink solutions for one-dimensional scalar fields. Jointly with others, the PI recently demonstrated how to treat Klein-Gordon equations with a nongeneric potential and long-range nonlinearities. The basic scalar field equations each exhibit a nongeneric potential with a threshold resonance and are therefore of the type to be considered in this project. The PI's work on quasi-periodic localization and its ramifications will be continued. A particularly challenging novel perspective will be offered by nonuniformly hyperbolic dynamical systems, where there are strong indications that it is possible to bring Anderson localization techniques to bear.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.