Spectral Theory and Nonlinear Waves

本计画致力于研究波在结构性与无序性介质中的传播。现代社会广泛依赖于涉及波的传输的工程应用。手机通信、卫星数据传输和互联网上的信息--都是沿着沿着数千英里的玻璃纤维电缆发送的--都是由描述波运动的数学控制的。波传播的一个显著特征，也是这个项目的主题，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，以及微观尺度上的波，如原子在激光器中以电磁辐射形式发射的波，都受到完全相同的数学理论的支配。因此，对整个社会的福祉和安全来说，在PI的研究领域培训年轻专家具有极其重要的战略意义;这也是该项目的目标之一。经验表明，只有在知识前沿积极工作的科学家才能实现这种更广泛的影响。PI最近完成了一个微扰分析的等变临界波映射到2-球，没有任何对称性假设的扰动。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普赖斯定律的背景下开发的。这种新技术的应用计划到其他非线性发展方程，其中承认分离变量和减少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扮演。另一个领域中，谱理论现在是脱颖而出是迅速发展领域的渐近稳定性的拓扑孤子，特别是扭结解决方案的一维标量场。PI最近与其他人一起演示了如何处理具有非通用势和长程非线性的Klein-Gordon方程。基本的标量场方程，每个表现出一个非通用的潜力与阈值共振，因此在这个项目中要考虑的类型。PI的准周期定位及其衍生物的工作将继续进行。一个特别具有挑战性的新的视角将提供非均匀双曲动力系统，有强烈的迹象表明，它是可能的，使安德森本地化技术bear.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.