Spectral Theory and Nonlinear Waves

谱理论和非线性波

• 批准号: 2054841
• 负责人: Wilhelm Schlag
• 金额: $ 40.02万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054841&HistoricalAwards=false
• 关键词: Spectral; Theory; Nonlinear; Waves

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.

该项目致力于研究结构化和无序培养基中的波传播。现代社会广泛依赖于涉及波传输的工程应用。手机通信，卫星数据传输和Internet上的信息 - 全部沿着数千英里的玻璃纤维电缆发送 - 由描述波动运动的数学控制。波浪传播的一个引人注目的特征和该项目的主题是其普遍性。实际上，在天文尺度上传播的波（例如Ligo检测到的引力波）以及微观尺度上的波，例如原子在激光器中以电磁辐射的形式发射的波，由相同的数学理论控制。因此，这对于整个社会的福祉和安全至关重要，可以在PI的研究领域培训年轻专家；这也是该项目的目标之一。经验表明，这种更广泛的影响只有在认识前沿积极工作的科学家才能实现。 PI最近完成了等效临界波映射到2个球体的扰动分析，而没有对扰动的任何对称假设。这种非线性分析基于Bessel型电势的波函数的半古典表示，PI十多年前在普遍相对论的价格定律中开发了超过十多年。该新技术的应用计划在其他非线性演化方程中，该方程的接受度将变量分离，并将其减少到无限的半古典波方程系统中。半古典参数H的作用通常由角动量的倒数扮演。光谱理论现在脱颖而出的另一个领域是拓扑结构的不对称稳定性的快速发展领域，特别是针对一维标量场的扭结溶液。 PI与其他人共同展示了如何以非元素潜力和远程非线性处理klein-gordon方程。基本标量场方程每个都以阈值共振暴露了非元素电位，因此是该项目中要考虑的类型。 PI在准周期定位及其后果的工作将继续进行。不一致的双曲动力系统将提供一个特别具有挑战性的新颖观点，在这种情况下，有很强的迹象表明有可能将安德森本地化技术带到熊状态。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来诚实地认为，通过评估来诚实地支持。