Dynamics of Nonlinear and Disordered Systems

非线性和无序系统的动力学

• 批准号: 2350356
• 负责人: Wilhelm Schlag
• 金额: $ 46.02万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350356&HistoricalAwards=false
• 关键词: Dynamics; Nonlinear; Disordered; Systems

Observations of solitary waves that maintain their shape and velocity during their propagation were recorded around 200 years ago. First by Bidone in Turin in 1826, and then famously by Russell in 1834 who followed a hump of water moving at constant speed along a channel for several miles. Today these objects are known as solitons. Lying at the intersection of mathematics and physics, they have been studied rigorously since the 1960s. For completely integrable wave equations, many properties of solitons are known, such as their elastic collisions, their stability properties, as well as their role as building blocks in the long-time description of waves. The latter is particularly important, as it for example predicts how waves carrying information decompose into quantifiable units. In quantum physics, quantum chemistry, and material science, these mathematical tools allow for a better understanding of the movement of electrons in various media. This project aims to develop the mathematical foundations which support these areas in applied science, which are of great importance to industry and society at large. The project provides research training opportunities for graduate students.The project’s goal is to establish both new results and new techniques in nonlinear evolution partial differential equations on the one hand, and the spectral theory of disordered systems on the other hand. The long-range scattering theory developed by Luhrmann and the Principal Investigator (PI) achieved the first results on potentials which exhibit a threshold resonance in the context of topological solitons. This work is motivated by the fundamental question about asymptotic kink stability for the phi-4 model. Asymptotic stability of Ginzburg-Landau vortices in their own equivariance class is not understood. The linearized problem involves a non-selfadjoint matrix operator, and the PI has begun to work on its spectral theory. With collaborators, the PI will engage on research on bubbling for the harmonic map heat flow and attempt to combine the recent paper on continuous-in-time bubbling with a suitable modulation theory. The third area relevant to this project is the spectral theory of disordered systems. More specifically, the PI will continue his work on quasiperiodic symplectic cocycles which arise in several models in condensed matter physics such as in graphene and on non-perturbative methods to analyze them.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.

大约在200年前，记录了对它们在传播过程中保持其形状和速度的恒定波的观察。 1826年首先由Bidone在都灵的Bidone首先，然后在1834年由Russell著名，后者跟随沿着通道的持续速度移动几英里的水。如今，这些物体被称为实体。自1960年代以来，它们就在数学和物理学的交汇处进行了严格的研究。对于完全可以集成的波动方程，已知实体的许多属性，例如它们的弹性碰撞，其稳定性以及它们在长期描述波中作为构建块的作用。后者尤其重要，因为例如预测波如何将信息分解为可量化的单元。在量子物理，量子化学和材料科学中，这些数学工具可以更好地了解各种媒体中电子的运动。该项目旨在开发支持这些领域的数学基础，这些基础在应用科学领域，这对整个行业和社会至关重要。该项目为研究生提供了研究培训机会。该项目的目标是一方面建立新的结果和新技术和新技术，另一方面，以及无序系统的光谱理论。卢曼（Luhrmann）和主要研究者（PI）开发的远程散射理论获得了对拓扑结构中表现出阈值共鸣的电位的第一个结果。这项工作是由PHI-4模型的不对称纠结稳定性的基本问题激发的。 Ginzburg-landau涡旋在其同等类别中的不对称稳定性尚不清楚。线性化问题涉及非频谱矩阵运算符，PI已开始在其光谱理论上发挥作用。与合作者一起，PI将参与有关谐波热流量的气泡研究的研究，并试图将有关连续性气泡的最新论文与合适的调制理论相结合。与该项目相关的第三个领域是无序系统的光谱理论。更具体地说，PI将继续他在静态合成生伴生物中的工作，这些合成在浓缩物理学的几种模型中出现，例如在石墨烯和非扰动方法中，以分析它们。该奖项反映了NSF的法规任务，并认为通过基金会的知识优点和广泛的crietia criperia criperia criperia the Insportaucation the Priews take the Interia the奖励。