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 200 年前，比多内 (Bidone) 在都灵首次记录了在传播过程中保持其形状和速度的孤立波的观测结果，随后罗素 (Russell) 在 1834 年跟踪了沿河道以恒定速度移动的水峰数次。今天，这些物体被称为孤子，它们处于数学和物理学的交叉点，自 20 世纪 60 年代以来就一直在研究完全可积波。方程中，孤子的许多特性都是已知的，例如它们的弹性碰撞、稳定性特性以及它们在波的长期描述中的作用，后者尤其重要，因为它可以预测波如何传播。在量子物理学、量子化学和材料科学中，这些数学工具可以更好地理解电子在各种介质中的运动，该项目旨在开发支持这些领域的应用科学的数学基础。哪些对工业非常重要该项目为研究生提供研究培训机会。该项目的目标一方面是建立非线性演化偏微分方程的新成果和新技术，另一方面是无序系统的谱理论。 Luhrmann 和首席研究员 (PI) 开发的长程散射理论在拓扑孤子背景下表现出阈值共振的势方面取得了第一个结果。这项工作的动机是关于渐近扭结稳定性的基本问题。 phi-4 模型。Ginzburg-Landau 涡旋在其自身等方差类中的渐近稳定性尚不清楚。线性化问题涉及非自共轭矩阵算子，PI 已开始与合作者一起研究其谱理论。从事谐波图热流冒泡的研究，并尝试将最近关于连续时间冒泡的论文与合适的调制理论结合起来。与该项目相关的第三个领域是。更具体地说，PI 将继续研究准周期辛共循环，这些共循环出现在凝聚态物理的多个模型中，例如石墨烯，以及分析它们的非微扰方法。该奖项反映了 NSF 的法定使命，并具有通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。