Dynamics of Nonlinear and Disordered Systems

在传播过程中保持其形状和速度的孤立波的观测记录在大约200年前。首先是1826年在都灵的比多内，然后是1834年著名的罗素，他沿着沿着一条通道以恒定速度移动的一个驼峰走了几英里。今天，这些物体被称为孤子。它们位于数学和物理的交叉点，自20世纪60年代以来一直受到严格的研究。对于完全可积的波动方程，孤子的许多性质是已知的，例如它们的弹性碰撞，它们的稳定性，以及它们在长时间描述波中的作用。后者尤其重要，因为它可以预测携带信息的波如何分解成可量化的单位。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应用科学领域的数学基础，这些领域对工业和整个社会都非常重要。该项目为研究生提供研究训练机会。该项目的目标是一方面建立非线性发展偏微分方程的新结果和新技术，另一方面建立无序系统的光谱理论。由Luhrmann和主要研究者（PI）开发的长程散射理论在拓扑孤子的背景下首次获得了关于表现出阈值共振的势的结果。这项工作的动机是基本问题的φ-4模型的渐近扭结稳定性。Ginzburg-Landau涡在它们自己的等方差类中的渐近稳定性还不清楚。线性化问题涉及非自伴矩阵算子，PI已经开始研究其谱理论。与合作者一起，PI将从事谐波映射热流的起泡研究，并尝试将最近关于连续时间起泡的论文与合适的调制理论联合收割机结合起来。第三个与这个项目有关的领域是无序系统的光谱理论。更具体地说，PI将继续他在凝聚态物理学（如石墨烯）的几个模型中出现的准周期辛上循环以及分析它们的非微扰方法方面的工作。该奖项反映了NSF的法定使命，并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，被认为值得支持。

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.