CAREER: New Frontiers in the Dynamics of Topological Solitons

非线性波在自然界中无处不在，从量子粒子的动力学到电磁辐射和引力波的传播。在数学上，许多这样的波传播现象可以用非线性色散方程来描述。虽然波通常会扩散和衰减，但这些非线性演化方程的一个显著特征是，它们可能允许粒子状解，通常称为孤子，其形状随着时间的推移而持续存在。对它们动力学的数学理解还远未完成。该项目的主要研究目标是在数学物理中出现的经典拓扑场论的背景下，研究非线性波如何形成粒子状结构以及这些结构如何相互作用。该项目的教育部分旨在通过组织与项目研究有关的小型课程和讲习班以及通过提供侧重于演讲技巧的专业发展机会来加强对研究生和博士后的培训，该项目侧重于数学物理学中几个著名的经典拓扑场论的孤子动力学。拓扑孤子的三个主要例子是在调查的中心：扭结，涡，和skyrmions在一个，两个和三个空间维度，分别。启发式地，这些孤子欠他们的拓扑基础的稳定性。然而，对这种直觉的数学论证仍然知之甚少，而且大多是开放的。该项目的首要目标是建立这些经典拓扑孤子的渐近稳定性结果，从而严格证明其稳定性的物理学。在该项目的过程中，研究人员还计划在这些和相关设置中研究多孤子配置。除了对该项目中心的基本问题的内在兴趣之外，这些问题的解决将对在许多其他非线性色散方程的背景下分析孤子和辐射之间的强非线性相互作用产生重大影响。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

Nonlinear waves are ubiquitous in nature, ranging from the dynamics of quantum particles to the propagation of electromagnetic radiation and gravitational waves. Mathematically, many such wave propagation phenomena can be described in terms of nonlinear dispersive equations. While waves typically spread out and decay, a striking feature of these nonlinear evolution equations is that they may admit particle-like solutions, often called solitons, whose shapes persist as time goes by. The mathematical understanding of their dynamics is still far from complete. The main research goal of this project is to investigate, in the context of classical topological field theories that arise in mathematical physics, how nonlinear waves can form particle-like structures and how these structures interact with each other. The educational component of the project seeks to enhance the training of graduate students and postdocs by organizing minicourses and workshops related to the research of the project and by providing professional development opportunities with an emphasis on presentation skills.This project focuses on soliton dynamics for several well-known classical topological field theories in mathematical physics. Three prime examples of topological solitons are at the center of the investigation: kinks, vortices, and skyrmions in one, two, and three space dimensions, respectively. Heuristically, these solitons owe their stability to their topological underpinnings. However, the mathematical justification of this intuition is still rather poorly understood and mostly open. The overarching goal of the project is to establish asymptotic stability results for these classical topological solitons, and thus to rigorously justify the heuristics for their stability. Over the course of the project the investigator also plans to move towards studying multi-soliton configurations in these and related settings. Beyond the intrinsic interest in the fundamental problems at the center of this project, their resolution will have significant impact on the analysis of strong nonlinear interactions between solitons and radiation in the context of many other nonlinear dispersive equations.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.