CAREER: New Frontiers in the Dynamics of Topological Solitons

职业：拓扑孤子动力学的新领域

• 批准号: 2235233
• 负责人: Jonas Luhrmann
• 金额: $ 43.78万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2028-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2235233&HistoricalAwards=false
• 关键词: CAREER; New; Frontiers; Dynamics; Topological

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.

非线性波本质上是普遍存在的，从量子颗粒的动力学到电磁辐射和重力波的传播。从数学上讲，许多这样的波传播现象可以用非线性分散方程来描述。尽管波通常会散布并衰减，但这些非线性演化方程的引人注目的特征是它们可能会允许类似粒子的溶液，通常称为孤子，随着时间的流逝，它们的形状持续存在。对它们动态的数学理解还远非完整。该项目的主要研究目标是在数学物理学中出现的经典拓扑领域理论的背景下，非线性波如何形成类似粒子的结构以及这些结构如何相互相互作用。该项目的教育组成部分旨在通过组织与该项目的研究相关的微型训练和讲习班，并通过提供专业发展机会，重点介绍演讲技巧，从而增强研究生和博士后的培训。该项目着重于索利顿动力学，以用于几种数学物理学中的几种知名古典拓扑理论。拓扑孤子的三个主要例子是调查的中心：分别在一个，两个和三个空间维度中的扭结，涡流和天空。这些孤儿启发性地归功于它们的拓扑基础。但是，这种直觉的数学理由仍然很糟糕，并且大多是开放的。该项目的总体目标是为这些古典拓扑孤独子建立渐近稳定性结果，从而严格地证明其稳定性的启发式法。在项目的过程中，研究人员还计划朝着研究这些和相关设置的多苏里顿配置迈进。除了对该项目中心的基本问题的内在兴趣之外，在许多其他非线性分散方程式的背景下，他们的解决方案将对对孤子和辐射之间的强烈非线性相互作用的分析产生重大影响。该奖项反映了NSF的法定任务，并通过使用基金会的知识优点和广泛的criperia来评估，通过评估值得评估，这是NSF的法定任务。