Multi-soliton Dynamics for Dispersive Partial Differential Equations

色散偏微分方程的多孤子动力学

• 批准号: 2247290
• 负责人: Andrew Lawrie
• 金额: $ 26.7万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247290&HistoricalAwards=false
• 关键词: Multi; soliton; Dynamics; Dispersive; Partial

The natural world is governed by wave equations: the electricity on a circuit board, the light in fiber-optic cables, the elementary particles inside atoms, and even the black hole in the center of the galaxy all propagate by wave dynamics. Though ubiquitous, wave-type equations are far from well-understood. The goal of this project is to understand how waves are affected by interference with themselves or with their environment. The research seeks to learn when and why some waves disperse, other waves persist, and still others collapse. Knowing how waves behave drives technological progress - smaller microchips, faster data transmission, and deeper insights into the fundamental physics of the universe. The project provides research training opportunities for undergraduate students, graduate students, and postdoctoral researchers.The investigator studies the long-time dynamics of solutions to nonlinear wave and dispersive equations, focusing on equations that admit topological solitons, which are used to model the physical phenomena described above. Solitons are localized solitary waves with a nontrivial topological invariant. They were introduced by Skyrme in the 1960s as candidates for particles in classical field theories. They have properties required from a particle in classical mechanics - one can define their position, momentum, and energy - and viewed from a distance, configurations of multiple solitons resemble systems of interacting particles. The investigator's work on multi-soliton dynamics makes this connection with classical mechanics explicit, reducing the dynamics of strongly interacting solitons to underlying n-body problems for their positions, momenta, scales, etc. A guiding principle in the analysis of soliton dynamics is the Soliton Resolution Conjecture, which predicts that generic solutions decompose near the final time of existence into a superposition of finitely many solitons and a term capturing the radiation, often a solution to the underlying linear equation. The investigator will work towards proving the conjecture in certain settings and going beyond it in others by considering three categories of problems: (1) the soliton resolution conjecture for evolution equations without symmetry assumptions, starting with the harmonic map heat flow in two dimensions, which is a long-standing open problem; (2) the unique continuation problem for singular nonlinear waves past the blow-up time; and (3) the question of giving asymptotic descriptions of multi-soliton solutions and their collisions.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.

自然世界受波动方程的控制：电路板上的电能，光纤电缆中的光，原子内部的基本颗粒，甚至是银河系中心的黑洞都通过波动力学传播。尽管无处不在，但波型方程远非众所周知。该项目的目的是了解如何通过干扰自己或环境的干扰影响波浪。该研究试图学习何时以及为什么有些海浪散布，其他海浪持续存在，而另一些则崩溃了。知道波浪的行为如何推动技术进步 - 较小的微芯片，更快的数据传输以及对宇宙基本物理学的更深入的见解。该项目为本科生，研究生和博士后研究人员提供了研究培训机会。研究人员研究了非线性波浪和分散方程的解决方案的长期动态，重点关注允许拓扑孤独子的方程，这些方程用于建模上述物理现象。孤子是局部孤立波，具有非平凡的拓扑不变。 Skyrme在1960年代将其引入了古典田地理论中的颗粒候选者。它们具有经典力学中粒子所需的特性 - 一个人可以定义其位置，动量和能量 - 并从远处查看，多个孤子的构型类似于相互作用的粒子系统。研究者在多苏泳动态上的工作使这种联系与经典的机制明确，从而将强烈相互作用的孤儿的动态降低到其位置，动量，尺度等的基本N体问题的动态。在分析Soliton动态分析的指导原理中，孤独动态的指导原则是唯一解决方案，该统一的术语预测了一般的术语，这是一般的一般解决方案，这是一定时间的一般时间，该计划的一般时间是一定的时间，该计划是一定的时间，该计划的一般时间是一定的时间，该计划是一定的时间，该计划是一定的时间，该计划的一定时间是一定的时间，该计划是一定的时间，该计划是一定的时间，该计划的一定时间是一定的，一定是一定的时间，一定是一定的，一定的时间是一定的时间，该计划的一定时间是一定的时间，这是一定时间的一定时间。捕获辐射，通常是基础线性方程的解决方案。研究者将通过考虑三类问题来证明某些情况下的猜想，并在其他环境中超越其他问题：（1）孤子解决方案的孤儿解决方程无需对称假设，从二维中的谐波映射热流开始，这是一个长期存在的开放问题； （2）超过爆炸时间的奇异非线性波的独特延续问题； （3）对多苏里顿解决方案及其碰撞的渐近描述的问题。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被视为值得通过评估来提供支持。