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.

自然界是由波动方程控制的：电路板上的电流，光纤电缆中的光，原子内的基本粒子，甚至星系中心的黑洞都是通过波动力学传播的。虽然普遍存在，但波型方程远未被很好地理解。该项目的目标是了解波是如何受到自身或环境干扰的影响。这项研究试图了解一些波何时以及为什么会分散，其他波会持续存在，还有一些波会崩溃。了解波的行为推动了技术进步-更小的微芯片，更快的数据传输，以及对宇宙基本物理学的更深入了解。本项目为本科生、研究生、博士后研究人员提供研究训练机会。研究者以上述物理现象的模型化中包含拓扑孤子的方程为中心，研究非线性波动方程和色散方程的解的长时间动态。孤子是具有非平凡拓扑不变量的局域孤立波。它们是由Skyrme在20世纪60年代引入的，作为经典场论中粒子的候选者。它们具有经典力学中粒子所需的性质--人们可以定义它们的位置、动量和能量--从远处看，多个孤子的结构类似于相互作用的粒子系统。研究人员在多孤子动力学方面的工作使这种与经典力学的联系变得明确，将强相互作用孤子的动力学减少到其位置，动量，尺度等的基本n体问题。孤子动力学分析的指导原则是孤子分辨率猜想，其预测，一般解在接近存在的最终时刻分解成多个孤子和捕获辐射的项的叠加，通常是基本线性方程的解。研究者将致力于在某些情况下证明该猜想，并通过考虑三类问题在其他情况下超越该猜想：（1）没有对称性假设的演化方程的孤子分辨率猜想，从二维调和映射热流开始，这是一个长期存在的开放问题;（2）爆破时间后奇异非线性波的唯一延拓问题;以及（3）给出多孤子解及其碰撞的渐近描述的问题。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，被认为值得支持。