Multi-soliton Dynamics for Dispersive Partial Differential Equations

自然界是由波动方程控制的：电路板上的电、光纤电缆中的光、原子内的基本粒子，甚至星系中心的黑洞，都是通过波动动力学传播的。尽管波型方程无处不在，但人们对它的理解还远远不够。这个项目的目标是了解波是如何受到自身或环境干扰的影响。这项研究旨在了解何时以及为什么有些波浪会消散，有些波浪会持续存在，还有一些波浪会崩溃。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的更深入的了解。本项目为本科生、研究生和博士后提供科研训练机会。研究者研究了非线性波动和色散方程解的长期动力学，重点研究了允许拓扑孤子的方程，这些方程用于模拟上述物理现象。孤子是具有非平凡拓扑不变量的局域孤立波。它们是由斯基米在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要求的性质——人们可以定义它们的位置、动量和能量——从远处看，多孤子的构型类似于相互作用的粒子系统。研究者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潜在的n体问题，因为它们的位置，动量，尺度等。孤子动力学分析的一个指导原则是孤子分辨率猜想，它预测一般解在存在的最后时刻分解成有限多个孤子的叠加和一个捕获辐射的项，通常是底层线性方程的解。研究者将通过考虑三大类问题，努力在某些情况下证明这一猜想，并在其他情况下超越它：(1)无对称假设的演化方程的孤子解析猜想，从二维调和图热流开始，这是一个长期存在的开放问题；(2)奇异非线性波爆破后的唯一延拓问题；(3)给出多孤子解及其碰撞的渐近描述问题。该奖项反映了美国国家科学基金会的法定使命，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。

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.