Soliton Dynamics for Non-Linear Wave 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，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，但人们还远没有很好地理解它。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，以及对宇宙形成的更深入的了解。本项目为研究生提供研究训练机会。研究者将研究允许拓扑孤子的非线性波动方程，拓扑孤子用于模拟上述物理现象。从技术上讲，这些是具有非平凡拓扑不变量的相干孤立波。典型的例子包括标量场理论中的扭结，作为平稳波图的调和图，测量金兹堡-朗道理论中的涡旋，磁单极子，斯基米子和杨-米尔斯瞬子。目标是了解拓扑孤子如何影响动力学，并解决两个长期存在的开放性问题。首先，研究者将尝试证明，非线性波可以唯一地延续过在有限时间内形成的奇点，方法是在孤子中集中能量（冒泡）。其次，研究者试图证明多孤子碰撞对于非可积波方程（如phi-4标量场模型和波映射方程）来说必然是非弹性的。本程序的关键部分是有限时间冒泡和全局实时多孤子动力学解的存在性和唯一性证明。研究者正在开发的解决这些问题的技术将在其他相关环境中有用。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, 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 formation of the universe. The project provides research training opportunities for graduate students.The investigator will study nonlinear wave equations that admit topological solitons, which are used to model the physical phenomena described above. Technically, these are coherent solitary waves with a nontrivial topological invariant. Canonical examples include kinks in scalar field theories, harmonic maps as stationary wave maps, vortices in gauged Ginzburg-Landau theory, magnetic monopoles, Skyrmions, and Yang-Mills instantons. The goal is to understand how topological solitons influence the dynamics, and to resolve two long-standing, open questions. First, the investigator will try to prove that nonlinear waves can be uniquely continued past a singularity that develops in finite time by concentrating energy (bubbling) in a soliton. Second, the investigator seeks to show that multi-soliton collisions are necessarily inelastic for non-integrable wave equations such as the phi-4 scalar field model and the wave maps equation. Crucial parts of this program are existence and uniqueness proofs of solutions exhibiting finite time bubbling and global-in-time multi-soliton dynamics. The techniques the investigator is developing to address these problems will be useful in other related contexts.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.