Soliton Dynamics for Non-Linear Wave Equations

自然界是由波动方程控制的：电路板上的电流，光纤电缆中的光，甚至银河系中心的黑洞都是通过波动力学传播的。虽然普遍存在，但波型方程还远未得到很好的理解。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，以及对宇宙形成的更深入了解。该项目为研究生提供研究培训机会。研究员将研究用于模拟上述物理现象的包含拓扑孤子的非线性波动方程。从技术上讲，这些是具有非平凡拓扑不变量的相干孤立波。典型的例子包括纯量场论中的扭结、作为驻波映射的调和映射、规范金斯堡-朗道理论中的涡旋、磁单极子、Skyrmions和杨-米尔斯瞬子。我们的目标是了解拓扑孤子如何影响动力学，并解决两个长期悬而未决的问题。首先，研究人员将试图证明，非线性波可以唯一地继续过去的奇点，在有限的时间内发展的能量集中（冒泡）在一个孤立子。其次，研究人员试图表明，多孤子碰撞必然是非弹性的不可积的波动方程，如φ-4标量场模型和波映射方程。这个程序的关键部分是有限时间冒泡和全球时间多孤子动力学的解决方案的存在性和唯一性证明。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.