Soliton Dynamics for Non-Linear Wave Equations

非线性波动方程的孤子动力学

• 批准号: 1954455
• 负责人: Andrew Lawrie
• 金额: $ 24.9万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954455&HistoricalAwards=false
• 关键词: Soliton; Dynamics; Non; Linear; Wave

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.

自然世界受波动方程的控制：电路板上的电力，光纤电缆中的光，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形成的更深入的见解。该项目为研究生提供了研究培训机会。研究人员将研究允许拓扑孤子的非线性波方程，这些方程用于对上述物理现象进行建模。从技术上讲，这些是与非平凡拓扑不变的连贯的孤立波。规范的示例包括标量场理论中的扭结，作为固定波图的谐波图，衡量的金茨堡 - 兰道理论中的涡流，磁性单极管，天空和扬米尔斯instantons。目的是了解拓扑孤子如何影响动态，并解决两个长期以来的开放问题。首先，研究人员将试图证明非线性波可以通过在孤子中浓缩能量（冒泡）在有限时间内发展出奇异性。其次，研究者试图表明，对于不可融合的波动方程，例如PHI-4标量场模型和波图方程，多索群碰撞必然是无弹性的。该程序的关键部分是存在有限的时间冒泡和全球多苏利顿动力学的解决方案的存在和独特的证明。研究人员正在开发解决这些问题的技术在其他相关背景下将很有用。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。