Nonlinear Dynamics in Models of Wave Propagation and Domain Coarsening

这项工作的目的是通过发展和应用非线性动力学方法来理解波动现象的重要方面-系统在时间和空间上变化的数学。一个主要的目标是解释孤子的鲁棒性或稳定性，孤子是一类重要的波，它表现出类似粒子的性质，能够进行非常深刻和明确的数学描述。波动动力学的数学对于材料科学中一个基本问题的进展也出乎意料地重要。为了了解合金的微观结构如何随时间变粗，研究人员正在开发一种新的数学理解，用于奇异质量分布的波状输运。这项工作试图从数学上解释各种波的一些鲜为人知的特征，流体表面波是一个主要的例子。这些特征包括：孤子（孤立存在的波）的非凡稳定性；现代快速汽车轮渡在一定条件下产生的海啸般的尾流；以及海浪集中和破碎同时产生的强大力量的本质。贯穿这些研究的共同主线是通过发展新的数学方法来分析动态变化，从而有效地理解物理现象。

The aim of this work is to understand significant aspects of wave phenomena through developing and applying the methods of nonlinear dynamics --- the mathematics of systems that change in time and space. A main goal is to explain the robustness or stability of solitons, an important class of waves which exhibit particle-like properties and are capable of a remarkably deep and explicit mathematical description. The mathematics of wave dynamics is also unexpectedly important for prograss concerning a fundamental problem in materials science. To understand how an alloy's microscopic structure coarsens in time, the investigator is developing a new mathematical understanding of wave-like transport for singular mass distributions.This work seeks to explain mathematically a number of poorly understood characteristics of waves of various kinds, surface waves on fluids being a prime example. Such characteristics include: the extraordinary stability of solitons (waves that persist in isolation); the tsunami-like qualities of wakes produced in certain conditions by modern fast car ferries; and the nature of the powerful forces created when waves focus and break simultaneously. The common thread that runs through these investigations is the effective understanding of physical phenomena through the development of new mathematical methods for analyzing dynamic change.