Nonlinear Dispersive Waves

这个项目的主要目标是提高我们对一系列偏微分方程解的理解，这些偏微分方程都可以描述为非线性波。这些方程是作为物理现象的模型出现的，这些物理现象在我们的世界中发挥着重要作用，有时甚至在我们的日常生活中也发挥着重要作用。关键的例子从海洋表面的水波，多粒子系统的量子力学相互作用，到气态恒星的表面动力学。所有这些现象的共同特征是线性波和非线性相互作用之间的相互作用，它们的平衡影响着短程和长程动力学。该项目的不同部分旨在研究短时间现象，如奇点形成，以及它们的长时间行为，如散射。该项目为研究生提供了研究培训机会。这项拟议的研究涵盖了一系列关于非线性偏微分方程的研究课题。要研究的问题都与非线性色散方程领域有关，但也与几何、调和分析、复分析和微局部分析等相关领域有很强的联系。在很大程度上，这些问题是高度非线性的，而且基本上也是基于流体动力学、电磁学和广义相对论等领域的物理模型。简而言之，人们可以认为这项工作的目标是理解非线性波相互作用，从短时间尺度开始，继续到长时间尺度，一直到散射和爆炸现象。这个项目的主要研究领域之一是流体动力学，更准确地说是水波，以及更广泛的自由边界问题。近年来，这一直是一个引起人们强烈兴趣的领域，尤其是因为它具有巨大的应用潜力，从海洋学到医学科学，再到恒星动力学。但最有趣的问题仍未解决，他们的研究带来了一系列极其困难的问题。另一个目标是研究完全可积系统，这通常是作为流体动力学中更困难的非线性问题的模型出现的，特别是水波。其主要目的是获得对长时间动力学的足够准确的理解，长时间动力学将孤子、色散激波和非线性散射结合在一个复杂的模式中。几何非线性波动方程的研究是本项目的第三个目标。在最近证明了波图和Yang-Mills系统的阈值猜想之后，目前的工作是按照孤子解猜想的预测，对爆破解和非散射解进行完全分类。最后，拟线性波和薛定谔演化提出了一些在非线性偏微分方程中最具挑战性的问题，了解它们的短期和长期动力学是该项目的另一个主要目标。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The main objective of this project is to improve our understanding of the solutions for a broad array of partial differential equations that can be all described as Nonlinear Waves. These equations arise as models for physical phenomena, which play important roles in our world and sometimes even in our daily lives. Key examples range from the water waves at the surface of our oceans, the quantum mechanical interaction of multiparticle systems, to the surface dynamics of gaseous stars. The common feature in all of these phenomena is the interplay between linear waves and nonlinear interactions, whose balance affects both short and long range dynamics. Different parts of the project aim to study both short time phenomena, such as singularity formation, as well as their long time behavior, such as scattering. The project provides research training opportunities for graduate students. The proposed research spans a broad array of research topics in nonlinear partial differential equations. The problems to be investigated are all associated with the field of nonlinear dispersive equations, but also carry strong connections to related areas such as geometry, harmonic analysis, complex analysis and microlocal analysis. For the most part, these problems are strongly nonlinear, and also fundamentally based on physical models from areas such as fluid dynamics, electromagnetism, and general relativity. In a nutshell, one can view the objective of this work to be the understanding of nonlinear wave interactions, beginning with short time scales, continuing with long time scales, all the way to scattering and blow-up phenomena. One of the main research areas targeted by this project is in fluid dynamics, more precisely water waves, as well as a broader class of free boundary problems. This has been an area of intense interest in recent years, not in the least because of its great potential for applications, ranging from oceanography to medical science and to stellar dynamics. But the most interesting problems are still unresolved, and their study is bringing forth an array of extremely difficult questions. Another goal is the study of completely integrable systems, which often arise as models for more difficult nonlinear problems in fluid dynamics in general, and water waves in particular. The main objective is to gain a sufficiently accurate understanding of the long time dynamics, which combine solitons, dispersive shocks and nonlinear scattering in a complex pattern. The study of geometric nonlinear wave equations is a third objective of this project. After the recent proof of the Threshold Conjecture for Wave Maps and Yang-Mills systems, the current work is directed toward a full classification of blow-up solutions and of non-scattering solutions, as predicted by the Soliton Resolution Conjecture. Finally, quasilinear wave and Schrodinger evolutions bring forth some of the most challenging problems in nonlinear partial differential equations, and understanding both their short and long time dynamics is another major goal of the project.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.