Collaborative Research: Dynamics of Nonlinear Partial Differential Equations: Integrating Deterministic and Probabilistic Methods

我们每时每刻都在与波相互作用。当我们听音乐时，当我们使用手机时，当我们用微波炉加热晚餐时，当我们看天上的星星时，当我们在阳光明媚的海滩上放松时。但是，当地震震动和传播、海啸形成或核辐射失控时，波浪现象也可能影响数百万人的生活。事实上，波浪自然产生于各种物理系统中，如非线性光学、大气和海浪、量子力学和等离子体。波的研究是理解小尺度现象（如玻色-爱因斯坦凝聚）和大尺度现象（如星系的合谋）的基础。这些自然的表达从来都不太流畅，也很少太简单：小波的相互作用可以产生非常大的结果，比如反常波，而像孤子这样的复杂物体在交叉时几乎看不到对方。像这样的现象是非线性波相互作用的副产品，在给定波系统的初始状态下，理解可能的结果是预测和控制它的基础，希望对我们有利。在这项NSF支持的研究中，pi提出了一系列处于非线性波现象研究前沿的项目，其中经典的基于谐波和傅立叶分析的确定性方法与概率方法一起实现，以捕获波现象的基本特性。近年来，确定性方法和概率方法自然地相互促进，当它们结合在一起时，不仅有助于我们的理解，而且还打开了向各个方向推进研究的新范式的大门。更准确地说，pi在非线性演化方程的前沿提出了四个项目，其中确定性和概率方法的相互作用是取得进展的关键。这些问题的范围从色散和流体方程的弱湍流研究到可积结构的分析，从吉布斯测度的定义到某些零形式非线性几何流的概率存在性和稳定性。在过去的几年中，pi的工作中的概率成分在连接色散和波动非线性方程社区与专门研究随机偏微分方程的社区方面做出了贡献。这种互动在这两个社区的成员之间创造了持续的合作。pi、他们的学生和合作者在解决本项目中描述的问题时所做的工作将进一步巩固这两个充满活力的社区之间的互动。该项目的更广泛影响部分旨在促进美国博士研究生和初级研究人员的培训，从而从根本上为STEM劳动力做出贡献。它还将加强传播和合作研究。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。

We interact with waves all the time and everywhere. When we listen to music, when we use our cell phones, when we warm up a dinner in a microwave, when we look at the stars in the sky and when we relax on a sunny beach. But wave phenomena may also affect the lives of millions of people when earthquakes shake and propagate, tsunamis form or nuclear radiations get out of control. Indeed, waves naturally arise occur in a variety of physical systems such as nonlinear optics, atmosphere and ocean waves, quantum mechanics and plasmas. The study of waves is fundamental for the understanding of phenomena at both a very small scale, such as the Bose-Einstein Condensate, and at a very large one, such as collusion of galaxies. These expressions of nature are never too smooth and rarely too simple: interactions of small waves can produce very large outcomes, such as freak waves, while complicated objects such as solitons almost do not see each other when they cross. Phenomena such as these are the byproduct of nonlinear wave interactions, and understanding what are the possible outcomes, given the initial state of a system of waves, is fundamental to predict and to control it, hopefully to our advantage. In this NSF supported research the PIs present a series of projects at the cutting edge of research in nonlinear wave phenomena in which deterministic approaches, classically based on harmonic and Fourier analysis, are implemented alongside probabilistic ones to capture basic properties of wave phenomena. It has become clear in recent years that deterministic methods and probabilistic ones naturally feed off each other and when combined not only contribute to our understanding but also open the door to new paradigms to move research forward in various directions. More precisely, the PIs propose four projects at the forefront of nonlinear evolution equations, where the interplay of deterministic and probabilistic approaches is the key to make progress. The problems range from the study of weak turbulence for dispersive and fluid equations to the analysis of integrable structures, from the definition of Gibbs type measures to the probabilistic existence and stability of certain geometric flows enjoying null form nonlinearities. The probabilistic component of PIs' work in the last few years has contributed in bridging the dispersive and wave nonlinear equations community with that specialized in stochastic partial differential equations. This interaction has created ongoing collaborations between members of these two communities. The work that the PIs, their students and collaborators will generate in solving the problems described in this project will further solidify the interactions between these two vibrant communities. The broader impact component of the project aims at fostering the training of doctoral graduate students and junior researchers in the US, thus fundamentally contributing to the STEM workforce. It will also enhance dissemination and collaborative research.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.