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

我们随时随地都在与海浪互动。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，当我们看着天空中的星星时，当我们在阳光明媚的海滩上放松时。但当地震震动和传播、海啸形成或核辐射失控时，波动现象也可能影响数百万人的生活。事实上，波的自然产生发生在各种物理系统中，如非线性光学、大气和海浪、量子力学和等离子体。对波的研究对于理解像玻色-爱因斯坦凝聚态这样的非常小的尺度和像星系共谋这样的非常大的尺度上的现象都是基本的。这些对自然的表达永远不会太流畅，也很少会太简单：小波的相互作用可以产生非常大的结果，比如反常的波，而像孤子这样的复杂物体在交叉时几乎看不到对方。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的情况下，了解可能的结果是预测和控制它的基础，希望这对我们有利。在这项由美国国家科学基金会资助的研究中，PI提出了一系列处于非线性波动现象研究前沿的项目，其中确定性方法与概率方法一起实施，这些方法通常基于调和和傅立叶分析，以捕捉波浪现象的基本属性。近年来已经很清楚的是，确定性方法和概率方法自然是相互促进的，当它们结合在一起时，不仅有助于我们的理解，而且还打开了新范式的大门，以推动研究在不同方向上向前发展。更准确地说，PI提出了四个处于非线性发展方程前沿的项目，其中确定性和概率方法的相互作用是取得进展的关键。这些问题从色散和流体方程的弱湍流的研究到可积结构的分析，从Gibbs测度的定义到具有零形非线性的某些几何流动的概率存在性和稳定性。过去几年PI的工作中的概率部分在弥散和波动非线性方程与专门研究随机偏微分方程组的社区之间架起了一座桥梁。这种互动在这两个社区的成员之间创造了持续的合作。PIs、他们的学生和合作者在解决本项目中描述的问题方面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。该项目更广泛的影响部分旨在培养美国的博士研究生和初级研究人员，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。这一奖项反映了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。

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.