CAREER: New Approaches to Mathematical Wave Turbulence

非线性色散方程可以模拟量子力学、等离子体物理、非线性光学、海洋学以及广义相对论中出现的物理现象。该项目的目的是通过以下方式提高对这些方程的数学和科学理解：A)旨在研究这些方程解的长期行为的具体研究项目；B)旨在向新一代多样化的数学家介绍这些问题的教育组成部分。主要的重点是研究所谓的非线性色散方程的“非平衡行为”。这种行为在许多重要的物理现象中都表现出来，比如海浪中的湍流或光信号的传输，这里仅举几个例子。虽然这种湍流很容易观察，并且在自然界中有很强的表现，但对它的科学认识相当差，基于模型方程对所涉及现象的数学证明相当困难，而且大部分仍然是开放的。本项目旨在严格证明某些非线性色散模型的湍流现象。非线性色散偏微分方程（PDE）在紧域上长时间行为的最重要特征是失平衡行为或长时间不稳定性。这意味着解在平衡解或构型附近不表现出长时间的稳定性。这些问题可以从动力系统的角度和统计物理的角度来解决，在物理文献中通常被称为“波动湍流理论”。拟议的项目从两个角度解决了这个广泛的问题。在动力学方面，目标是构建显示所谓“能量级联”现象的解，在这种现象中，色散系统的能量在不同长度尺度之间移动其集中区域。从统计物理的角度，提出了旨在证明波浪湍流基本方程的项目。这些是物理文献中从启发式考虑中推导出的有效动力学方程，如果它们得到严格证明，则对色散偏微分方程的长期行为具有非常重要的意义。该项目还包括一个教育部分，旨在通过讲习班、研讨会、REU项目等提高年轻一代研究人员对这些基本问题的兴趣。

Nonlinear dispersive equations model physical phenomena arising in quantum mechanics, plasma physics, nonlinear optics, oceanography, all the way to general relativity. The purpose of this project is to improve the mathematical and scientific understanding of those equations via A) concrete research projects aimed at studying the long-time behavior of solutions to such equations, and B) an educational component aimed at introducing such problems to a new and diverse generation of mathematicians. The main focus is on the study of the so-called "non-equilibrium behavior" of nonlinear dispersive equations. Such a behavior is exhibited in many physically important phenomena, like turbulence in ocean waves or in the transmission of optical signals, to mention only a couple. While this turbulence is easy to observe and has very strong manifestations in nature, its scientific understanding is rather poor and the mathematical justification of the involved phenomena based on the model equations is rather difficult and is mostly still open. This project is aimed at rigorously proving turbulence phenomenon for certain nonlinear dispersive models.The most important feature of the long-time behavior of nonlinear dispersive partial differential equations (PDE) on compact domains is out-of-equilibrium behavior or long-time instability. This means that solutions do not exhibit long-time stability near equilibriums solutions or configurations. Such questions can be addressed from a dynamical systems perspective and from a statistical physics perspective, in what is often called in the physics literature as "wave turbulence theory". The proposed projects address this broad problematic from both perspectives. In terms of dynamics, the aim is to construct solutions exhibiting the so-called "energy cascade" phenomenon, in which the energy of a dispersive system moves its concentration zone between characteristically different length scales. On the statistical physics perspective, projects aimed at justifying the fundamental equations of wave turbulence are proposed. These are effective equations for the dynamics that are derived in the physics literature from heuristic considerations, and have very important implications on the long-time behavior of dispersive PDE, provided they are rigorously justified. The project also includes an educational component aimed at raising the interest of a younger generation of researchers in those fundamental problems through workshops, seminars, REU projects, etc.