CAREER: New Approaches to Mathematical Wave Turbulence

非线性色散方程模型的物理现象出现在量子力学，等离子体物理学，非线性光学，海洋学，一直到广义相对论。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）旨在研究此类方程的长期解的具体研究项目，以及B）旨在向新一代和多样化的数学家介绍此类问题的教育组件。主要集中在研究所谓的非线性色散方程的“非平衡行为”。这种行为表现在许多重要的物理现象中，例如海浪中的湍流或光信号的传输，仅举两例。虽然这种湍流很容易观察到，并且在自然界中有很强的表现，但其科学理解相当差，并且基于模型方程的所涉及现象的数学证明相当困难，并且大多数仍然是开放的。本项目旨在严格证明某些非线性色散模型的湍流现象，紧致区域上的非线性色散偏微分方程（PDE）的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。这意味着溶液在平衡点附近不表现出长时间的稳定性。这些问题可以从动力学系统的角度和统计物理学的角度来解决，在物理学文献中通常称为“波湍流理论”。拟议的项目从这两个角度解决这一广泛的问题。在动力学方面，目的是构建表现出所谓的“能量级联”现象的解决方案，其中分散系统的能量在特征性不同的长度尺度之间移动其浓度区。从统计物理的角度，提出了一些旨在证明波动湍流基本方程的方案。这些都是有效的方程的动力学推导出的物理文献中的启发式考虑，并有非常重要的影响，色散偏微分方程的长期行为，只要他们是严格合理的。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.