Wave Turbulence and Stability of Solitary Waves

许多物理系统的基本模型可以用数学术语描述为“非线性色散方程”。例如，广义相对论、等离子体物理、大气与海洋科学、非线性光学等方程都是如此。虽然这些系统在许多方面非常不同，但共同的数学特征导致了相似的行为。该项目将研究的现象一方面是孤立波，另一方面是波浪湍流。以水面上的波浪为例，前者是海啸（一种刚性行为），后者是无序涟漪（一种混沌行为）。该项目的目的是研究这些在基础科学和技术中有许多应用的现象。该项目还将为研究生提供研究培训机会。该项目将深入了解非线性色散方程的两个基本特征：孤波和波湍流。对于前者，畸变傅立叶变换将用于分析孤立波的稳定性，特别是它们的非线性共振（在动力系统的意义上），从而导致可能的最佳稳定性结果。对于后者，将有两方面的进展：关于波浪湍流的动力学描述的推导，以及关于描述波浪湍流的动力学方程的分析。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。

Fundamental models for a number of physical systems can be described in mathematical terms as "nonlinear dispersive equations". This is, for instance, the case for the equations of General Relativity, Plasma Physics, Atmosphere and Ocean Science, Nonlinear optics. While these systems are very different in many respects, the common mathematical features are responsible for similar behaviors. The phenomena that the project will investigate are, on the one hand, solitary waves, and on the other hand, wave turbulence. In the case of waves on a surface of water, think of tsunami for the former (a rigid behavior), and disordered ripples for the latter (a chaotic behavior). The aim of this project is to investigate these phenomena, which have many applications to basic science and technology. The project will also provide research training opportunities for graduate students.The project will reach a deeper understanding of two basic features of nonlinear dispersive equations: solitary waves and wave turbulence. For the former, the distorted Fourier transform will be used to analyze the stability of solitary waves, in particular their nonlinear resonances (in the sense of dynamical systems), leading to possibly optimal stability results. For the latter, the progress will be twofold: on the derivation of the kinetic description of wave turbulence, and on the analysis of the kinetic equation describing wave turbulence.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.