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、等离子体物理学、大气和海洋科学、非线性光学的方程就是这种情况。虽然这些系统在许多方面都有很大的不同，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。在水面上的波浪的情况下，对于前者，可以想到海啸（刚性行为），对于后者，可以想到无序的涟漪（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。该项目将加深对非线性色散方程的两个基本特征的理解：孤立波和波动湍流。对于前者，扭曲傅立叶变换将被用来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），导致可能的最佳稳定性结果。对于后者，进展将是双重的：在推导波湍流的动力学描述上，以及在描述波湍流的动力学方程的分析上。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.