Wave Turbulence and Stability of Solitary Waves

点击翻译按钮获取中文摘要

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.