Analytical Challenges Near Dynamical Thresholds for Nonlinear Wave and Fluid Equations

A vast number of physical systems, from water flowing down a canal to light traveling through an optical fiber, involve the nonlinear interaction of waves. These interactions are typically described by scientists and engineers using partial differential equations. Frequently, a given equation will exhibit multiple different qualitative behaviors depending on the model’s parameters or data. This project seeks to understand solutions to these equations that live on the verge of two or more different dynamical regimes. Solutions near these “dynamical thresholds” are key to understanding when a given equation is a faithful representation of some underlying physical system, and when it is not. The results of this investigation will not only yield new mathematical insights but also help to elucidate the possible uses and limitations of these models. In addition to these research goals, this project includes several educational and outreach activities, including training opportunities for both graduate and undergraduate students. This project consists of three distinct problems. While the notion of what constitutes a dynamical threshold in each problem varies, what ties them together is a common set of mathematical tools with their foundations in real and harmonic analysis, spectral theory, and dynamical systems. The first problem considers solutions to integrable partial differential equations. Using recent developments in the analysis of these equations, the principal investigator (PI) will study dispersive estimates and soliton stability up to the threshold of ill-posedness. The second problem concerns certain toy models for wave turbulence. By incorporating tools from probability and combinatorics, the PI will seek to describe the effective dynamics up to the timescale at which the kinetic description breaks down. The final problem considers vortex filaments — fluid configurations with vorticity concentrated along a curve. As the circulation of a vortex filament increases, the evolution transitions from being dominated by dissipation to being dominated by transport. Informed by previous work, the PI will make progress toward rigorously justifying these dynamics. 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.