Universality and Integrability in Nonlinear Waves and Random Matrices

This project concerns a theoretical study which lies at the interface between different disciplines in science and engineering: nonlinear wave formation and fluids (soliton theory), certain probabilistic models (random matrices), and statistical mechanics (interacting particle systems). All the models of interest are foundational instances of integrable systems. Integrable systems are a special class of dynamical systems, describing a large array of physical models and characterized by a rich mathematical structure: many physical observables are conserved as the system evolves, and there exist techniques which in principle allow one to derive explicit, exact solutions for any initial data and precisely track their behavior. In certain settings, integrable systems display remarkable universal patterns, meaning that their solutions become independent on the initial data and, in some cases, on the governing equations, even in the presence of randomness. Analyzing universality properties of a system -or of a class of systems- allows to better understand the physical structure of the model(s) and to predict its ’expected’ behavior in applications and experiments. This project focuses on the study of asymptotic behaviors and critical phenomena for the aforementioned integrable models, and the emergence of universality properties. It aims to resolve open questions in these fields, based on classical tools and new approaches developed by the investigator. The project has significant applications in several physics models of current interest: growth phenomena (e.g. crystals, cancer cells), polymers, optical fibers, superconductors (e.g. Bose-Einstein condensate) and fluids (most notably, rogue waves and turbulence). In particular, one of the main objects of study in this project is the analysis of properties of special types of solutions, the so-called soliton gasses, whose existence in nature is supported by recent experimental evidence. The research agenda is organized into the following parallel directions of research. (1) Investigate asymptotic behavior and universal profiles of dispersive integrable partial differential equations in the presence of randomness: in particular, random soliton ensembles and soliton gasses. The research activity includes describing and classifying the wave patterns and coherent structures emerging as the dynamic of the random solutions evolves. (2) Analyze universality properties of statistical quantities of determinantal point processes and novel random matrix ensembles within certain asymptotic regimes, and their geometrical interpretation. The study involves using and further developing tools from complex and asymptotic analysis. The main strategy is the reformulation of each problem in terms of a particular boundary value problem, the so-called Riemann-Hilbert (RH) problem. The next concurrent steps are (a) establishing a geometrical connection between the RH problem, and Painlevé-type equations and isomonodromic tau functions, via a linear system of ordinary differential equations (Lax Pair), as well as (b) performing RH analysis with steepest descent methods. 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.