Probabilistic and Deterministic Aspects of Nonlinear Dispersive and Wave Equations

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Nonlinear dispersive and wave equations model wave propagation phenomena for many physical systems, from water waves to the dynamics of quantum gases. Understanding the "typical" behavior of these systems has many applications, such as problems in material science involving signal degeneration in optic fibers. For the last few decades, research on these equations has centered around questions on the existence of solutions, their long time behavior, and the possibility of singularity formation. Fundamental progress has been made in many settings, yet in some regimes, the nonlinear interactions are so strong compared to the dispersion of the waves that typical methods break down. In those regimes, probabilistic tools have been instrumental in analyzing the behavior of these systems, enabling researchers to answer new and exciting questions in a variety of settings. This project aims to investigate the behavior of solutions to nonlinear wave and Schrodinger equations, and more generally Hamiltonian equations in infinite dimensions. More specifically, the PI will address several problems, including probabilistic existence and long-time behavior for power-type nonlinear wave and Schrodinger equations, the analysis of integrable structures, and the definition of invariant Gibbs-type measures. These problems have several common goals, many of them motivated by recent developments in the study of nonlinear dispersive and wave equations via probabilistic techniques. The PI will refine existing probabilistic tools and develop new ones in order to treat new settings, such as certain geometric flows. Additionally, to tackle these problems, the PI will employ a combination of tools from harmonic analysis, probability theory, spectral theory and the theory of Hamiltonian systems.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.