Probabilistic Aspects of Dispersive and Wave Equations

For centuries, partial differential equations (PDE) have played a fundamental role in understanding physical and natural phenomena. Dispersive/wave equations model wave propagation phenomena which are ubiquitous in nature. They also describe the basic laws of quantum physics, which is one of the greatest achievements of the 20th century. This project studies fundamental questions about dispersive and wave equations by introducing ideas from probability theory. The results of the project will advance the mathematical theory of wave turbulence, which has important applications to plasma physics, nonlinear optics, and oceanography, and the analysis of Gibbs measures for Hamiltonian systems, which plays a key role in quantum field theory and statistical physics. Due to its scope and connections to physics and science, the project will also promote interdisciplinary interactions. As part of the project, the Principal Investigator (PI) is training junior researchers and contributes to maintaining the diversity in STEM disciplines at University of Southern California. This award supports work on five research projects (A-E). The first three projects are concerned with the mathematical theory of wave turbulence. In Project A, the PI extends the short kinetic time derivation of wave kinetic equation to longer kinetic times. This is a major step in the development of the theory, as it goes beyond the perturbative regime and will also shed light on the longstanding open problem of the long-time derivation of the Boltzmann equation. In Project B, the PI plans to generalize this derivation to cover the full range of conjectured scaling laws, which is physically well motivated and also leads to new mathematically interesting structures. New significant combinatorial structures and cancellations which are not present in the physics literature are expected to be discovered. Project C considers the wave turbulence problem for water waves, which has been studied since the 1960s by physicists. Mathematically, it is a quasilinear equation and substantial new ideas are required to obtain results similar to the ones available in the semilinear case. The last two projects concern Gibbs and other invariant measures in statistical physics and quantum field theory. Project D concerns the Gibbs measure for the 2D hyperbolic sine-Gordon equation, which is an important model that contains near-critical scenarios. Here, the goal is to further develop the random tensor theory introduced by the PI in earlier work. Project E investigates, through a combination of techniques from probability theory and integrable systems, the invariance of the white noise measure for the one-dimensional cubic nonlinear Schrödinger equation, which is critical but also completely integrable. 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.