Collaborative Research: Dynamics of Nonlinear Partial Differential Equations: Integrating Deterministic and Probabilistic Methods

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

We interact with waves all the time and everywhere. When we listen to music, when we use our cell phones, when we warm up a dinner in a microwave, when we look at the stars in the sky and when we relax on a sunny beach. But wave phenomena may also affect the lives of millions of people when earthquakes shake and propagate, tsunamis form or nuclear radiations get out of control. Indeed, waves naturally arise occur in a variety of physical systems such as nonlinear optics, atmosphere and ocean waves, quantum mechanics and plasmas. The study of waves is fundamental for the understanding of phenomena at both a very small scale, such as the Bose-Einstein Condensate, and at a very large one, such as collusion of galaxies. These expressions of nature are never too smooth and rarely too simple: interactions of small waves can produce very large outcomes, such as freak waves, while complicated objects such as solitons almost do not see each other when they cross. Phenomena such as these are the byproduct of nonlinear wave interactions, and understanding what are the possible outcomes, given the initial state of a system of waves, is fundamental to predict and to control it, hopefully to our advantage. In this NSF supported research the PIs present a series of projects at the cutting edge of research in nonlinear wave phenomena in which deterministic approaches, classically based on harmonic and Fourier analysis, are implemented alongside probabilistic ones to capture basic properties of wave phenomena. It has become clear in recent years that deterministic methods and probabilistic ones naturally feed off each other and when combined not only contribute to our understanding but also open the door to new paradigms to move research forward in various directions. More precisely, the PIs propose four projects at the forefront of nonlinear evolution equations, where the interplay of deterministic and probabilistic approaches is the key to make progress. The problems range from the study of weak turbulence for dispersive and fluid equations to the analysis of integrable structures, from the definition of Gibbs type measures to the probabilistic existence and stability of certain geometric flows enjoying null form nonlinearities. The probabilistic component of PIs' work in the last few years has contributed in bridging the dispersive and wave nonlinear equations community with that specialized in stochastic partial differential equations. This interaction has created ongoing collaborations between members of these two communities. The work that the PIs, their students and collaborators will generate in solving the problems described in this project will further solidify the interactions between these two vibrant communities. The broader impact component of the project aims at fostering the training of doctoral graduate students and junior researchers in the US, thus fundamentally contributing to the STEM workforce. It will also enhance dissemination and collaborative research. 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.