Harmonic Analysis Challenges in Nonlinear Dispersive Partial Differential Equations

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This research project explores the mathematical properties of a class of equations known as nonlinear dispersive equations. Equations of this type arise in models of several physical phenomena, including propagation of light in optical fibers, Bose-Einstein condensates, small-amplitude water waves, and waves in plasmas. Despite their common occurrence, understanding of the behavior of solutions to such equations is limited by their mathematical complexity. This project aims to extend current theoretical understanding of this important class of equations. Training of junior researchers and students in this area of research is also an integrated part of the project.The problems to be investigated are nonlinear dispersive partial differential equations with broken symmetries and/or non-constant coefficients. While some of these equations are very physical, such as the cubic-quintic nonlinear Schrodinger equation with non-zero boundary conditions, others have been carefully selected by the investigator to highlight certain deficiencies in our mathematical understanding of the underlying linear propagator. A major thrust of this project is therefore to resolve questions in harmonic analysis related to the linear propagator in various geometries. These range from proving Strichartz estimates powerful enough to resolve the small data energy-critical problem on high-dimensional tori to shepherding recent progress on the restriction conjecture into the realm of non-constant coefficients, which is a key step toward resolving mass-critical problems outside the translation-invariant setting.