Grant: Discovery Launch Supplement

The proposed program aims to advance our understanding of Dispersive Nonlinear Partial Differential Equations (PDEs). Equations of this type emerge as asymptotic models in a plethora of physical phenomena, such as quantum mechanics, plasma physics, nonlinear optics, oceanography, and general relativity. More specifically, the program focuses on questions that lie in the intersection of dispersive PDEs and complete integrability. Equations that fall under this framework include some of the most widely studied dispersive equations, such as the Korteweg-de Vries equation. Indeed, understanding the behavior of solutions to these equations, especially in a critical regime, has been one of the driving forces of the field for the past thirty years. Integrable PDEs are characterized by their rich structure: they possess an infinite family of conservation laws and they admit an inverse scattering theory. While both of these distinctive features have been instrumental in the study of completely integrable PDE, they have limitations. The conserved quantities are not well-suited to control very rough solutions, while the inverse scattering transform often requires spatial decay. A defining theme of this program is the study of problems at low - and even critical - regularity, beyond the threshold of efficacy of the above approaches. Over the past few years new methods have emerged, which I have helped develop, that led to a full understanding of the well-posedness of several PDEs. This program seeks to build upon these successes by further developing and refining this novel toolbox to attack a wide range of problems that were beyond our reach until recently. This includes extending our understanding of well-posedness with rough, non-decaying initial data to more integrable PDEs, with models in higher spatial dimensions being of particular interest, as well as employing these new insights to the investigation of other properties, such as the behavior of ensembles of trajectories and the long-time asymptotic behavior of solutions. The overarching goal of this program is to deepen our understanding of what it means for a PDE to be completely integrable. While the complete integrability property is rare among the wealth of nonlinear dispersive PDEs, several non-integrable equations have been shown to display similar qualitative and asymptotic behavior. This leads one to suspect that integrable systems describe generic dynamics in some sense and can provide hypotheses for non-integrable equations as well. Illuminating these connections is another one of the long-term goals of this program.