Well-Posedness for Integrable Dispersive Partial Differential Equations

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The Korteweg-de Vries equation was introduced in the 1890s to explain the experimental observation of solitary waves on the surface of shallow channels of water. These waves travel large distances while maintaining their profile. Still more astounding is the fact that such waves undergo particle-like interactions; this prompted researchers to introduce the term "soliton" to describe them. The goal of this project is to deepen our understanding of other equations that exhibit the same phenomenology. One particular challenge that the project seeks to overcome is the fact that existing methods are poorly suited to the study of waves that are not well localized in space. The project provides research training opportunities for graduate students.The problems to be studied lie at the intersection of nonlinear dispersive PDE, completely integrable systems, and probability theory. These problems are of well-established interest and chosen both because they have resisted previous technology and because the principal investigator believes that the new techniques she helped develop make them finally accessible. Among the topics that will be investigated as part of the project are large-data sharp well-posedness for the derivative nonlinear Schrodinger equation, well-posedness for periodic and tidal completely integrable systems, constructing Gibbs dynamics for the Landau-Lifshitz model, and understanding the long-time behavior of solutions to both defocusing and focusing completely integrable 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.