Invited Speakers

Methods for Sharp Well-posedness of Completely Integrable Systems

I will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE. Using the Korteweg-de Vries equation as our prototypical model, we first discuss the notions of macroscopic and microscopic conservation laws. We will place a particular emphasis on our new approach to these topics that remains viable for solutions with little regularity and poor  spatial decay.  We will then discuss important consequences of the conservation laws, including equicontinuity and local smoothing.

In the second talk, we review the notion of well-posedness, including a discussion of how continuity properties of the data-to-solution map inform our notion of non-perturbative well-posedness.  The first-generation method of commuting flows will then be introduced and it will be shown how this method allowed us to complete the story of well-posedness for the Korteweg-de Vries equation.  We will then elaborate on why a second-generation method was needed to complete the story for the cubic Schrödinger and the modified Korteweg-de Vries equations.

The Derivative NLS on the Line

The derivative nonlinear Schrödinger equation arises as an effective equation for the dynamics of Alfvén waves in plasmas. While this model has been analyzed in depth since the 1970s, the last two years have seen significant progress on the behavior of solutions in Sobolev spaces. In this talk we discuss some of these recent results, culminating with a proof of well-posedness on the line for all initial data with finite mass. This is joint work with Rowan Killip, Maria Ntekoume, and Monica Vişan.

Invariance of White Noise for KdV on the Line

We consider the problem of invariance of white noise for KdV on the real line.  This problem was previously studied on the torus by Oh, Quastel, and Valkó.  In the real line case, white noise fails to belong to H^{-1} (the optimal space for well-posedness of KdV).  To address this difficulty, we pass through a sequence of suitable Hamiltonian approximations that are closely connected to the complete integrability of KdV.  This is joint work with R. Killip and M. Visan.