Sharp well-posedness for the cubic NLS and mKdV on the line

Abstract: The 1d cubic nonlinear Schrödinger equation (NLS) and the modified Korteweg-de Vries equation (mKdV) are two of the most intensively studied nonlinear dispersive equations. Not only are they important physical models, arising, for example, from the study of fluid dynamics and nonlinear optics, but they also have a rich mathematical structure: they are both members of the ZS-AKNS hierarchy of integrable equations. In this talk, we discuss an optimal well-posedness result for the cubic NLS and mKdV on the line. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, which in turn rests on the discovery of a one-parameter family of microscopic conservation laws. This is joint work with Rowan Killip and Monica Vișan.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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