Global Dynamics of small data solutions to the Derivative Nonlinear Schrödinger equation

In this paper, we consider the derivative nonlinear Schrödinger (DNLS) equation. While the existence theory has been intensely studied, properties like dispersive estimates for the solutions have not yet been investigated. Here we address this question for the problem with small and localized data, and show that a dispersive estimate for the solution holds globally in time. For the proof of our result we use vector field methods combined with the \emph{testing by wave packets method}, whose implementation in this problem is novel.