Dispersive Asymptotics for Linear and Integrable Equations by the $\overline{\partial}$ Steepest Descent Method

We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-$t$ limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of $\overline{\partial}$-problems. Expanding upon prior work of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schr\"odinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data.