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This paper has been accepted for publication in the Fields Institute Communications. To download a preprint of this paper, just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1812625. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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

Momar Dieng and Kenneth D. T.-R. McLaughlin

Department of Mathematics, University of Arizona

Peter D. Miller

Department of Mathematics, University of Michigan

Abstract:

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 Schrödinger 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.