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

TL;DR

This paper introduces a new method based on $ar{ ext{∂}}$-steepest descent for analyzing large-time asymptotics of dispersive linear and integrable equations, providing sharper results with minimal initial data regularity.

Contribution

It develops a simplified $ar{ ext{∂}}$-steepest descent approach for dispersive equations, improving asymptotic accuracy for the nonlinear Schrödinger equation under minimal regularity.

Findings

Sharper asymptotics for the defocusing nonlinear Schrödinger equation.

Method applicable to linear and integrable dispersive equations.

Minimal regularity assumptions on initial data.

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 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.