Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces

TL;DR

This paper analyzes the long-time behavior of solutions to the defocusing modified KdV equation in weighted Sobolev spaces, employing advanced analytical methods to extend asymptotic results to lower regularity initial data.

Contribution

It introduces a novel approach combining nonlinear steepest descent and PDE techniques to establish long-time asymptotics for solutions with less regular initial conditions.

Findings

Established long-time asymptotics for MKdV in weighted Sobolev spaces.

Extended asymptotic analysis to solutions with lower regularity initial data.

Applied a combination of nonlinear steepest descent and PDE approximation methods.

Abstract

The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through $\overline{\partial}$-derivatives. To extend the asymptotics to solutions with initial data in lower regularity spaces, we apply a global approximation via PDE techniques.