A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems

TL;DR

This paper develops a $ar{ ext{d}}$-steepest descent method to analyze the long-time asymptotics of solutions to integrable equations via Riemann-Hilbert problems, covering different asymptotic regions.

Contribution

It introduces a novel $ar{ ext{d}}$-steepest descent approach for RHPs in the AKNS hierarchy, enabling detailed asymptotic analysis for solutions with $H^{1,1}( )$ initial data.

Findings

Derived formulas for long-time asymptotics in different regions

Applicable to NLS, mKdV, and higher-order integrable equations

Provides a unified framework for asymptotic analysis of integrable systems

Abstract

We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations. Our analysis is based on the $\dbar$-steepest descent method. We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(\R)$ initial data. The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painlev\'e region). The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations.