A nonlinear stationary phase method for oscillatory Riemann-Hilbert problems

TL;DR

This paper extends the nonlinear steepest descent method for oscillatory Riemann-Hilbert problems to a broader class of non-analytic phase functions, using real variable techniques to analyze asymptotic behavior in integrable PDEs.

Contribution

It introduces a generalized nonlinear stationary phase method for non-analytic phases in Riemann-Hilbert problems, expanding the applicability of the Deift-Zhou approach.

Findings

Extended the nonlinear steepest descent method to more general non-analytic phases.

Provided asymptotic analysis for oscillatory Riemann-Hilbert problems in integrable systems.

Utilized real variable methods to handle non-analytic phase functions.

Abstract

We study the asymptotic behavior of oscillatory Riemann-Hilbert problems arising in the AKNS hierarchy of integrable nonlinear PDE's. Our method is based on the Deift-Zhou nonlinear steepest descent method in which the given Riemann-Hilbert problem localizes to small neighborhoods of stationary phase points. In their original work, Deift and Zhou only considered analytic phase functions. Subsequently Varzugin extended the Deift-Zhou method to a certain restricted class of non-analytic phase functions. In this paper, we extend Varzugin's method to a substantially more general class of non-analytic phase functions. In our work real variable methods play a key role.