The nonlinear steepest descent method for Riemann-Hilbert problems of low regularity

TL;DR

This paper develops a nonlinear steepest descent method tailored for Riemann-Hilbert problems with low regularity and slow decay, enabling analysis of long-time asymptotics for the mKdV equation with limited initial data.

Contribution

It introduces a new theorem for Riemann-Hilbert problems with low regularity, expanding the applicability of asymptotic analysis techniques.

Findings

Established a nonlinear steepest descent theorem for low regularity problems.

Derived long-time asymptotics for the mKdV equation with limited initial data.

Demonstrated the theorem's effectiveness through specific examples.

Abstract

We prove a nonlinear steepest descent theorem for Riemann-Hilbert problems with Carleson jump contours and jump matrices of low regularity and slow decay. We illustrate the theorem by deriving the long-time asymptotics for the mKdV equation in the similarity sector for initial data with limited decay and regularity.