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

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. Our approach is slightly different from the original approach of Deift and Zhou: By isolating the dominant contributions of the critical points directly in an appropriately rescaled Riemann-Hilbert problem, we find the asymptotics using Cauchy's formula. This hopefully leads to a more transparent presentation.