Asymptotics of KdV shock waves via the Riemann-Hilbert approach

Abstract

This paper discusses some general aspects and techniques associated with the long-time asymptotics of steplike solutions of the Korteweg--de Vries (KdV) equation via vector Riemann--Hilbert problems. We also elaborate on an ill-posedness of the matrix Riemann--Hilbert problem for the KdV case in the class of matrices with square integrable singularities. Furthermore, we refine the asymptotics for the shock wave in the Whitham zone derived previously and rigorously justify it for a more general class of initial data. In particular, we clarify the influence of resonances and of the discrete spectrum on the leading asymptotics.