# Asymptotics of KdV shock waves via the Riemann-Hilbert approach

## TL;DR

This paper analyzes the long-time behavior of KdV shock waves using the Riemann-Hilbert method, addressing ill-posedness issues and refining asymptotic descriptions for broader initial data classes.

## Contribution

It advances the understanding of KdV shock wave asymptotics by refining previous results and addressing ill-posedness in the Riemann-Hilbert framework.

## Key findings

- Refined asymptotics for KdV shock waves in the Whitham zone.
- Identification of ill-posedness issues in the matrix Riemann-Hilbert problem.
- Clarification of the effects of resonances and discrete spectrum on asymptotics.

## 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.