Parametrix problem for the Korteweg--de Vries equation with steplike initial data

TL;DR

This paper investigates the long-time behavior of solutions to the Korteweg--de Vries equation with steplike initial data, introducing a novel approach based on resolvent comparison to analyze shock wave formation.

Contribution

It presents an alternative method to the Riemann--Hilbert problem for studying KdV asymptotics, addressing cases where a global parametrix does not exist.

Findings

Describes shock wave formation between asymptotic and soliton regions.

Develops a resolvent comparison technique for KdV asymptotics.

Provides insights into the limitations of existing Riemann--Hilbert methods.

Abstract

In this paper we study the asymptotics of solutions to the Korteweg--de Vries equation with steplike initial data, which lead to shock waves in the region between the asymptotically constant region and the soliton region, as $t \rightarrow \infty$. To achieve this, we present an alternative approach to the usual argument involving a small norm Riemann--Hilbert problem, which is based instead on the direct comparison of resolvents related to the corresponding Riemann--Hilbert problems. The motivation for this approach stems from the fact that an invertible holomorphic global parametrix solution for our problem does not exist for certain discrete times.