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

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→∞. 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 outer parametrix solution for our problem does not exist for certain discrete times.