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

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 & RARR; & INFIN;. 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.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

In this paper, the asymptotic behavior of solutions to the Korteweg-de Vries equation with steplike initial data, leading to shock waves, is studied. An alternative approach is presented, which involves the direct comparison of resolvents related to the corresponding Riemann-Hilbert problems, instead of the usual argument involving a small norm Riemann-Hilbert problem. The motivation for this approach arises from the absence of an invertible holomorphic outer parametrix solution for our problem at certain discrete times.