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

Mateusz Piorkowski

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arXiv · 1908.11340

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

Mateusz Piorkowski

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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.

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Mateusz Piorkowski. 2019-08-29. Parametrix problem for the Korteweg--de Vries equation with steplike initial data. https://arxiv.org/abs/1908.11340

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