4.7 Article

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

Journal

JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 372, Issue -, Pages 280-314

Publisher

ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2023.06.052

Keywords

Riemann-Hilbert problem; KdV equation; Shock wave

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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.
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.

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