Uniqueness of the solution of the Riemann — Hilbert problem for a rarefaction wave of the Korteweg — de Vries equation

1Andreiev, KM, Egorova, IY
1B.I. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv
Dopov. Nac. akad. nauk Ukr. 2017, 11:3-9
https://doi.org/10.15407/dopovidi2017.11.003
Section: Mathematics
Language: Russian
Abstract: 

An essential aspect in the asymptotic analysis of solutions for nonlinear completely integrable equations by the method of steepest descent is the study of the uniqueness of the corresponding Riemann–Hilbert problem. We establish the uniqueness of the solution for the Riemann — Hilbert problem associated the left scattering da ta for the Korteweg — de Vries equation with the steplike initial data, which correspond to a rarefaction wave. Such a problem allows us to investigate the asymptotic behavior of the solution behind the back wave front. The proof of the uniqueness is done for the nonresonant and resonant cases.

Keywords: Korteweg — de Vries equation, rarefaction wave, Riemann — Hilbert problem
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