Uniqueness of the solution of the Riemann — Hilbert problem for a rarefaction wave of the Korteweg — de Vries equation
DOI:
https://doi.org/10.15407/dopovidi2017.11.003Keywords:
Korteweg — de Vries equation, rarefaction wave, Riemann — Hilbert problemAbstract
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.
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Zakharov, V. E., Manakov, S. V., Novikov, S. P. & Pitaevskii, L. P. (1980). Solitons theory: Inverse problem method. Moscow: Nauka (in Russian).
Leach, J. A. & Needham, D. J. (2008). The large–time development of the solution to an initial-value problem for the Korteweg–de Vries equation. I. Initial data has a discontinuous expansive step. Nonlinearity, 21, pp. 2391-2408. https://doi.org/10.1088/0951-7715/21/10/010
Andreiev, K., Egorova, I., Lange, T.-L. & Teschl, G. (2016). Rarefaction waves of the Korteweg — de Vries equation via nonlinear steepest descent. J. Differ. Equat., 261, pp. 5371-5410. https://doi.org/10.1016/j.jde.2016.08.009
Gladka, Z. N. (2015). On solutions of the Korteweg — de Vries equation with initial data of the step type. Dopov. Nac. akad. nauk Ukr., No. 2, pp. 7-14 (in Russian). https://doi.org/10.15407/dopovidi2015.02.007
Buslaev, V. S. & Fomin, V. N. (1962). An inverse scattering problem for one-dimentional Schrodinger equation on the entire axis. Vestn. Leningr. Univ., 17, No. 1, pp. 56-64 (in Russian).
Khruslov, E. Ya. (1976). Asymptotics of the solution of the Cauchy problem for the Korteweg — de Vries equation with initial data of step type. Math. USSR Sb., 28, pp. 229-248. https://doi.org/10.1070/SM1976v028n02ABEH001649
Egorova, I., Gladka, Z., Lange, T.-L. & Teschl, G. (2015). Inverse scattering theory for Schrödinger operators with steplike potentials. Zh. Mat. Fiz. Anal. Geom., 11, pp. 123-158. https://doi.org/10.15407/mag11.02.123
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