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
Section: Mathematics
Language: Russian

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
  1. Zakharov, V. E., Manakov, S. V., Novikov, S. P. & Pitaevskii, L. P. (1980). Solitons theory: Inverse problem method. Moscow: Nauka (in Russian).
  2. 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
  3. 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
  4. 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
  5. 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).
  6. 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
  7. 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