Geodesic
On the long-time asymptotics for the Korteweg–de Vries equation with steplike initial data associated with rarefaction waves
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017), pp. 325-343.

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We discuss an asymptotical behavior of the rarefaction wave for the KdV equation in the region behind the wave front. The first and the second terms of the asymptotical expansion for such a solution with respect to large time were derived without detailed analysis in [1]. In the present work, we correct the formula for the second term by investigating the corresponding parametrix problem. We also study an influence of the resonance on the asymptotical behavior of the solution. 
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     title = {On the long-time asymptotics for the {Korteweg{\textendash}de} {Vries} equation with steplike initial data associated with rarefaction waves},
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K. Andreiev; I. Egorova. On the long-time asymptotics for the Korteweg–de Vries equation with steplike initial data associated with rarefaction waves. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017), pp. 325-343. http://geodesic.mathdoc.fr/item/JMAG_2017_13_a0/

[1] K. Andreiev, I. Egorova, T.-L. Lange, G. Teschl, “Rarefaction Waves of the Korteweg–de Vries Equation via Nonlinear Steepest Descent”, J. Differential Equations, 261 (2016), 5371–5410 | DOI | MR | Zbl

[2] K. Andreiev, I. Egorova, “Uniqueness of the Solution of the Riemann–Hilbert Problem for the Rarefaction Wave of the Korteweg–de Vries Equation”, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 2017, no. 11, 3–9 | DOI

[3] R. Beals, P. Deift, C. Tomei, Direct and Inverse Scattering on the Line, Mathematical Surveys and Monographs, 28, Amer. Math. Soc., Providence, RI, 1988 | DOI | MR | Zbl

[4] V.S. Buslaev, V.N. Fomin, “An Inverse Scattering Problem for One-Dimentional Schrödinger Equation on the Entire Axis”, Vestnik Leningrad. Univ., 17 (1962), 56–64 (Russian) | MR | Zbl

[5] A. Cohen, T. Kappeler, “Scattering and Inverse Scattering for Steplike Potentials in the Schrödinger Equation”, Indiana Univ. Math. J., 34 (1985), 127–180 | DOI | MR | Zbl

[6] P. Deift, X. Zhou, “A Steepest Descent Method for Oscillatory Riemann–Hilbert Problems”, Ann. of Math., 137 (1993), 295–368 | DOI | MR | Zbl

[7] I. Egorova, Z. Gladka, T.-L. Lange, G. Teschl, “Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials”, Zh. Mat. Fiz. Anal. Geom., 11 (2015), 123–158 | DOI | MR | Zbl

[8] I. Egorova, G. Teschl, “On the Cauchy Problem for the Korteweg–de Vries Equation with Steplike Finite-Gap Initial Data II. Perturbations with Finite Moments”, J. Anal. Math., 115 (2011), 71–101 | DOI | MR | Zbl

[9] K. Grunert, G. Teschl, “Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent”, Math. Phys. Anal. Geom., 12 (2009), 287–324 | DOI | MR | Zbl

[10] A.V. Gurevich, L.P. Pitaevskii, “Decay of Initial Discontinuity in the Korteweg–de Vries Equation”, JETP Lett., 17:5 (1973), 193–195

[11] E.Ya. Khruslov, “Asymptotics of the Solution of the Cauchy Problem for the Korteweg–de Vries Equation with Initial Data of Step Type”, Mat. Sb. (N.S.), 99 (141):2 (1976), 261–281 (Russian) | MR | Zbl

[12] J.A. Leach, D.J. Needham, “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 (2008), 2391–2408 | DOI | MR | Zbl

[13] J. Lenells, The Nonlinear Steepest Descent Method for Riemann–Hilbert Problems of Low Regularity, 35 pp., arXiv: 1501.05329 [nlin.SI] | MR

[14] G. Teschl, Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, Graduate Studies in Mathematics, 99, Amer. Math. Soc., Providence, RI, 2009 | DOI | MR | Zbl

[15] V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaevskii, Theory of Solitons. The Method of the Inverse Problem, Nauka, M., 1980 (Russian) | MR | Zbl