Geodesic
Dispersive rarefaction wave with a large initial gradient
Ural mathematical journal, Tome 3 (2017) no. 1, pp. 33-43 Cet article a éte moissonné depuis la source Math-Net.Ru

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Consider the Cauchy problem for the Korteweg-de Vries equation with a small parameter at the highest derivative and a large gradient of the initial function. Numerical and analytical methods show that the obtained using renormalization formal asymptotics, corresponding to rarefaction waves, is an asymptotic solution of the KdV equation. The graphs of the asymptotic solutions are represented, including the case of non-monotonic initial data.
Keywords: The Korteweg-de Vries, Cauchy problem, Asymptotic behavior, Rarefaction wave. 
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Alexander E. Elbert; Sergey V. Zakharov. Dispersive rarefaction wave with a large initial gradient. Ural mathematical journal, Tome 3 (2017) no. 1, pp. 33-43. http://geodesic.mathdoc.fr/item/UMJ_2017_3_1_a2/

[1] Gurevich A.V., Pitaevskii L., “Nonstationary structure of a collisionless shock wave”, Sov. Phys.- JETP, 38:2 (1974)

[2] Gurevich A.V., Krylov A.L., “El' G.A. Breaking of a Riemann wave in dispersive hydrodynamics”, JETP Lett., 54:2 (1991), 102–107

[3] Krylov A.L., Khodorovskii V.V., El' G.A., “Evolution of a nonmonotonic perturbation in Korteweg-de Vries hydrodynamics”, JETP Lett., 56:6 (1992), 323–327 | MR

[4] Mazur N.G., “Quasiclassical asymptotics of the inverse scattering solutions of the KdV equation and the solution of Whitham's modulation equations”, Theoret. and Math. Phys., 106:1 (1996), 35–49 | DOI | MR | Zbl

[5] Khruslov E.Ya., “Asymptotics of the solutions of the Cauchy problem for the Korteweg-de Vries equation with initial data of step type”, Math. USSR-Sb., 28:2 (1976), 229–248 | DOI | MR

[6] Cohen A., “Solutions of the Korteweg-de Vries equation with steplike initial profile”, Comm. Partial Diff. Eq., 9:8 (1984), 751—806 | DOI | MR | Zbl

[7] Venakides S. Long time asymptotics of the Korteweg-de Vries equation, Transactions of AMS, 293:1 (1986), 411–419 | DOI | MR | Zbl

[8] Suleimanov B.I., “Solution of the Korteweg-de Vries equation which arises near the breaking point in problems with a slight dispersion”, JETP Lett., 58:11 (1993), 849–854 | MR

[9] Suleimanov B.I., “Asymptotics of the Gurevich–Pitaevskii universal special solution of the Korteweg–de Vries equation as $|x|\to\infty$”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 137–145 | DOI | MR

[10] Kappeler T., “Solutions of the Korteweg–de Vries equation with steplike initial data”, J. Diff. Eq., 63:3 (1986), 306–331 | DOI | MR | Zbl

[11] Bondareva I.N., “The Korteweg–de Vries equation in classes of increasing functions with prescribed asymptotics as $|x|\to\infty$”, Math. USSR-Sb, 50:1 (1985.), 125–135 | DOI | Zbl

[12] Zakharov S.V. Renormalization in the Cauchy problem for the Korteweg –de Vries equation, Theoret. and Math. Phys., 175:2 (2013), 592–595 | DOI | MR | Zbl

[13] Teodorovich E.V., “Renormalization group method in the problems of mechanics”, J. Appl. Math. Mech., 68:2 (2004), 299–326 | DOI | MR | Zbl

[14] Il'in A.M., Matching of asymptotic expansions of solutions of boundary value problems, ed. AMS, 1992, 281 pp.

[15] Zakharov S.V., “The Cauchy problem for a quasilinear parabolic equation with two small parameters”, Dokl. Math., 78:2 (2008), 769–770 | DOI | MR | Zbl

[16] Zakharov S.V. The Cauchy problem for a quasilinear parabolic equation with a large initial gradient and low viscosity, Comput. Math. Math. Phys., 50:4 (2010), 665–672 | DOI | MR | Zbl

[17] Egorova I., Gladka Z., Lange T.L., Teschl G., On the inverse scattering transform method for the Korteweg–de Vries equation with steplike initial data, Prepr., University of Vienna, Wien, 2014 | MR

[18] Egorova I., Grunert K., Teschl G., “On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data I. Schwartz-type perturbations”, Nonlinearity, 22 (2009), 1431–1457 | DOI | MR | Zbl

[19] Egorova I., Teschl G., “On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data II. Perturbations with finite moments”, J. d'Analyse Math., 115:1 (2011), 71–101 | DOI | MR | Zbl

[20] Grunert K. Teschl G., “Long-time asymptotics for the Korteweg–de Vries equation via nonlinear steepest descent”, Math. Phys. Anal. Geom., 12:3 (2009), 287–324 | DOI | MR | Zbl

[21] Kotlyarov V.P., Minakov A.M., “Riemann-Hilbert problem to the modified Korteveg–de Vries equation: Long-time dynamics of the step-like initial data”, J. Math. Phys., 51:9 (2010.), 093506 | DOI | MR | Zbl

[22] Leach J.A., Needham D.J., “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, 21:10 (2008), 2391–2408 | DOI | MR | Zbl

[23] Novokshenov V.Yu., “Time asymptotics for soliton equations in problems with step initial conditions”, J. Math. Sci. (N.Y.), 125:5 (2005), 717–749 | DOI | MR

[24] Baranetskii V.B., Kotlyarov V.P., “Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the threshold type”, Theoret. and Math. Phys., 126:2 (2001), 175–186. | DOI | MR | Zbl

[25] Brekhovskikh V.V., Gorev V.V., “Collisionless damping of soliton solutions of Korteweg–de Vries equation, the modified Korteweg - de Vries equation and nonlinear Schrödinger equation”, Izvestiya vuzov. Povolzhskiy region. Physical-mathematical sciences, 2015, no. 2, 190–202 (in Russian)

[26] Gladka Z.N., “On solutions of the Korteweg–de Vries equation with initial data of step-type”, Dop. National Academy of Sciences of Ukraine, 2 (2015) (in Russian)

[27] Gladka Z.N., “On the reflection coeffcient of the Schrö dinger operator with a smooth potential”, Dop. National Academy of Sciences of Ukraine, 9 (2014) (in Russian) | Zbl

[28] Berezin Yu.A., Karpman V.I., “Nonlinear evolution of disturbances in plasmas and other dispersive media”, JETP, 24:5 (1967.), 1049–1056.

[29] Fogaca D.A., Navarra F.S., Ferreira Filho L.G., “KdV solitons in a cold quark gluon plasma”, Physical Review D., 84 (2011), 054011 | DOI

[30] Frank Verheest, Carel Olivier, Willy A. Hereman, “Modified Korteweg–de Vries solitons at supercritical densities in two-electron temperature plasmas”, J. of Plasma Physics, 82 (2016), 905820208, 13 pp. | DOI

[31] Misra A.P., Barman Arnab, “Landau damping of Gardner solutons in a dusty bi-ion plasma”, Phys. Plasmas, 22 (2015), 073708 | DOI

[32] Dutykh D., Tobisch E., Observation of the Inverse Energy Cascade in the modified Korteweg-de Vries Equation, arXiv: 1406.3784

[33] Zakharov S.V., Elbert A.E., “Modelling compression waves with a large initial gradient in the Korteweg–de Vries hydrodynamics”, Ufa Math. J., 9:1 (2017), 41–53 | DOI

[34] Ablowitz M.J., Baldwin D.E., Hoefer M.A., “Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction”, Physical Review E, 80 (2009.), 016603 | DOI | MR

[35] Kyrylo Andreiev, Iryna Egorova, Till Luc Lange, Gerald Teschl., Rarefaction waves of the Korteweg-de Vries equation via nonlinear steepest descent, arXiv: 1602.02427 | MR