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TMF, 1989, Volume 78, Number 3, Pages 345–356 (Mi tmf4790)  

This article is cited in 16 scientific papers (total in 17 papers)

Asymptotics at $t\to\infty$ of the solution to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as $x\to\pm\infty$

R. F. Bikbaev, R. A. Sharipov


Abstract: The Cauchy problem for the KdV equation is considered in the class of functions approaching at $x\to\pm\infty$ two different finite-gap solutions of this equation which correspond to the same Riemann surface. With the help of the inverse scattering method the large time asymptotics of the Cauchy problem solution is considered.

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English version:
Theoretical and Mathematical Physics, 1989, 78:3, 244–252

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Received: 29.08.1987

Citation: R. F. Bikbaev, R. A. Sharipov, “Asymptotics at $t\to\infty$ of the solution to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as $x\to\pm\infty$”, TMF, 78:3 (1989), 345–356; Theoret. and Math. Phys., 78:3 (1989), 244–252

Citation in format AMSBIB
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\by R.~F.~Bikbaev, R.~A.~Sharipov
\paper Asymptotics at~$t\to\infty$ of~the solution to~the Cauchy problem for the Korteweg--de~Vries equation in~the class of~potentials with finite-gap behavior as~$x\to\pm\infty$
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\yr 1989
\vol 78
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\pages 345--356
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=996221}
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\jour Theoret. and Math. Phys.
\yr 1989
\vol 78
\issue 3
\pages 244--252
\crossref{https://doi.org/10.1007/BF01017661}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. F. Bikbaev, “Korteweg–de Vries equation with finite-gap boundary conditions, and the Whitham deformations of Riemann surfaces”, Funct. Anal. Appl., 23:4 (1989), 257–266  mathnet  crossref  mathscinet  zmath  isi
    2. P. G. Grinevich, “Rapidly decreasing potentials on a background of finite-zone potentials and the $\partial$-problem on Riemann spaces”, Funct. Anal. Appl., 23:4 (1989), 321–322  mathnet  crossref  mathscinet  zmath  isi
    3. R. F. Bikbaev, “Distribution of magnetization in an easy-plane ferromagnet”, Theoret. and Math. Phys., 80:3 (1989), 1004–1006  mathnet  crossref  mathscinet  isi
    4. R. F. Bikbaev, “Large-time asymptotics of the solution of the nonlinear Schrödinger equation with boundary conditions of step type”, Theoret. and Math. Phys., 81:1 (1989), 1011–1017  mathnet  crossref  mathscinet  zmath  isi
    5. B. M. Levitan, A. B. Khasanov, “Estimation of the Cauchy function for finite-zone nonperiodic potentials”, Funct. Anal. Appl., 26:2 (1992), 91–98  mathnet  crossref  mathscinet  zmath  isi
    6. Theoret. and Math. Phys., 99:2 (1994), 599–605  mathnet  crossref  mathscinet  zmath  isi
    7. R. F. Bikbaev, “Complex Whitham deformations in the problems with “integrable instability””, Theoret. and Math. Phys., 104:3 (1995), 1078–1097  mathnet  crossref  mathscinet  zmath  isi
    8. A. I. Bobenko, A. M. Il'in, S. Yu. Dobrokhotov, A. R. Its, L. A. Kalyakin, V. B. Matveev, V. Yu. Novokshenov, A. B. Shabat, “Ramil' Faritovich Bikbaev (obituary)”, Russian Math. Surveys, 51:1 (1996), 129–133  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    9. V. Yu. Novokshenov, “Whitham Gap Dynamics of the Real-Valued Solution of the Sine-Gordon Equation with Finite-Gap Boundary Conditions”, Funct. Anal. Appl., 30:4 (1996), 246–256  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Egorova, I, “On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data: I. Schwartz-type perturbations”, Nonlinearity, 22:6 (2009), 1431  crossref  isi
    12. Egorova I., Teschl G., “On the Cauchy Problem for the Kortewegde Vries Equation With Steplike Finite-Gap Initial Data II. Perturbations With Finite Moments”, J Anal Math, 115 (2011), 71–101  crossref  isi
    13. I. Egorova, Z. Gladka, T. L. Lange, G. Teschl, “Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials”, Zhurn. matem. fiz., anal., geom., 11:2 (2015), 123–158  mathnet  crossref  mathscinet
    14. Kotlyarov V., Minakov A., “Modulated Elliptic Wave and Asymptotic Solitons in a Shock Problem To the Modified Korteweg-de Vries Equation”, J. Phys. A-Math. Theor., 48:30 (2015), 305201  crossref  isi
    15. I. Egorova, Z. Gladka, G. Teschl, “On the form of dispersive shock waves of the Korteweg–de Vries equation”, Zhurn. matem. fiz., anal., geom., 12:1 (2016), 3–16  mathnet  crossref  mathscinet
    16. Rustem R. Aydagulov, Alexander A. Minakov, “Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis”, SIGMA, 14 (2018), 119, 19 pp.  mathnet  crossref
    17. V. Yu. Novokshenov, “Parametricheskii rezonans v integriruemykh sistemakh i usrednenie na rimanovykh poverkhnostyakh”, Differentsialnye uravneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 163, VINITI RAN, M., 2019, 65–80  mathnet  mathscinet
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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