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Mat. Sb. (N.S.), 1988, Volume 135(177), Number 2, Pages 261–268 (Mi msb1700)  

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

On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function

N. E. Firsova


Abstract: A scheme is presented for solving the Cauchy problem for the KdV equation with initial data a sum of a periodic function $p(x)$ and a rapidly decreasing function $q(x)$. The scattering theory constructed earlier by the author for the pair of operators $H_0=-d^2/dx^2+p(x)$ and $H=H_0+q(x)$ is used to solve this problem. Evolution formulas for the scattering data are found. The solution $p(x,t)$ of the KdV equation with a periodic initial condition obtained by V. A. Marchenko and S. P. Novikov is assumed known.
Bibliography: 11 titles.

Full text: PDF file (409 kB)
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English version:
Mathematics of the USSR-Sbornik, 1989, 63:1, 257–265

Bibliographic databases:

UDC: 517.946
MSC: Primary 35Q20; Secondary 34B25
Received: 11.06.1986

Citation: N. E. Firsova, “On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function”, Mat. Sb. (N.S.), 135(177):2 (1988), 261–268; Math. USSR-Sb., 63:1 (1989), 257–265

Citation in format AMSBIB
\Bibitem{Fir88}
\by N.~E.~Firsova
\paper On solution of the Cauchy problem for the Korteweg--de~Vries equation with initial data the sum of a~periodic and a~rapidly decreasing function
\jour Mat. Sb. (N.S.)
\yr 1988
\vol 135(177)
\issue 2
\pages 261--268
\mathnet{http://mi.mathnet.ru/msb1700}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=937811}
\zmath{https://zbmath.org/?q=an:0669.35106|0659.35088}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 63
\issue 1
\pages 257--265
\crossref{https://doi.org/10.1070/SM1989v063n01ABEH003272}


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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. F. Gesztesy, B. Simon, G. Teschl, “Spectral deformations of one-dimensional Schrödinger operators”, J Anal Math, 70:1 (1996), 267  crossref  mathscinet  zmath  isi
    3. 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
    4. Ag. Kh. Khanmamedov, “Transformation operators for the perturbed Hill difference equation and one of their applications”, Siberian Math. J., 44:4 (2003), 729–738  mathnet  crossref  mathscinet  zmath  isi
    5. Egorova I., Michor J., Teschl G., “Inverse Scattering Transform for the Toda Hierarchy with Quasi-Periodic Background”, Proc. Amer. Math. Soc., 135:6 (2007), 1817–1827  crossref  mathscinet  zmath  isi
    6. A. Kh. Khanmamedov, “The solution of Cauchy's problem for the Toda lattice with limit periodic initial data”, Sb. Math., 199:3 (2008), 449–458  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Khanmamedov, AK, “Initial-boundary value problem for the Volterra lattice on a half-line with zero boundary condition”, Doklady Mathematics, 78:3 (2008), 848  crossref  isi
    8. de Monvel A.B. Egorova I. Teschl G., “Inverse Scattering Theory for One-Dimensional Schrodinger Operators with Steplike Finite-Gap Potentials”, J. Anal. Math., 106 (2008), 271–316  crossref  mathscinet  zmath  isi
    9. Iryna Egorova, Katrin Grunert, Gerald Teschl, “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  elib
    10. I. Egorova, G. Teschl, “A Paley–Wiener theorem for periodic scattering with applications to the Korteweg–de Vries equation”, Zhurn. matem. fiz., anal., geom., 6:1 (2010), 21–33  mathnet  mathscinet  zmath  elib
    11. Katrin Grunert, “The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds”, Journal of Differential Equations, 250:8 (2011), 3534  crossref
    12. Iryna Egorova, Gerald Teschl, “On the Cauchy problem for the Kortewegde Vries equation with steplike finite-gap initial data II. Perturbations with finite moments”, JAMA, 115:1 (2011), 71  crossref
    13. Mikikits-Leitner A. Teschl G., “Trace Formulas for Schrodinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds”, Spectral Theory and Analysis, Operator Theory Advances and Applications, 214, ed. Janas J. Kurasov P. Laptev A. Naboko S. Stolz G., Birkhauser Verlag Ag, 2011, 107–124  mathscinet  isi
    14. 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
    15. M. G. Makhmudova, A. Kh. Khanmamedov, “Asymptotic periodic solution of the Cauchy problem for the Langmuir lattice”, Comput. Math. Math. Phys., 55:12 (2015), 2008–2013  mathnet  crossref  crossref  mathscinet  isi  elib
    16. A. V. Vestyak, H. A. Matevossian, “Behavior of the Solution of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients”, Math. Notes, 100:5 (2016), 751–754  mathnet  crossref  crossref  mathscinet  isi  elib
    17. A. V. Vestyak, H. A. Matevossian, “On the Behavior of the Solution of the Cauchy Problem for an Inhomogeneous Hyperbolic Equation with Periodic Coefficients”, Math. Notes, 102:3 (2017), 424–428  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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