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Zh. Mat. Fiz. Anal. Geom., 2016, Volume 12, Number 1, Pages 3–16 (Mi jmag626)  

This article is cited in 1 scientific paper (total in 1 paper)

On the form of dispersive shock waves of the Korteweg–de Vries equation

I. Egorovaa, Z. Gladkaa, G. Teschlbc

a B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Nauki Ave., Kharkiv, 61103, Ukraine
b Faculty of Mathematics University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
c International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

Abstract: We show that the long-time behavior of solutions to the Korteweg–de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schrödinger operator) depends only on the size of the step of the initial data and on the direction, $\frac{x}{t}=$const, along which we determine the asymptotic behavior of the solution. In turn, the phase shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the scattering data, and is computed explicitly via the Jacobi inversion problem.

Key words and phrases: KdV equation, steplike, dispersive shock wave.

Funding Agency Grant Number
Austrian Science Fund V120
Research supported by the Austrian Science Fund (FWF) under Grant V120.


DOI: https://doi.org/10.15407/mag12.01.003

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Bibliographic databases:

MSC: Primary 37K40, 35Q53; Secondary 33E05, 35Q15
Received: 12.10.2015
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Citation: I. Egorova, Z. Gladka, G. Teschl, “On the form of dispersive shock waves of the Korteweg–de Vries equation”, Zh. Mat. Fiz. Anal. Geom., 12:1 (2016), 3–16

Citation in format AMSBIB
\Bibitem{EgoGlaTes16}
\by I.~Egorova, Z.~Gladka, G.~Teschl
\paper On the form of dispersive shock waves of the Korteweg–de Vries equation
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2016
\vol 12
\issue 1
\pages 3--16
\mathnet{http://mi.mathnet.ru/jmag626}
\crossref{https://doi.org/10.15407/mag12.01.003}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3477947}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000368348000001}


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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. A. Rybkin, “KdV equation beyond standard assumptions on initial data”, Physica D, 365 (2018), 1–11  crossref  mathscinet  zmath  isi  scopus
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