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TMF, 1994, Volume 99, Number 3, Pages 471–477 (Mi tmf1612)  

This article is cited in 4 scientific papers (total in 4 papers)

Integrable and nonintegrable cases of the Lax equations with a source

V. K. Mel'nikov

Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics

Abstract: The Korteweg–de Vries equation with a source given as a Fourier integral over eigenfunctions of the so-called generating operator is considered. It is shown that depending on the choice of a basis of eigenfunctions we have the following three possibilities: 1) evolution equations for the scattering data are nonintegrable; 2) evolution equations for the scattering data are integrable but the solution of the Cauchy problem for the Korteweg–de Vries equation with a source at some $t'>t_0$ leaves the considered class of functions decreasing rapidly enough as $x\to \pm \infty$; 3) evolution equations for the scattering data are integrable and the solution of the Cauchy problem for the Korteweg–de Vries equation with a source exists at all $t>t_0$. All these possibilities are widespread and occur in other Lax equations with a source.

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English version:
Theoretical and Mathematical Physics, 1994, 99:3, 733–737

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Citation: V. K. Mel'nikov, “Integrable and nonintegrable cases of the Lax equations with a source”, TMF, 99:3 (1994), 471–477; Theoret. and Math. Phys., 99:3 (1994), 733–737

Citation in format AMSBIB
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\by V.~K.~Mel'nikov
\paper Integrable and nonintegrable cases of the Lax equations with a source
\jour TMF
\yr 1994
\vol 99
\issue 3
\pages 471--477
\mathnet{http://mi.mathnet.ru/tmf1612}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1308813}
\zmath{https://zbmath.org/?q=an:0850.35100}
\transl
\jour Theoret. and Math. Phys.
\yr 1994
\vol 99
\issue 3
\pages 733--737
\crossref{https://doi.org/10.1007/BF01017060}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PW12900016}


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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. B. Khasanov, G. U. Urazboev, “The solution of general KdV equation in a class of steplike functions”, J. Math. Sci. (N. Y.), 136:1 (2006), 3625–3640  mathnet  crossref  mathscinet  zmath  elib
    2. Khasanov, AB, “On the general Korteweg-de Vries equation with a source in the class of steplike functions”, Doklady Mathematics, 70:1 (2004), 512  mathscinet  isi
    3. A. B. Khasanov, G. U. Urazboev, “On the sine-Gordon equation with a self-consistent source”, Siberian Adv. Math., 19:1 (2009), 13–23  mathnet  crossref  mathscinet
    4. I. I. Baltaeva, G. U. Urazboev, “About the Camassa–Holm equation with a self-consistent source”, Ufa Math. J., 3:2 (2011), 10–18  mathnet  zmath
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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