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Mat. Sb., 1995, Volume 186, Number 8, Pages 133–141 (Mi msb64)  

This article is cited in 6 scientific papers (total in 6 papers)

The Dirac operator with elliptic potential

A. O. Smirnov

St. Petersburg State Academy of Aerospace Equipment Construction

Abstract: The Dirac operator with elliptic finite-gap potential
$$ -\mathrm i\begin{pmatrix}1&0
0&-1\end{pmatrix}\Psi _x +\begin{pmatrix}0&p
q&0\end{pmatrix}\Psi =\lambda\Psi . $$
is considered. An Ansatz for the Krichever curves associated with elliptic (in $x$) finite-gap solutions of the 'decomposed' non-linear Schrödinger equation
$$ \begin{cases} \mathrm ip_t+p_{xx}-2p^2q=0,
iq_t-q_{xx}+2pq^2=0 \end{cases} $$
and of the modified $KdV$ ($mKdV$) equation
$$ \begin{cases} p_t+p_{xxx}-6pqp_x=0,
q_t+q_{xxx}-6pqq_x=0. \end{cases} $$
is presented. Examples of two- and three-sheeted coverings associated with the one- and twogap Dirac potential are discussed.

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English version:
Sbornik: Mathematics, 1995, 186:8, 1213–1221

Bibliographic databases:

UDC: 517.5
MSC: Primary 35F25; Secondary 35J10, 35Q20
Received: 27.07.1994

Citation: A. O. Smirnov, “The Dirac operator with elliptic potential”, Mat. Sb., 186:8 (1995), 133–141; Sb. Math., 186:8 (1995), 1213–1221

Citation in format AMSBIB
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\by A.~O.~Smirnov
\paper The Dirac operator with elliptic potential
\jour Mat. Sb.
\yr 1995
\vol 186
\issue 8
\pages 133--141
\mathnet{http://mi.mathnet.ru/msb64}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1357361}
\zmath{https://zbmath.org/?q=an:0863.35029}
\transl
\jour Sb. Math.
\yr 1995
\vol 186
\issue 8
\pages 1213--1221
\crossref{https://doi.org/10.1070/SM1995v186n08ABEH000064}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995TX11200016}


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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. O. Smirnov, “On some set of elliptic solutions of the Boussinesq equation”, Theoret. and Math. Phys., 109:3 (1996), 1515–1522  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. O. Smirnov, “On a class of elliptic potentials of the Dirac operator”, Sb. Math., 188:1 (1997), 115–135  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Fritz Gesztesy, Rudi Weikard, “A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy”, Acta Math, 181:1 (1998), 63  crossref  mathscinet  zmath  isi
    4. Gesztesy, F, “Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - An analytic approach”, Bulletin of the American Mathematical Society, 35:4 (1998), 271  crossref  mathscinet  zmath  isi
    5. Gökçe Başar, Gerald V. Dunne, “Twisted kink crystal in the chiral Gross-Neveu model”, Phys Rev D, 78:6 (2008), 065022  crossref  isi
    6. Francisco Correa, Gerald V. Dunne, Mikhail S. Plyushchay, “The Bogoliubov–de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry”, Annals of Physics, 324:12 (2009), 2522  crossref  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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