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TMF, 1985, Volume 65, Number 2, Pages 271–284 (Mi tmf5099)  

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

Nonlinear model of Schrödinger type: Conservation laws, Hamiltonian structure, and complete integrability

N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, A. M. Kurbatov, V. G. Samoilenko


Abstract: A method is proposed for finding Lax type representations for nonlinear evolution (one-dimensional) equations of mathematical physics. It is shown that the Schrödinger type nonlinear model $\psi_t-i\psi_{xx}+2|\psi|^2\psi_x=0$ admits a Lax-type representation and is a Hamiltonian completely integrable dynamical system. Exact quasiperiodic (finite-gap, i.e  having only a finite number of stability bands in its spectrum) solutions of this system are obtained in terms of Riemann theta functions.

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English version:
Theoretical and Mathematical Physics, 1985, 65:2, 1154–1164

Bibliographic databases:

Received: 26.12.1984

Citation: N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, A. M. Kurbatov, V. G. Samoilenko, “Nonlinear model of Schrödinger type: Conservation laws, Hamiltonian structure, and complete integrability”, TMF, 65:2 (1985), 271–284; Theoret. and Math. Phys., 65:2 (1985), 1154–1164

Citation in format AMSBIB
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\by N.~N.~Bogolyubov (Jr.), A.~K.~Prikarpatskii, A.~M.~Kurbatov, V.~G.~Samoilenko
\paper Nonlinear model of Schr\"odinger type: Conservation laws, Hamiltonian structure, and complete integrability
\jour TMF
\yr 1985
\vol 65
\issue 2
\pages 271--284
\mathnet{http://mi.mathnet.ru/tmf5099}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=823666}
\zmath{https://zbmath.org/?q=an:0618.35113}
\transl
\jour Theoret. and Math. Phys.
\yr 1985
\vol 65
\issue 2
\pages 1154--1164
\crossref{https://doi.org/10.1007/BF01017940}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1985C929000011}


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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. N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, “Complete integrability of the nonlinear ito and Benney–Kaup systems: Gradient algorithm and lax representation”, Theoret. and Math. Phys., 67:3 (1986), 586–596  mathnet  crossref  mathscinet  zmath  isi
    2. N. N. Bogolyubov (Jr.), A. K. Prikarpatskii, “Quantum current lie algebra as the universal algebraic structure of the symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics”, Theoret. and Math. Phys., 75:1 (1988), 329–339  mathnet  crossref  mathscinet  zmath  isi
    3. Yakhshimuratov A., “The Nonlinear Schrodinger Equation with a Self-consistent Source in the Class of Periodic Functions”, Math Phys Anal Geom, 14:2 (2011), 153–169  crossref  isi
    4. A. B. Yakhshimuratov, “Integration of a higher-order nonlinear Schrödinger system with a self-consistent source in the class of periodic functions”, Theoret. and Math. Phys., 202:2 (2020), 137–149  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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