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TMF, 1979, Volume 38, Number 1, Pages 26–35 (Mi tmf2538)  

This article is cited in 33 scientific papers (total in 33 papers)

Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet

V. E. Zakharov, L. A. Takhtadzhyan


Abstract: The concept of gauge equivalence is introduced for nonlinear equations that can be integrated by the inverse scattering technique. It is shown that the nonlinear Schrödinger equation is equivalent to a continuous isotropic chain of Heisenberg spins.

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English version:
Theoretical and Mathematical Physics, 1979, 38:1, 17–23

Bibliographic databases:

Received: 06.02.1978

Citation: V. E. Zakharov, L. A. Takhtadzhyan, “Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet”, TMF, 38:1 (1979), 26–35; Theoret. and Math. Phys., 38:1 (1979), 17–23

Citation in format AMSBIB
\Bibitem{ZakTak79}
\by V.~E.~Zakharov, L.~A.~Takhtadzhyan
\paper Equivalence of the nonlinear Schr\"odinger equation and the equation of a~Heisenberg ferromagnet
\jour TMF
\yr 1979
\vol 38
\issue 1
\pages 26--35
\mathnet{http://mi.mathnet.ru/tmf2538}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=525848}
\transl
\jour Theoret. and Math. Phys.
\yr 1979
\vol 38
\issue 1
\pages 17--23
\crossref{https://doi.org/10.1007/BF01030253}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. S. Gerdjikov, M. I. Ivanov, P. P. Kulish, “Quadratic bundle and nonlinear equations”, Theoret. and Math. Phys., 44:3 (1980), 784–795  mathnet  crossref  mathscinet  zmath  isi
    2. B. A. Putko, “Reduction of Kählerian chiral model”, Theoret. and Math. Phys., 50:1 (1982), 69–75  mathnet  crossref  mathscinet  isi
    3. V. O. Tarasov, L. A. Takhtadzhyan, L. D. Faddeev, “Local Hamiltonians for integrable quantum models on a lattice”, Theoret. and Math. Phys., 57:2 (1983), 1059–1073  mathnet  crossref  mathscinet  isi
    4. V. E. Zakharov, A. V. Mikhailov, “Method of the inverse scattering problem with spectral parameter on an algebraic curve”, Funct. Anal. Appl., 17:4 (1983), 247–251  mathnet  crossref  mathscinet  zmath  isi
    5. V. V. Nesterenko, “Nonlinear $\sigma$ model for the Dodd–Bullough equation”, Theoret. and Math. Phys., 58:2 (1984), 126–131  mathnet  crossref  mathscinet  zmath  isi
    6. B. M. Barbashov, V. V. Nesterenko, A. M. Chervyakov, “Reduction in the model of a relativistic string for arbitrary dimension of Minkowski space”, Theoret. and Math. Phys., 59:2 (1984), 458–465  mathnet  crossref  mathscinet  zmath  isi
    7. E. V. Doktorov, M. V. Milovanov, “Connection between the Einstein–Maxwell equations and the self-duality equations for gauge fields”, Theoret. and Math. Phys., 75:3 (1988), 599–604  mathnet  crossref  mathscinet  isi
    8. I. S. Vaklev, M. I. Ivanov, “Gauge transformation and generating operators for a quadratic bundle”, Theoret. and Math. Phys., 77:1 (1988), 1044–1055  mathnet  crossref  mathscinet  isi
    9. V. D. Lipovskii, A. V. Shirokov, “Example of gauge equivalence of multidimensional integrable equations”, Funct. Anal. Appl., 23:3 (1989), 225–226  mathnet  crossref  mathscinet  zmath  isi
    10. O. I. Mokhov, E. V. Ferapontov, “Hamiltonian Pairs Associated with Skew-Symmetric Killing Tensors on Spaces of Constant Curvature”, Funct. Anal. Appl., 28:2 (1994), 123–125  mathnet  crossref  mathscinet  zmath  isi
    11. R. Myrzakulov, A. K. Danlybaeva, G. N. Nugmanova, “Geometry and multidimensional soliton equations”, Theoret. and Math. Phys., 118:3 (1999), 347–356  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. R. Balakrishnan, S. Murugesh, “Kinematics of the Three Moving Space Curves Associated with the Nonlinear Schrödinger Equation”, Theoret. and Math. Phys., 133:3 (2002), 1609–1618  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. A. V. Marshakov, “Semiclassical geometry and integrability of the ads/cft correspondence”, Theoret. and Math. Phys., 142:2 (2005), 222–236  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. A. V. Marshakov, “Matrix models, complex geometry, and integrable systems: II$^*$”, Theoret. and Math. Phys., 147:3 (2006), 777–820  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    17. Aristophanes Dimakis, Folkert Müller-Hoissen, “Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions”, SIGMA, 6 (2010), 055, 27 pp.  mathnet  crossref  mathscinet
    18. Gerdjikov V.S., Grahovski G.G., “Two Soliton Interactions of BD.I Multicomponent NLS Equations and Their Gauge Equivalent”, Application of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings, 1301, 2010, 561–572  isi
    19. Alexandar B. Yanovski, Gaetano Vilasi, “Geometric Theory of the Recursion Operators for the Generalized Zakharov–Shabat System in Pole Gauge on the Algebra $\mathrm{sl}(n,\mathbb C)$ with and without Reductions”, SIGMA, 8 (2012), 087, 23 pp.  mathnet  crossref  mathscinet
    20. Grahovski G.G., “The Generalised Zakharov-Shabat System and the Gauge Group Action”, J. Math. Phys., 53:7 (2012), 073512  crossref  isi
    21. Aminov G. Arthamonov S. Smirnov A. Zotov A., “Rational TOP and Its Classical R-Matrix”, J. Phys. A-Math. Theor., 47:30 (2014), 305207  crossref  isi
    22. Myrzakulov R. Mamyrbekova G.K. Nugmanova G.N. Yesmakhanova K.R. Lakshmanan M., “Integrable Motion of Curves in Self-Consistent Potentials: Relation To Spin Systems and Soliton Equations”, Phys. Lett. A, 378:30-31 (2014), 2118–2123  crossref  isi
    23. A. V. Mikhailov, “Formal diagonalisation of Lax–Darboux schemes”, Model. i analiz inform. sistem, 22:6 (2015), 795–817  mathnet  crossref  mathscinet  elib
    24. Zhou T. Stone M., “Solitons in a Continuous Classical Haldane-Shastry Spin Chain”, Phys. Lett. A, 379:43-44 (2015), 2817–2825  crossref  isi
    25. Yan Zh. Gegenhasi, “On a integrable deformations of Heisenberg supermagnetic model”, J. Nonlinear Math. Phys., 23:3 (2016), 335–342  crossref  mathscinet  isi  elib  scopus
    26. Demontis F. Ortenzi G. Sommacal M., “Heisenberg Ferromagnetism as An Evolution of a Spherical Indicatrix: Localized Solutions and Elliptic Dispersionless Reduction”, Electron. J. Differ. Equ., 2018, 106  isi
    27. Demontis F. Lombardo S. Sommacal M. van der Mee C. Vargiu F., “Effective Generation of Closed-Form Soliton Solutions of the Continuous Classical Heisenberg Ferromagnet Equation”, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 35–65  crossref  isi
    28. Yurov A.V. Yurov V.A., “The Landau-Lifshitz Equation, the NLS, and the Magnetic Rogue Wave as a By-Product of Two Colliding Regular “Positons””, Symmetry-Basel, 10:4 (2018), 82  crossref  isi
    29. Yanovski A.B. Valchev T.I., “Hermitian and Pseudo-Hermitian Reduction of the Gmv Auxiliary System. Spectral Properties of the Recursion Operators”, Advanced Computing in Industrial Mathematics (Bgsiam 2017), Studies in Computational Intelligence, 793, ed. Georgiev K. Todorov M. Georgiev I., Springer International Publishing Ag, 2019, 433–446  crossref  mathscinet  isi  scopus
    30. Vasilyev M. Zotov A., “On Factorized Lax Pairs For Classical Many-Body Integrable Systems”, Rev. Math. Phys., 31:6 (2019), 1930002  crossref  isi
    31. Demontis F. Ortenzi G. Sommacal M. van der Mee C., “The Continuous Classical Heisenberg Ferromagnet Equation With in-Plane Asymptotic Conditions. i. Direct and Inverse Scattering Theory”, Ric. Mat., 68:1 (2019), 145–161  crossref  isi
    32. Demontis F. Ortenzi G. Sommacal M. van der Mee C., “The Continuous Classical Heisenberg Ferromagnet Equation With in-Plane Asymptotic Conditions. II. Ist and Closed-Form Soliton Solutions”, Ric. Mat., 68:1 (2019), 163–178  crossref  isi
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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