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TMF, 2000, Volume 124, Number 1, Pages 62–71 (Mi tmf626)  

This article is cited in 29 scientific papers (total in 29 papers)

Multicomponent generalization of the hierarchy of the Landau–Lifshitz equation

I. Z. Golubchika, V. V. Sokolovb

a Bashkir State Pedagogical University
b Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: We construct a second-order $2N$-component integrable system (with arbitrary $N$) whose spectral parameter lies on a curve of genus $g=1+(N-3)2^{N-2}$. The odd-order flows admit $N$-component reductions, which for $N=3$ coincide with the odd-order flows of the hierarchy of the Landau–Lifshitz equation.


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Theoretical and Mathematical Physics, 2000, 124:1, 909–917

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Received: 31.01.2000

Citation: I. Z. Golubchik, V. V. Sokolov, “Multicomponent generalization of the hierarchy of the Landau–Lifshitz equation”, TMF, 124:1 (2000), 62–71; Theoret. and Math. Phys., 124:1 (2000), 909–917

Citation in format AMSBIB
\by I.~Z.~Golubchik, V.~V.~Sokolov
\paper Multicomponent generalization of the hierarchy of the Landau--Lifshitz equation
\jour TMF
\yr 2000
\vol 124
\issue 1
\pages 62--71
\jour Theoret. and Math. Phys.
\yr 2000
\vol 124
\issue 1
\pages 909--917

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    This publication is cited in the following articles:
    1. V. E. Adler, “Discretizations of the Landau–Lifshits equation”, Theoret. and Math. Phys., 124:1 (2000), 897–908  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type”, Funct. Anal. Appl., 36:3 (2002), 172–181  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Meshkov, AG, “Integrable evolution equations on the N-dimensional sphere”, Communications in Mathematical Physics, 232:1 (2002), 1  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. Skrypnyk, T, “Spin generalizations of Clebsch and Neumann integrable systems”, Journal of Physics A-Mathematical and General, 36:15 (2003), 4407  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. A. G. Meshkov, V. V. Sokolov, “Classification of Integrable Divergent $N$-Component Evolution Systems”, Theoret. and Math. Phys., 139:2 (2004), 609–622  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Skrypnyk, T, “Deformations of loop algebras and integrable systems: hierarchies of integrable equations”, Journal of Mathematical Physics, 45:12 (2004), 4578  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Skrypnyk, T, “Deformations of loop algebras and classical integrable systems: Finite-dimensional Hamiltonian systems”, Reviews in Mathematical Physics, 16:7 (2004), 823  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Sokolov, VV, “On decompositions of the loop algebra over so(3) into a sum of two subalgebras”, Doklady Mathematics, 70:1 (2004), 568  mathscinet  zmath  isi
    9. Skrypnyk, T, “'Doubled' generalized Landau-Lifshitz hierarchies and special quasigraded Lie algebras”, Journal of Physics A-Mathematical and General, 37:31 (2004), 7755  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    10. M. Yu. Balakhnev, “A class of integrable evolutionary vector equations”, Theoret. and Math. Phys., 142:1 (2005), 8–14  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. T. V. Skrypnik, “Quasigraded lie algebras, Kostant–Adler scheme, and integrable hierarchies”, Theoret. and Math. Phys., 142:2 (2005), 275–288  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Balakhnev MJ, “The vector generalization of the Landau-Lifshitz equation: Backlund transformation and solutions”, Applied Mathematics Letters, 18:12 (2005), 1363–1372  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. Anatoly G. Meshkov, Maxim Ju. Balakhnev, “Integrable Anisotropic Evolution Equations on a Sphere”, SIGMA, 1 (2005), 027, 11 pp.  mathnet  crossref  mathscinet  zmath
    14. Skrypnyk, T, “New integrable Gaudin-type systems, classical r-matrices and quasigraded Lie algebras”, Physics Letters A, 334:5–6 (2005), 390  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    15. M. Yu. Balakhnev, “Superposition formulas for integrable vector evolution equations”, Theoret. and Math. Phys., 154:2 (2008), 220–226  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    16. Balakhnev, MJ, “On a classification of integrable vectorial evolutionary equations”, Journal of Nonlinear Mathematical Physics, 15:2 (2008), 212  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    17. Gerdjikov V.S., Mikhailov A.V., Valchev T.I., “Reductions of integrable equations on A.III-type symmetric spaces”, J. Phys. A: Math. Theor., 43:43 (2010), 434015  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    18. M. Yu. Balakhnev, “First-Order Differential Substitutions for Equations Integrable on $\mathbb S^n$”, Math. Notes, 89:2 (2011), 184–193  mathnet  crossref  crossref  mathscinet  isi
    19. V. S. Gerdjikov, G. G. Grahovski, A. V. Mikhailov, T. I. Valchev, “Rational bundles and recursion operators for integrable equations on A.III-type symmetric spaces”, Theoret. and Math. Phys., 167:3 (2011), 740–750  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    20. Andrei V. Zotov, “$1+1$ Gaudin Model”, SIGMA, 7 (2011), 067, 26 pp.  mathnet  crossref  mathscinet
    21. Vladimir S. Gerdjikov, Georgi G. Grahovski, Alexander V. Mikhailov, Tihomir I. Valchev, “Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces”, SIGMA, 7 (2011), 096, 48 pp.  mathnet  crossref  mathscinet
    22. Valchev T.I., “On Certain Reductions of Integrable Equations on Symmetric Spaces”, International Workshop on Complex Structures, Integrability and Vector Fields, AIP Conference Proceedings, 1340, 2011, 154–164  crossref  mathscinet  zmath  adsnasa  isi
    23. Song Chong, Yu Jie, “The Cauchy Problem of Generalized Landau-Lifshitz Equation Into S (N)”, Sci. China-Math., 56:2 (2013), 283–300  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    24. Igonin S., van de Leur J., Manno G., Trushkov V., “Infinite-Dimensional Prolongation Lie Algebras and Multicomponent Landau-Lifshitz Systems Associated with Higher Genus Curves”, J. Geom. Phys., 68 (2013), 1–26  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    25. Sun XiaoWei, Wang YouDe, “New Geometric Flows on Riemannian Manifolds and Applications To Schrodinger-Airy Flows”, Sci. China-Math., 57:11 (2014), 2247–2272  crossref  mathscinet  zmath  isi  scopus  scopus
    26. Meshkov A. Sokolov V., “Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries”, Lett. Math. Phys., 104:3 (2014), 341–360  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    27. Skrypnyk T., “Reduction in Soliton Hierarchies and Special Points of Classical R-Matrices”, J. Geom. Phys., 130 (2018), 260–287  crossref  mathscinet  zmath  isi  scopus  scopus
    28. Valchev I T. Yanovski A.B., “Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions”, J. Nonlinear Math. Phys., 25:3 (2018), 442–461  crossref  mathscinet  zmath  isi  scopus
    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, eds. Georgiev K., Todorov M., Georgiev I., Springer International Publishing Ag, 2019, 433–446  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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