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Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 6, Pages 1123–1141 (Mi izv1334)  

This article is cited in 22 scientific papers (total in 22 papers)

Integrable dynamical systems associated with the KdV equation

O. I. Bogoyavlenskii


Abstract: An isospectral deformation representation is constructed for a countable set of dynamical systems with a quadratic nonlinearity, which become the Korteweg–de Vries equation in the continuum limit. Integrable reductions of certain dynamical systems with an arbitrary degree of nonlinearity are obtained. The dynamics of the components of the scattering matrix are integrated for these infinitedimensional dynamical systems. An isospectral deformation representation is indicated for certain nonhomogeneous finite-dimensional dynamical systems. A new construction of integrable dynamical systems associated with simple Lie algebras and generalizing the discrete KdV equations is found.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 31:3, 435–454

Bibliographic databases:

Document Type: Article
UDC: 517.91
MSC: Primary 58F07; Secondary 35Q20, 39A12
Received: 15.07.1987

Citation: O. I. Bogoyavlenskii, “Integrable dynamical systems associated with the KdV equation”, Izv. Akad. Nauk SSSR Ser. Mat., 51:6 (1987), 1123–1141; Math. USSR-Izv., 31:3 (1988), 435–454

Citation in format AMSBIB
\Bibitem{Bog87}
\by O.~I.~Bogoyavlenskii
\paper Integrable dynamical systems associated with the KdV equation
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1987
\vol 51
\issue 6
\pages 1123--1141
\mathnet{http://mi.mathnet.ru/izv1334}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=933958}
\zmath{https://zbmath.org/?q=an:0679.58025|0648.58015}
\transl
\jour Math. USSR-Izv.
\yr 1988
\vol 31
\issue 3
\pages 435--454
\crossref{https://doi.org/10.1070/IM1988v031n03ABEH001084}


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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. O. I. Bogoyavlenskii, “The Lax representation with a spectral parameter for certain dynamical systems”, Math. USSR-Izv., 32:2 (1989), 245–268  mathnet  crossref  mathscinet  zmath
    2. O. I. Bogoyavlenskii, “Algebraic constructions of certain integrable equations”, Math. USSR-Izv., 33:1 (1989), 39–65  mathnet  crossref  mathscinet  zmath
    3. O. I. Bogoyavlenskii, “A theorem on two commuting automorphisms, and integrable differential equations”, Math. USSR-Izv., 36:2 (1991), 263–279  mathnet  crossref  mathscinet  zmath  adsnasa
    4. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. O. I. Bogoyavlenskii, “Breaking solitons. VI. Extension of systems of hydrodynamic type”, Math. USSR-Izv., 39:2 (1992), 959–973  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. V. A. Yurko, “Integrations of nonlinear dynamic systems with the method of inverse spectral problems”, Math. Notes, 57:6 (1995), 672–675  mathnet  crossref  mathscinet  zmath  isi  elib
    8. Yuri B. Suris, “Integrable discretizations of the Bogoyavlensky lattices”, J Math Phys (N Y ), 37:8 (1996), 3982  crossref  mathscinet  zmath  adsnasa  elib
    9. V. A. Yurko, “On discrete operators of higher order”, Russian Math. Surveys, 51:3 (1996), 578–580  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    10. Vassilios G. Papageorgiou, Frank W. Nijhoff, “On some integrable discrete-time systems associated with the Bogoyavlensky lattices”, Physica A: Statistical Mechanics and its Applications, 228:1-4 (1996), 172  crossref
    11. Yuri B. Suris, “Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices”, J Math Phys (N Y ), 38:8 (1997), 4179  crossref  mathscinet  zmath  adsnasa
    12. A. S. Osipov, “Integration of Non-Abelian Langmuir Type Lattices by the Inverse Spectral Problem Method”, Funct. Anal. Appl., 31:1 (1997), 67–70  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. V. N. Sorokin, “Completely integrable nonlinear dynamical systems of the Langmuir chains type”, Math. Notes, 62:4 (1997), 488–500  mathnet  crossref  crossref  mathscinet  zmath  isi
    14. V. A. Yurko, “Integrable dynamical systems associated with higher-order difference operators”, Russian Math. (Iz. VUZ), 42:10 (1998), 69–79  mathnet  mathscinet  zmath  elib
    15. Wen-Xiu Ma, Benno Fuchssteiner, “Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations”, J Math Phys (N Y ), 40:5 (1999), 2400  crossref  mathscinet  zmath  adsnasa  elib
    16. Vladimir Sorokin, Jeannette Van Iseghem, “Matrix Hermite–Padé problem and dynamical systems”, Journal of Computational and Applied Mathematics, 122:1-2 (2000), 275  crossref
    17. A Dimakis, F Müller-Hoissen, J Phys A Math Gen, 34:43 (2001), 9163  crossref  mathscinet  zmath  adsnasa
    18. Jeannette Van Iseghem, “Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases”, Linear Algebra and its Applications, 384 (2004), 21  crossref
    19. A.K. Svinin, “Reductions of the Volterra lattice”, Physics Letters A, 337:3 (2005), 197  crossref  elib
    20. A K Svinin, “On some class of reductions for the Itoh–Narita–Bogoyavlenskii lattice”, J Phys A Math Theor, 42:45 (2009), 454021  crossref  isi  elib
    21. E. Parodi, “On classification of discrete, scalar-valued Poisson brackets”, Journal of Geometry and Physics, 2012  crossref
    22. A. I. Aptekarev, “Integriruemye poludiskretizatsii giperbolicheskikh uravnenii – “skhemnaya” dispersiya i mnogomernaya perspektiva”, Preprinty IPM im. M. V. Keldysha, 2012, 020, 28 pp.  mathnet
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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