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

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:

UDC: 517.91
MSC: Primary 58F07; Secondary 35Q20, 39A12

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
2. O. I. Bogoyavlenskii, “Algebraic constructions of certain integrable equations”, Math. USSR-Izv., 33:1 (1989), 39–65
3. O. I. Bogoyavlenskii, “A theorem on two commuting automorphisms, and integrable differential equations”, Math. USSR-Izv., 36:2 (1991), 263–279
4. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89
5. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64
6. O. I. Bogoyavlenskii, “Breaking solitons. VI. Extension of systems of hydrodynamic type”, Math. USSR-Izv., 39:2 (1992), 959–973
7. V. A. Yurko, “Integrations of nonlinear dynamic systems with the method of inverse spectral problems”, Math. Notes, 57:6 (1995), 672–675
8. Yuri B. Suris, “Integrable discretizations of the Bogoyavlensky lattices”, J Math Phys (N Y ), 37:8 (1996), 3982
9. V. A. Yurko, “On discrete operators of higher order”, Russian Math. Surveys, 51:3 (1996), 578–580
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
11. Yuri B. Suris, “Nonlocal quadratic Poisson algebras, monodromy map, and Bogoyavlensky lattices”, J Math Phys (N Y ), 38:8 (1997), 4179
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
13. V. N. Sorokin, “Completely integrable nonlinear dynamical systems of the Langmuir chains type”, Math. Notes, 62:4 (1997), 488–500
14. V. A. Yurko, “Integrable dynamical systems associated with higher-order difference operators”, Russian Math. (Iz. VUZ), 42:10 (1998), 69–79
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
16. Vladimir Sorokin, Jeannette Van Iseghem, “Matrix Hermite–Padé problem and dynamical systems”, Journal of Computational and Applied Mathematics, 122:1-2 (2000), 275
17. A Dimakis, F Müller-Hoissen, J Phys A Math Gen, 34:43 (2001), 9163
18. Jeannette Van Iseghem, “Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases”, Linear Algebra and its Applications, 384 (2004), 21
19. A.K. Svinin, “Reductions of the Volterra lattice”, Physics Letters A, 337:3 (2005), 197
20. A K Svinin, “On some class of reductions for the Itoh–Narita–Bogoyavlenskii lattice”, J Phys A Math Theor, 42:45 (2009), 454021
21. E. Parodi, “On classification of discrete, scalar-valued Poisson brackets”, Journal of Geometry and Physics, 2012
22. A. I. Aptekarev, “Integriruemye poludiskretizatsii giperbolicheskikh uravnenii – “skhemnaya” dispersiya i mnogomernaya perspektiva”, Preprinty IPM im. M. V. Keldysha, 2012, 020, 28 pp.
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