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Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 4, Pages 737–766 (Mi izv1317)  

This article is cited in 30 scientific papers (total in 30 papers)

Some constructions of integrable dynamical systems

O. I. Bogoyavlenskii


Abstract: New constructions of integrable dynamical systems are found that admit representation as Lax matrix equations. A countable set of integrable systems is constructed which in the continuous limit turn into the Korteweg–de Vries equation. For an arbitrary space $\mathscr M$ with finite measure $\mu$ and measure-preserving mapping $\alpha\colon\mathscr M\to\mathscr M$ differential equations are constructed on the space of measurable functions on $\mathscr M$. Here differentiation is with respect to time $t$ and the equations have a countable set of first integrals. Constructions are also given for first integrals of dynamical systems preserving certain differential forms, and new constructions of matrix differential equations having large families of first integrals.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 31:1, 47–75

Bibliographic databases:

UDC: 517.91
MSC: Primary 58F07; Secondary 35Q20
Received: 30.01.1987

Citation: O. I. Bogoyavlenskii, “Some constructions of integrable dynamical systems”, Izv. Akad. Nauk SSSR Ser. Mat., 51:4 (1987), 737–766; Math. USSR-Izv., 31:1 (1988), 47–75

Citation in format AMSBIB
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\by O.~I.~Bogoyavlenskii
\paper Some constructions of integrable dynamical systems
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1987
\vol 51
\issue 4
\pages 737--766
\mathnet{http://mi.mathnet.ru/izv1317}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=914858}
\zmath{https://zbmath.org/?q=an:0695.34045|0642.34048}
\transl
\jour Math. USSR-Izv.
\yr 1988
\vol 31
\issue 1
\pages 47--75
\crossref{https://doi.org/10.1070/IM1988v031n01ABEH001043}


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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. O. I. Bogoyavlenskii, “Integrable dynamical systems associated with the KdV equation”, Math. USSR-Izv., 31:3 (1988), 435–454  mathnet  crossref  mathscinet  zmath
    2. 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
    3. O. I. Bogoyavlenskii, “Algebraic constructions of certain integrable equations”, Math. USSR-Izv., 33:1 (1989), 39–65  mathnet  crossref  mathscinet  zmath
    4. O. I. Bogoyavlenskii, “Overturning solitons in new two-dimensional integrable equations”, Math. USSR-Izv., 34:2 (1990), 245–259  mathnet  crossref  mathscinet  zmath
    5. O. I. Bogoyavlenskii, “Breaking solitons. II”, Math. USSR-Izv., 35:1 (1990), 245–248  mathnet  crossref  mathscinet  zmath
    6. I. Bakas, “The structure of theW ∞ algebra”, Comm Math Phys, 134:3 (1990), 487  crossref  mathscinet  zmath  adsnasa
    7. 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
    8. 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
    9. 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
    10. O. I. Bogoyavlenskii, “Breaking solitons. V. Systems of hydrodynamic type”, Math. USSR-Izv., 38:3 (1992), 439–454  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. 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
    12. A. S. Piskunov, “A (3+1)-dimensional equation admitting a Lax representation”, Russian Acad. Sci. Izv. Math., 40:1 (1993), 225–233  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. Theoret. and Math. Phys., 92:3 (1992), 1024–1031  mathnet  crossref  mathscinet  zmath  isi
    14. 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
    15. 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
    16. 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
    17. 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
    18. A. Jipa, C. Beşliu, M. Iosif, R. Zaharia, “Some experimental results on unusual states in relativistic nuclear collisions”, Il Nuovo Cimento A Series 10, 112:3 (1999), 179  crossref
    19. YURI B. SURIS, “INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEM: LOCAL EQUATIONS OF MOTION AND THEIR HAMILTONIAN PROPERTIES”, Rev. Math. Phys, 11:06 (1999), 727  crossref
    20. Vladimir Sorokin, Jeannette Van Iseghem, “Matrix Hermite–Padé problem and dynamical systems”, Journal of Computational and Applied Mathematics, 122:1-2 (2000), 275  crossref
    21. A Dimakis, F Müller-Hoissen, J Phys A Math Gen, 34:43 (2001), 9163  crossref  mathscinet  zmath  adsnasa
    22. Maria Przybylska, “Additive generalizations of the lax equation”, Reports on Mathematical Physics, 48:3 (2001), 425  crossref
    23. Jeannette Van Iseghem, “Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases”, Linear Algebra and its Applications, 384 (2004), 21  crossref
    24. Mark S Hickman, “Leading order integrability conditions for differential-difference equations”, Journal of Nonlinear Mathematical Physics, 15:1 (2008), 66  crossref
    25. A K Svinin, “On some class of homogeneous polynomials and explicit form of integrable hierarchies of differential–difference equations”, J. Phys. A: Math. Theor, 44:16 (2011), 165206  crossref
    26. Andrei K Svinin, “On some integrable lattice related by the Miura-type transformation to the Itoh–Narita–Bogoyavlenskii lattice”, J. Phys. A: Math. Theor, 44:46 (2011), 465210  crossref
    27. E. Parodi, “On classification of discrete, scalar-valued Poisson brackets”, Journal of Geometry and Physics, 2012  crossref
    28. V. N. Sorokin, “Slabaya asimptotika sovmestnykh mnogochlenov Polacheka”, Preprinty IPM im. M. V. Keldysha, 2017, 026, 20 pp.  mathnet  crossref
    29. C.A. Evripidou, P. Kassotakis, P. Vanhaecke, “Integrable Deformations of the BogoyavlenskijItoh LotkaVolterra Systems”, Regul. Chaotic Dyn., 22:6 (2017), 721–739  mathnet  crossref  mathscinet
    30. Tobita A., Fukuda A., Ishiwata E., Iwasaki M., Nakamura Y., “Monotonic Convergence to Eigenvalues of Totally Nonnegative Matrices in An Integrable Variant of the Discrete Lotka-Volterra System”, Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing (Epasa 2015), Lecture Notes in Computational Science and Engineering, 117, eds. Sakurai T., Zhang S., Imamura T., Yamamoto Y., Kuramashi Y., Hoshi T., Springer-Verlag Berlin, 2017, 157–169  crossref  mathscinet  isi  scopus
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