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

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

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
\Bibitem{Bog87} \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
2. O. I. Bogoyavlenskii, “The Lax representation with a spectral parameter for certain dynamical systems”, Math. USSR-Izv., 32:2 (1989), 245–268
3. O. I. Bogoyavlenskii, “Algebraic constructions of certain integrable equations”, Math. USSR-Izv., 33:1 (1989), 39–65
4. O. I. Bogoyavlenskii, “Overturning solitons in new two-dimensional integrable equations”, Math. USSR-Izv., 34:2 (1990), 245–259
5. O. I. Bogoyavlenskii, “Breaking solitons. II”, Math. USSR-Izv., 35:1 (1990), 245–248
6. I. Bakas, “The structure of theW ∞ algebra”, Comm Math Phys, 134:3 (1990), 487
7. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89
8. O. I. Bogoyavlenskii, “A theorem on two commuting automorphisms, and integrable differential equations”, Math. USSR-Izv., 36:2 (1991), 263–279
9. O. I. Bogoyavlenskii, “Breaking solitons. VI. Extension of systems of hydrodynamic type”, Math. USSR-Izv., 39:2 (1992), 959–973
10. O. I. Bogoyavlenskii, “Breaking solitons. V. Systems of hydrodynamic type”, Math. USSR-Izv., 38:3 (1992), 439–454
11. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64
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21. A Dimakis, F Müller-Hoissen, J Phys A Math Gen, 34:43 (2001), 9163
22. Maria Przybylska, “Additive generalizations of the lax equation”, Reports on Mathematical Physics, 48:3 (2001), 425
23. Jeannette Van Iseghem, “Stieltjes continued fraction and QD algorithm: scalar, vector, and matrix cases”, Linear Algebra and its Applications, 384 (2004), 21
24. Mark S Hickman, “Leading order integrability conditions for differential-difference equations”, Journal of Nonlinear Mathematical Physics, 15:1 (2008), 66
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
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
27. E. Parodi, “On classification of discrete, scalar-valued Poisson brackets”, Journal of Geometry and Physics, 2012
28. V. N. Sorokin, “Slabaya asimptotika sovmestnykh mnogochlenov Polacheka”, Preprinty IPM im. M. V. Keldysha, 2017, 026, 20 pp.
29. C.A. Evripidou, P. Kassotakis, P. Vanhaecke, “Integrable Deformations of the Bogoyavlenskij–Itoh Lotka–Volterra Systems”, Regul. Chaotic Dyn., 22:6 (2017), 721–739
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
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