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 TMF, 1999, Volume 118, Number 2, Pages 217–228 (Mi tmf695)

Integrable lattices

V. G. Marikhin, A. B. Shabat

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We propose a method for constructing integrable lattices starting from dynamic systems with two different parameterizations of the canonical variables and hence two independent Bäcklund flows. We construct integrable lattices corresponding to generalizations of the nonlinear Schrödinger equation. We discuss the Toda, Volterra, and Heisenberg models in detail. For these systems, as well as for the Landau–Lifshitz model, we obtain totally discrete Lagrangians. We also discuss the relation of these systems to the Hirota equations.

DOI: https://doi.org/10.4213/tmf695

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English version:
Theoretical and Mathematical Physics, 1999, 118:2, 173–182

Bibliographic databases:

Citation: V. G. Marikhin, A. B. Shabat, “Integrable lattices”, TMF, 118:2 (1999), 217–228; Theoret. and Math. Phys., 118:2 (1999), 173–182

Citation in format AMSBIB
\Bibitem{MarSha99} \by V.~G.~Marikhin, A.~B.~Shabat \paper Integrable lattices \jour TMF \yr 1999 \vol 118 \issue 2 \pages 217--228 \mathnet{http://mi.mathnet.ru/tmf695} \crossref{https://doi.org/10.4213/tmf695} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1702860} \zmath{https://zbmath.org/?q=an:0984.37093} \elib{https://elibrary.ru/item.asp?id=13311853} \transl \jour Theoret. and Math. Phys. \yr 1999 \vol 118 \issue 2 \pages 173--182 \crossref{https://doi.org/10.1007/BF02557310} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000079807100004} 

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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. A. B. Shabat, “Third version of the dressing method”, Theoret. and Math. Phys., 121:1 (1999), 1397–1408
2. Marikhin V.G., Shabat A.B., “Hamiltonian theory of Backlund transformations”, Optical Solitons: Theoretical Challenges and Industrial Perspectives, Centre de Physique Des Houches, no. 12, 1999, 19–29
3. V. E. Adler, “Discretizations of the Landau–Lifshits equation”, Theoret. and Math. Phys., 124:1 (2000), 897–908
4. V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125:3 (2000), 1603–1661
5. Marikhin, VG, “Self-similar solutions of equations of the nonlinear Schrodinger type”, Journal of Experimental and Theoretical Physics, 90:3 (2000), 553
6. Adler, VE, “On the structure of the Backlund transformations for the relativistic lattices”, Journal of Nonlinear Mathematical Physics, 7:1 (2000), 34
7. V. G. Marikhin, “Coulomb Gas Representation for Rational Solutions of the Painlevé Equations”, Theoret. and Math. Phys., 127:2 (2001), 646–663
8. Marikhin, VG, “Integrable systems with quadratic nonlinearity in Fourier space”, Physics Letters A, 310:1 (2003), 60
9. Ustinov, NV, “The lattice equations of the Toda type with an interaction between a few neighbourhoods”, Journal of Physics A-Mathematical and General, 37:5 (2004), 1737
10. Vsevolod E. Adler, Alexey B. Shabat, “On the One Class of Hyperbolic Systems”, SIGMA, 2 (2006), 093, 17 pp.
11. M. V. Demina, N. A. Kudryashov, “Special polynomials and rational solutions of the hierarchy of the second Painlevé equation”, Theoret. and Math. Phys., 153:1 (2007), 1398–1406
12. Kudryashov, NA, “The generalized Yablonskii-Vorob'ev polynomials and their properties”, Physics Letters A, 372:29 (2008), 4885
13. V. G. Marikhin, “Action as an invariant of Bäcklund transformations for Lagrangian systems”, Theoret. and Math. Phys., 184:1 (2015), 953–960
14. V. G. Marikhin, “Three-dimensional lattice of Bäcklund transformations of integrable cases of the Davey–Stewartson system”, Theoret. and Math. Phys., 189:3 (2016), 1718–1725
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