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

This article is cited in 14 scientific papers (total in 14 papers)

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

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Received: 21.07.1997

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
\by V.~G.~Marikhin, A.~B.~Shabat
\paper Integrable lattices
\jour TMF
\yr 1999
\vol 118
\issue 2
\pages 217--228
\jour Theoret. and Math. Phys.
\yr 1999
\vol 118
\issue 2
\pages 173--182

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    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  isi
    3. V. E. Adler, “Discretizations of the Landau–Lifshits equation”, Theoret. and Math. Phys., 124:1 (2000), 897–908  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Marikhin, VG, “Self-similar solutions of equations of the nonlinear Schrodinger type”, Journal of Experimental and Theoretical Physics, 90:3 (2000), 553  crossref  mathscinet  adsnasa  isi  scopus  scopus  scopus
    6. Adler, VE, “On the structure of the Backlund transformations for the relativistic lattices”, Journal of Nonlinear Mathematical Physics, 7:1 (2000), 34  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    7. V. G. Marikhin, “Coulomb Gas Representation for Rational Solutions of the Painlevé Equations”, Theoret. and Math. Phys., 127:2 (2001), 646–663  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. Marikhin, VG, “Integrable systems with quadratic nonlinearity in Fourier space”, Physics Letters A, 310:1 (2003), 60  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    10. Vsevolod E. Adler, Alexey B. Shabat, “On the One Class of Hyperbolic Systems”, SIGMA, 2 (2006), 093, 17 pp.  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Kudryashov, NA, “The generalized Yablonskii-Vorob'ev polynomials and their properties”, Physics Letters A, 372:29 (2008), 4885  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    13. V. G. Marikhin, “Action as an invariant of Bäcklund transformations for Lagrangian systems”, Theoret. and Math. Phys., 184:1 (2015), 953–960  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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