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TMF, 1997, Volume 112, Number 2, Pages 179–194 (Mi tmf1038)  

This article is cited in 25 scientific papers (total in 25 papers)

Generalized Legendre transformations

V. E. Adlera, A. B. Shabatb

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We discuss a general theory of the integrable Toda lattices which are considered as Lagrangian dynamical systems with one continuous and one discrete time. The invariance with respect to an analog of the classical Legendre transformations implies their integrability.

DOI: https://doi.org/10.4213/tmf1038

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English version:
Theoretical and Mathematical Physics, 1997, 112:2, 935–948

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Received: 04.06.1997

Citation: V. E. Adler, A. B. Shabat, “Generalized Legendre transformations”, TMF, 112:2 (1997), 179–194; Theoret. and Math. Phys., 112:2 (1997), 935–948

Citation in format AMSBIB
\by V.~E.~Adler, A.~B.~Shabat
\paper Generalized Legendre transformations
\jour TMF
\yr 1997
\vol 112
\issue 2
\pages 179--194
\jour Theoret. and Math. Phys.
\yr 1997
\vol 112
\issue 2
\pages 935--948

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    3. V. G. Marikhin, A. B. Shabat, “Integrable lattices”, Theoret. and Math. Phys., 118:2 (1999), 173–182  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. A. B. Shabat, “Third version of the dressing method”, Theoret. and Math. Phys., 121:1 (1999), 1397–1408  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. 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
    6. 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
    7. V. E. Adler, “Legendre Transforms on a Triangular Lattice”, Funct. Anal. Appl., 34:1 (2000), 1–9  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. Calogero, F, “A novel solvable many-body problem with elliptic interactions”, International Mathematics Research Notices, 2000, no. 15, 775  crossref  mathscinet  zmath  isi
    9. Bruschi, M, “Solvable and/or integrable and/or linearizable N-body problems in ordinary (three-dimensional) space. I”, Journal of Nonlinear Mathematical Physics, 7:3 (2000), 303  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    10. Krichever, I, “Elliptic analog of the Toda lattice”, International Mathematics Research Notices, 2000, no. 8, 383  crossref  mathscinet  zmath  isi
    11. 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
    12. V. E. Adler, V. G. Marikhin, A. B. Shabat, “Lagrangian Chains and Canonical Bäcklund Transformations”, Theoret. and Math. Phys., 129:2 (2001), 1448–1465  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. I. M. Krichever, S. P. Novikov, “Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles”, Russian Math. Surveys, 58:3 (2003), 473–510  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. R. I. Yamilov, “Relativistic Toda Chains and Schlesinger Transformations”, Theoret. and Math. Phys., 139:2 (2004), 623–635  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    15. 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
    16. Suris Y.B., “Discrete Lagrangian models”, Discrete Integrable Systems, Lecture Notes in Physics, 644, 2004, 111–184  crossref  mathscinet  zmath  adsnasa  isi
    17. Vsevolod E. Adler, Alexey B. Shabat, “On the One Class of Hyperbolic Systems”, SIGMA, 2 (2006), 093, 17 pp.  mathnet  crossref  mathscinet  zmath
    18. Yamilov, R, “Symmetries as integrability criteria for differential difference equations”, Journal of Physics A-Mathematical and General, 39:45 (2006), R541  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    19. Boll R., Suris Yu.B., “Non-symmetric discrete Toda systems from quad-graphs”, Appl Anal, 89:4 (2010), 547–569  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    20. Atkinson J., Joshi N., “Singular-Boundary Reductions of Type-Q Abs Equations”, Int. Math. Res. Notices, 2013, no. 7, 1451–1481  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    21. Perez Teruel G.R., “An Alternative Formulation of Classical Mechanics Based on an Analogy with Thermodynamics”, Eur. J. Phys., 34:6 (2013), 1589–1599  crossref  zmath  isi  scopus  scopus  scopus
    22. 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
    23. Adler V.E., “Integrability Test For Evolutionary Lattice Equations of Higher Order”, J. Symbolic Comput., 74 (2016), 125–139  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    24. Zhang Yu., Zhou R.-G., “A Chain of Type II and Its Exact Solutions”, Chin. Phys. Lett., 33:11 (2016), 110203  crossref  isi  scopus
    25. Suris Yu.B., “Discrete Time Toda Systems”, J. Phys. A-Math. Theor., 51:33 (2018)  crossref  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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