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 Funktsional. Anal. i Prilozhen., 2000, Volume 34, Issue 1, Pages 1–11 (Mi faa278)

Legendre Transforms on a Triangular Lattice

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: We show that the condition of invariance with respect to generalized Legendre transforms effectively singles out a class of integrable difference equations on a triangular lattice; these equations are discrete analogs of relativistic Toda lattices. Some of these equations are apparently new. For one of them, higher symmetries are written out and the zero curvature representation is obtained.

DOI: https://doi.org/10.4213/faa278

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English version:
Functional Analysis and Its Applications, 2000, 34:1, 1–9

Bibliographic databases:

UDC: 517.962.24

Citation: V. E. Adler, “Legendre Transforms on a Triangular Lattice”, Funktsional. Anal. i Prilozhen., 34:1 (2000), 1–11; Funct. Anal. Appl., 34:1 (2000), 1–9

Citation in format AMSBIB
\Bibitem{Adl00} \by V.~E.~Adler \paper Legendre Transforms on a Triangular Lattice \jour Funktsional. Anal. i Prilozhen. \yr 2000 \vol 34 \issue 1 \pages 1--11 \mathnet{http://mi.mathnet.ru/faa278} \crossref{https://doi.org/10.4213/faa278} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1747820} \zmath{https://zbmath.org/?q=an:0976.37043} \transl \jour Funct. Anal. Appl. \yr 2000 \vol 34 \issue 1 \pages 1--9 \crossref{https://doi.org/10.1007/BF02467062} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000087490500001} 

• http://mi.mathnet.ru/eng/faa278
• https://doi.org/10.4213/faa278
• http://mi.mathnet.ru/eng/faa/v34/i1/p1

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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. Adler, VE, “Discrete equations on planar graphs”, Journal of Physics A-Mathematical and General, 34:48 (2001), 10453
2. Bobenko, AI, “Integrable systems on quad-graphs”, International Mathematics Research Notices, 2002, no. 11, 573
3. Bobenko, AI, “Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and lax equations on the regular triangular lattice”, International Mathematics Research Notices, 2002, no. 3, 111
4. Bobenko, AI, “Hexagonal circle patterns and integrable systems: Patterns with constant angles”, Duke Mathematical Journal, 116:3 (2003), 525
5. Adler, VE, “Q(4): Integrable master equation related to an elliptic curve”, International Mathematics Research Notices, 2004, no. 47, 2523
6. Suris Y.B., “Discrete Lagrangian models”, Discrete Integrable Systems, Lecture Notes in Physics, 644, 2004, 111–184
7. Yamilov, R, “Symmetries as integrability criteria for differential difference equations”, Journal of Physics A-Mathematical and General, 39:45 (2006), R541
8. Doliwa, A, “Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices”, Journal of Mathematical Physics, 48:11 (2007), 113506
9. Boll R., Suris Yu.B., “Non-symmetric discrete Toda systems from quad-graphs”, Appl Anal, 89:4 (2010), 547–569
10. Scimiterna C., Levi D., “Three-Point Partial Difference Equations Linearizable by Local and Nonlocal Transformations”, J. Phys. A-Math. Theor., 46:2 (2013), 025205
11. Boll R., Petrera M., Suris Yu.B., “Multi-Time Lagrangian 1-Forms for Families of Backlund Transformations: Toda-Type Systems”, J. Phys. A-Math. Theor., 46:27 (2013), 275204
12. Boll R., Petrera M., Suris Yu.B., “What Is Integrability of Discrete Variational Systems?”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 470:2162 (2014), 20130550
13. Vekslerchik V.E., “Explicit solutions for a nonlinear model on the honeycomb and triangular lattices”, J. Nonlinear Math. Phys., 23:3 (2016), 399–422
14. Lam W.Y., “Discrete Minimal Surfaces: Critical Points of the Area Functional From Integrable Systems”, Int. Math. Res. Notices, 2018, no. 6, 1808–1845
15. V. E. Adler, “Integrable seven-point discrete equations and second-order evolution chains”, Theoret. and Math. Phys., 195:1 (2018), 513–528
16. V. E. Vekslerchik, “Explicit Solutions for a Nonlinear Vector Model on the Triangular Lattice”, SIGMA, 15 (2019), 028, 17 pp.
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