Geodesic
Legendre Transforms on a Triangular Lattice
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 1-11.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FAA_2000_34_1_a0,
     author = {V. E. Adler},
     title = {Legendre {Transforms} on a {Triangular} {Lattice}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--11},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2000_34_1_a0/}
}
TY  - JOUR
AU  - V. E. Adler
TI  - Legendre Transforms on a Triangular Lattice
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2000
SP  - 1
EP  - 11
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2000_34_1_a0/
LA  - ru
ID  - FAA_2000_34_1_a0
ER  - 
%0 Journal Article
%A V. E. Adler
%T Legendre Transforms on a Triangular Lattice
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2000
%P 1-11
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2000_34_1_a0/
%G ru
%F FAA_2000_34_1_a0
V. E. Adler. Legendre Transforms on a Triangular Lattice. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 1-11. http://geodesic.mathdoc.fr/item/FAA_2000_34_1_a0/

[1] Toda M., Teoriya nelineinykh reshetok, Mir, M., 1984 | MR

[2] Adler V. E., Shabat A. B., “Obobschennye preobrazovaniya Lezhandra”, TMF, 112:2 (1997), 179–194 | DOI | MR | Zbl

[3] Hirota R., “Nonlinear partial difference equations. II: Discrete-time Toda equation”, J. Phys. Soc. Japan, 43 (1977), 2074–2078 | DOI | MR

[4] Suris Yu. B., “Discrete time generalized Toda lattices: complete integrability and relation with relativistic Toda lattices”, Phys. Lett. A, 145 (1990), 113–119 | DOI | MR

[5] Suris Yu. B., “Bi-Hamiltonian structure of the $qd$ algorithm and new discretizations of the Toda lattice”, Phys. Lett. A, 206 (1995), 153–161 | DOI | MR | Zbl

[6] Suris Yu. B., “A discrete-time relativistic Toda lattice”, J. Phys. A, 29 (1996), 451–465 | DOI | MR | Zbl

[7] Suris Yu. B., A collection of integrable systems of the Toda type in continuous and discrete time, with $2\times 2$ Lax representations, arXiv: /solv-int/9703004

[8] Suris Yu. B., “New integrable systems related to the relativistic Toda lattice”, J. Phys. A, 30 (1997), 1745–1761 | DOI | MR | Zbl

[9] Ramani A., Grammaticos B., Satsuma J., “Integrability of multidimensional discrete systems”, Phys. Lett. A, 169 (1992), 323–328 | DOI | MR