Geodesic
Graded Lie algebras, representation theory, integrable mappings, and integrable systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 2, pp. 251-271 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A new class of integrable mappings and chains is introduced. The corresponding $1+2$ integrable systems that are invariant under such integrable mappings are presented in an explicit form. Soliton-type solutions of these systems are constructed in terms of matrix elements of fundamental representations of semisimple $A_n$ algebras for a given group element. The possibility of generalizing this construction to the multidimensional case is discussed. 
@article{TMF_2000_122_2_a8,
     author = {A. N. Leznov},
     title = {Graded {Lie} algebras, representation theory, integrable mappings, and integrable systems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {251--271},
     year = {2000},
     volume = {122},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a8/}
}
TY  - JOUR
AU  - A. N. Leznov
TI  - Graded Lie algebras, representation theory, integrable mappings, and integrable systems
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2000
SP  - 251
EP  - 271
VL  - 122
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a8/
LA  - ru
ID  - TMF_2000_122_2_a8
ER  - 
%0 Journal Article
%A A. N. Leznov
%T Graded Lie algebras, representation theory, integrable mappings, and integrable systems
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2000
%P 251-271
%V 122
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a8/
%G ru
%F TMF_2000_122_2_a8
A. N. Leznov. Graded Lie algebras, representation theory, integrable mappings, and integrable systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 2, pp. 251-271. http://geodesic.mathdoc.fr/item/TMF_2000_122_2_a8/

[1] A. N. Leznov, “The internal symmetry group and methods of field theory for integrating exactly soluble dynamic systems”, Group Theoretical Methods in Physics, Proc. of International Seminar (Zvenigorod, 1982), eds. M. A. Markov, V. I. Manko, and A. E. Shabad, Harwood, New York, 1985, 443–457 | MR

[2] D. B. Fairlie, A. N. Leznov, Phys. Lett. A, 199 (1995), 360–364 | DOI | MR | Zbl

[3] A. N. Leznov, Two-dimensional Ultra-Toda integrable mappings and chains (Abelian case), ; А. Н. Лезнов, ТМФ, 117:1 (1998), 107–122 E-print hep-th/9703025 | DOI | MR | Zbl

[4] A. N. Leznov, I. A. Fedoseev, TMF, 53:3 (1982), 358–373 ; I. A. Fedoseev, A. N. Leznov, Phys. Lett. B, 141:1–2 (1984), 100 ; А. Н. Лезнов, М. А. Мухтаров, ТМФ, 71:1 (1987), 46–53 | MR | DOI | MR | MR

[5] A. N. Leznov, V. G. Smirnov, Lett. Math. Phys., 5 (1981), 31–36 | DOI | MR | Zbl

[6] V. B. Derjagin, A. N. Leznov, A. S. Sorin, Nucl. Phys. B, 527 (1998), 643 | DOI | MR | Zbl

[7] P. L. Eisenhart, Continuous Groups of Transformations, Princeton University Press, Princeton, NJ, 1933 | Zbl

[8] A. N. Leznov, M. V. Saveliev, Group-theoretical methods for integration of nonlinear dynamical systems, Progress in Physics, 15, Birkhauser-Verlag, Basel, 1992 | MR | Zbl

[9] V. B. Derjagin, A. N. Leznov, E. A. Yuzbashjan, The two-dimensional integrable mappings and hierarchies of integrable systems, Preprint MPI-96-40, Bonn, 1996 | MR | Zbl

[10] A. Davey, K. Stewartson, Proc. Roy. Soc. A, 338 (1974), 101–109 | DOI | MR

[11] A. N. Leznov, TMF, 42 (1980), 343–355 | MR

[12] A. N. Leznov, E. A. Yuzbashjan, Multi-soliton solution of two-dimensional matrix Davey–Stewartson equation, Preprint MPI-96-37, Bonn, 1996 ; Nucl. Phys. B, 496 [PM] (1997), 643–653 ; Lett. Math. Phys., 35 (1995), 345–349 | MR | DOI | Zbl | DOI | MR | Zbl

[13] I. Gelfand, V. Retakh, Funkts. analiz i ego prilozh., 25:2 (1991), 13–25 ; 26:4 (1992), 1–20 ; P. Etingof, I. Gelfand, V. Retakh, Math. Research Letters, 4 (1997), 413–425 | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[14] A. V. Razumov, M. V. Saveliev, Maximally Nonabelian Toda Systems, E-print hep-th/9612081 | MR

[15] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov: Metod obratnoi zadachi, Nauka, M., 1980 ; Л. А. Тахтаджян, Л. Д. Фаддеев, Гамильтонов подход в теории солитонов, Наука, М., 1986 ; П. Олвер, Приложения групп Ли к дифференциальным уравнениям, Мир, М., 1989 ; L. A. Dickey, Soliton equations and Hamiltonian systems, World Scientific, London, 1991 | MR | Zbl | MR | Zbl | MR | MR | Zbl