Geodesic
Action as an invariant of Bäcklund transformations for Lagrangian systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 1, pp. 71-78

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

We construct a general theory of Bäcklund transformations for Lagrangian systems under the condition that the action is preserved by these transformations. We discuss the known Bäcklund transformations for classical soliton equations from the standpoint of this approach and obtain a new Bäcklund transformation for the Tzitzéica equation.
Keywords: integrable system, canonical transformation. 
V. G. Marikhin. Action as an invariant of Bäcklund transformations for Lagrangian systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 1, pp. 71-78. http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a3/
@article{TMF_2015_184_1_a3,
     author = {V. G. Marikhin},
     title = {Action as an~invariant of {B\"acklund} transformations for {Lagrangian} systems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {71--78},
     year = {2015},
     volume = {184},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a3/}
}
TY  - JOUR
AU  - V. G. Marikhin
TI  - Action as an invariant of Bäcklund transformations for Lagrangian systems
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2015
SP  - 71
EP  - 78
VL  - 184
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a3/
LA  - ru
ID  - TMF_2015_184_1_a3
ER  - 
%0 Journal Article
%A V. G. Marikhin
%T Action as an invariant of Bäcklund transformations for Lagrangian systems
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2015
%P 71-78
%V 184
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a3/
%G ru
%F TMF_2015_184_1_a3

[1] V. E. Adler, V. G. Marikhin, A. B. Shabat, TMF, 129:2 (2001), 163–183 | DOI | DOI | MR | Zbl

[2] V. G. Marikhin, A. B. Shabat, “Hamiltonian theory of Bäcklund transformations”, Optical Solitons: Theoretical Challenges and Industrial Perspectives (Les Houches Workshop, September 28 – October 2, 1998), Centre de Physique des Houches, 12, eds. V. E. Zakharov, S. Wabnitz, Springer, Berlin, 1999, 19–29 | DOI

[3] V. G. Marikhin, Pisma v ZhETF, 66:11 (1997), 673–68 | DOI

[4] A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, UMN, 42:4(256) (1987), 3–53 | DOI | MR | Zbl

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

[6] A. N. Leznov, A. B. Shabat, R. I. Yamilov, Phys. Lett. A, 174:5–6 (1993), 397–402 | DOI | MR

[7] R. I. Yamilov, “Classification of Toda type scalar lattices”, Nonlinear Evolution Equations and Dynamical Systems (NEEDS'92) (Dubna, Russia, 6–17 July, 1992), eds. V. G. Makhankov, O. Pashaev, I. Puzynin, World Sci., Singapore, 1993, 423–431 | MR

[8] V. G. Marikhin, A. B. Shabat, TMF, 118:2 (1999), 217–228 | DOI | DOI | MR | Zbl

[9] A. B. Shabat, R. I. Yamilov, Algebra i analiz, 2:2 (1990), 183–208 | MR | Zbl

[10] V. E. Adler, TMF, 124:1 (2000), 48–61 | DOI | DOI | MR | Zbl

[11] E. K. Sklyanin, Funkts. analiz i ego pril., 16:4 (1982), 27–34 | DOI | MR | Zbl

[12] A. P. Veselov, A. B. Shabat, Funkts. analiz i ego pril., 27:2 (1993), 1–21 | DOI | MR | Zbl

[13] V. E. Adler, Internat. Math. Res. Notes, 1998, no. 1, 1–4 | DOI | MR | Zbl

[14] M. G. Tzitzéica, Rendiconti del Circolo Matematico di Palermo, 25:1 (1907), 180–187 | DOI

[15] A. V. Zhiber, A. B. Shabat, Dokl. AN SSSR, 247:5 (1979), 1103–1107 | MR

[16] S. S. Safin, R. A. Sharipov, TMF, 95:1 (1993), 146–159 | DOI | MR | Zbl

[17] A. V. Mikhailov, V. V. Sokolov, ustnoe soobschenie, 2015