Geodesic
On the theory of Lie–Bäcklund transformation groups
Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 205-226 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The attempt to apply the theory of Lie groups to the case of Lie–Bäcklund transformations, which are certain tangent transformations of infinite degree, leads to an infinite-dimensional analogue of Lie's equations. This constitutes the main difficulty in any attempt to construct an analytic theory of Lie–Bäcklund transformation groups. In this paper an algebraic solution of this difficulty by means of power series is suggested. A formal theory which preserves the principal features of Lie's theory of tangent transformations is constructed. Some applications of this theory to the group theoretic study of differential equations in which the use of Lie–Bäcklund transformations is essential are considered. Bibliography: 23 titles. 
@article{SM_1980_37_2_a3,
     author = {N. Kh. Ibragimov},
     title = {On the theory of {Lie{\textendash}B\"acklund} transformation groups},
     journal = {Sbornik. Mathematics},
     pages = {205--226},
     year = {1980},
     volume = {37},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1980_37_2_a3/}
}
TY  - JOUR
AU  - N. Kh. Ibragimov
TI  - On the theory of Lie–Bäcklund transformation groups
JO  - Sbornik. Mathematics
PY  - 1980
SP  - 205
EP  - 226
VL  - 37
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1980_37_2_a3/
LA  - en
ID  - SM_1980_37_2_a3
ER  - 
%0 Journal Article
%A N. Kh. Ibragimov
%T On the theory of Lie–Bäcklund transformation groups
%J Sbornik. Mathematics
%D 1980
%P 205-226
%V 37
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1980_37_2_a3/
%G en
%F SM_1980_37_2_a3
N. Kh. Ibragimov. On the theory of Lie–Bäcklund transformation groups. Sbornik. Mathematics, Tome 37 (1980) no. 2, pp. 205-226. http://geodesic.mathdoc.fr/item/SM_1980_37_2_a3/

[1] S. Lie, “Begründung einer Invariantentheorie der Berührungstransformationen”, Math. Ann., 8:2 (1874), 215–288 | DOI | MR

[2] A. V. Bäcklund, “Einiges über Gurven- und Flächentransformationen”, Lunds Univ. $\mathring{A}$rs-Skrift, X (1874), 1–12

[3] S. Lie, “Zur Theorie der Flächen Konstanter Krümmung. III”, Archiv Math. Naturvid, V:3 (1880), 282–305

[4] A. V. Backlund, “Om Ytor med konstant negative krökning”, Lunds Univ. $\mathring{A}$rs-Skrift, XIX (1883)

[5] “Bäcklund Transformations, the Inverse Scattering Method, Solitons, and their Applications”, NSF Research Workshop on Contact Transformations, v. 515, Lecture Notes in Math., ed. R. M. Miura, Springer-Verlag, 1976 | MR

[6] H. X. Ibragimov, Gruppovye svoistva nekotorykh differentsialnykh uravnenii, izd-vo “Nauka”, Novosibirsk, 1967

[7] N. X. Ibragimov, R. L. Anderson, “Gruppy kasatelnykh preobrazovanii Li–Beklunda”, DAN SSSR, 227:3 (1976), 539–542 | MR | Zbl

[8] N. H. Ibragimov, R. L. Anderson, “Lie-Bäcklund Tangent Transformations”, J. Math. Anal. Appl., 59:1 (1977), 145–162 | DOI | MR | Zbl

[9] R. L. Anderson, M. H. Ibragimov, “Lie-Bäcklund Transformations in Applications”, SIAM, 1978

[10] N. H. Ibragimov, “Group Theoretical Nature of Conservation Theorems”, Letters in Math. Phys., 1 (1977), 423–428 | DOI | MR | Zbl

[11] 3. V. Khukhunashvili, “Simmetriya differentsialnykh uravnenii teorii polya”, Izv. VUZov, Fizika, 1971, no. 3, 95–103

[12] A. M. Vinogradov, “Mnogoznachnye resheniya i printsip klassifikatsii nelineinykh differentsialnykh uravnenii”, DAN SSSR, 210:1 (1973), 11–14 | Zbl

[13] B. A. Kupershmidt, “Lagranzhev formalizm v variatsionnom ischislenii”, Funkts. analiz, 10:2 (1976), 77–78 | MR | Zbl

[14] V. N. Shapovalov, “Simmetriya differentsialnykh uravnenii”, Izv. VUZov, 1977, no. 6, 57–70

[15] V. I. Fuschich, “Gruppovye svoistva differentsialnykh uravnenii kvantovoi mekhaniki”, Problemy asimptoticheskoi teorii nelineinykh kolebanii, izd-vo “Naukova Dumka”, Kiev, 1977, 238–246

[16] E. Noether, “Invariante Variationsprobleme”, Nachr. Kgl. Ges. Wiss. Göttingen, Math. Phys., 1918, 235–257 | Zbl

[17] M. Kuranishi, “On the local theory of continuous infinite pseudogroups. I, II”, Nagoya Math. J., 15 (1959), 225–260 ; 19 (1961), 55–91 | MR | MR | Zbl

[18] I. M. Gelfand, L. A. Dikii, “Asimptotika rezolventy shturm – liuvillevskikh uravnenii i algebra uravnenii Kortevega–de Friza”, Uspekhi matem. nauk, XXX:5(185) (1975), 67–100 | MR

[19] L. V. Ovsyannikov, Gruppovoi analiz differentsialnykh uravnenii, izd-vo “Nauka”, Moskva, 1978 | MR

[20] L. V. Ovsyannikov, Gruppovye svoistva differentsialnykh uravnenii, izd-vo SO AN SSSR, Novosibirsk, 1962 | MR

[21] G. Birkgof, Gidrodinamika, IL, Moskva, 1963

[22] N. X. Ibragimov, “Zakony sokhraneniya v gidrodinamike”, DAN SSSR, 210:6 (1973), 1307–1309 | Zbl

[23] E. D. Terentev, Yu. D. Shmyglevskii, “Polnaya sistema divergentnykh uravnenii dinamiki sovershennogo gaza”, ZhVMF, 15:6 (1975), 1535–1544 | MR