Integrable differential equations and coverings of elliptic curves
Izvestiya. Mathematics , Tome 22 (1984) no. 2, pp. 357-377.

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

An algebraic theory is presented for differential equations with operator sheaf of elliptic type (in particular, with rational sheaf of operators); this theory includes local conservation laws, Bäcklund–Darboux transformations, and algebro-geometric (e.g., multisoliton) solutions. Bibliography: 27 titles.
@article{IM2_1984_22_2_a9,
     author = {I. V. Cherednik},
     title = {Integrable differential equations and coverings of elliptic curves},
     journal = {Izvestiya. Mathematics },
     pages = {357--377},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {1984},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a9/}
}
TY  - JOUR
AU  - I. V. Cherednik
TI  - Integrable differential equations and coverings of elliptic curves
JO  - Izvestiya. Mathematics 
PY  - 1984
SP  - 357
EP  - 377
VL  - 22
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a9/
LA  - en
ID  - IM2_1984_22_2_a9
ER  - 
%0 Journal Article
%A I. V. Cherednik
%T Integrable differential equations and coverings of elliptic curves
%J Izvestiya. Mathematics 
%D 1984
%P 357-377
%V 22
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a9/
%G en
%F IM2_1984_22_2_a9
I. V. Cherednik. Integrable differential equations and coverings of elliptic curves. Izvestiya. Mathematics , Tome 22 (1984) no. 2, pp. 357-377. http://geodesic.mathdoc.fr/item/IM2_1984_22_2_a9/

[1] Zakharov V. E., Mikhailov A. V., “Relyativistski-invariantnye dvumernye modeli teorii polya, integriruemye metodom obratnoi zadachi”, Zh. eksperim. i teor. fiz., 74:6 (1978), 1953–1973 | MR

[2] Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskii L. P., Teoriya solitonov, Nauka, M., 1980 | MR

[3] Sklyanin E. K., On complete integrability of the Landau–Lifshitz equation, Preprint, LOMI E-3-1979

[4] Cherednik I. V., “Ob integriruemosti uravneniya dvumernogo asimmetrichnogo kiralnogo $O(3)$-polya i ego kvantovogo analoga”, Yadernaya fizika, 33:1 (1981), 278–282

[5] Gelfand I. M., Dikii L. A., “Rezolventa i gamiltonovy sistemy”, Funkts. analiz, 11:2 (1977), 11–27 | MR

[6] Cherednik I. V., “Algebraicheskie aspekty dvumernykh kiralnykh polei. II”, Itogi nauki i tekhn. Ser. Algebra, topol., geom., 18, VINITI AN SSSR, M., 1981, 73–150 | MR | Zbl

[7] Zakharov V. E., Shabat A. B., “Integrirovanie nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi, II”, Funkts. analiz, 13:3 (1979), 13–22 | MR | Zbl

[8] Cherednik I. V., “O resheniyakh algebraicheskogo tipa asimmetrichnykh differentsialnykh uravnenii”, Funkts. analiz, 15:3 (1981), 93–94 | MR

[9] Krichever I. M., “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funkts. analiz, 11:1 (1977), 15–31 | Zbl

[10] Cherednik I. V., “O konechnozonnykh resheniyakh uravneniya dualnosti nad $S^4$ i dvumernykh relyativistski-invariantnykh sistem”, DAN SSSR, 246:3 (1979), 575–578 | MR

[11] Belavin A. A., “Diskretnye gruppy i integriruemost kvantovykh sistem”, Funkts. analiz, 14:4 (1980), 18–26 | MR | Zbl

[12] Cherednik I. V., “O svoistvakh faktorizovannykh $S$-matrits v ellipticheskikh funktsiyakh”, Yadernaya fizika, 36:2(8) (1982), 549–557 | MR

[13] Leng S., Vvedenie v algebraicheskie i abelevy funktsii, Mir, M., 1976 | MR | Zbl

[14] Faddeev L. D., “Obratnaya zadacha kvantovoi teorii rasseyaniya, II”, Sovr. problemy mat., 3, VINITI AN SSSR, M., 1974, 93–180

[15] Reiman A. G., Semenov-Tyan-Shanskii M. A., “Semeistvo gamiltonovykh struktur, ierarkhiya gamiltonianov i reduktsiya dlya matrichnykh differentsialnykh operatorov pervogo poryadka”, Funkts. analiz, 14:2 (1980), 77–78 | MR | Zbl

[16] Narasimkhan M. S., Sheshadri K. S., “Stabilnye i unitarnye rassloeniya na rimanovoi poverkhnosti”, Matematika, 13:1 (1969), 27–52

[17] Atiyah M. F., “Vector bundles over an elliptic curve”, Proc. London Math. Soc., 7 (1957), 414–452 | DOI | MR | Zbl

[18] Cherednik I. V., “Lokalnye zakony sokhraneniya glavnykh kiralnykh polei”, Teor. mat. fiz., 38:2 (1979), 179–185 | MR

[19] Ogielski A. T., Prasad H. K., Sinha A., Chau Wang Ling-Lie, “Bäcklund transformations and local conservation laws for principal chiral fields”, Phys. Letters, 91B:3,4 (1980), 387–391 | MR

[20] Cepp Zh., Algebraicheskie gruppy i polya klassov, Mir, M., 1968

[21] Krichever I. M., Novikov S. P., “Golomorfnye rassloeniya nad rimanovymi poverkhnostyami i uravnenie KP, I”, Funkts. analiz, 12:4 (1978), 41–52 | MR | Zbl

[22] Matveev V. B., “Darboux transformations and nonlinear equations”, Compt. Rend. RCP, 264 (1980), 247–264, Paris, 1980, ed. CNRS

[23] Cherednik I. V., “Differentsialnye uravneniya dlya funktsii Beikera–Akhiezera algebraicheskikh krivykh”, Funkts. analiz, 12:3 (1978), 45–54 | MR | Zbl

[24] Wilson G., “On two constructions of conservation laws for Lax equations”, Quart. J. Math. Oxford (2), 32 (1981), 491–512 | DOI | MR | Zbl

[25] Flaschka H., Construction of conservation laws for Lax equations: comments on a paper by G. Wilson, Preprint Univ. Arisona, 1982 | MR

[26] Choodnovsky D. V., Choodnovsky G. V., Hamiltonian structure of isospectral deformation equation, Preprint DPh-T-80-127, 1980

[27] Wahlquist H., “Backlund transformations of potentials of the Korteweg–de Vries equation”, Lect. Notes in Math., 515, 1972, 162–173 | MR