Commuting differential operators of rank~3, and nonlinear differential equations
Izvestiya. Mathematics , Tome 35 (1990) no. 3, pp. 629-655.

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

Complete solutions of the commutation equations of ordinary differential operators are obtained, to which there corresponds a three-dimensional vector bundle of common eigenfunctions over an elliptic curve. The deformation of the commuting pair by the Kadomtsev–Petviashvili equation is studied. The finite-zone solutions of the Kadomtsev–Petviashvili equation of rank 3 and genus 1 are explicitly expressed in terms of functional parameters satisfying a Boussinesq-type system of two evolution equations. Bibliography: 40 titles.
@article{IM2_1990_35_3_a5,
     author = {O. I. Mokhov},
     title = {Commuting differential operators of rank~3, and nonlinear differential equations},
     journal = {Izvestiya. Mathematics },
     pages = {629--655},
     publisher = {mathdoc},
     volume = {35},
     number = {3},
     year = {1990},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1990_35_3_a5/}
}
TY  - JOUR
AU  - O. I. Mokhov
TI  - Commuting differential operators of rank~3, and nonlinear differential equations
JO  - Izvestiya. Mathematics 
PY  - 1990
SP  - 629
EP  - 655
VL  - 35
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1990_35_3_a5/
LA  - en
ID  - IM2_1990_35_3_a5
ER  - 
%0 Journal Article
%A O. I. Mokhov
%T Commuting differential operators of rank~3, and nonlinear differential equations
%J Izvestiya. Mathematics 
%D 1990
%P 629-655
%V 35
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1990_35_3_a5/
%G en
%F IM2_1990_35_3_a5
O. I. Mokhov. Commuting differential operators of rank~3, and nonlinear differential equations. Izvestiya. Mathematics , Tome 35 (1990) no. 3, pp. 629-655. http://geodesic.mathdoc.fr/item/IM2_1990_35_3_a5/

[1] Ains E. L., Obyknovennye differentsialnye uravneniya, Gos. nauchno-tekhn. izd-vo Ukrainy, Kharkov, 1939

[2] Burchnall J. L., Chaundy I. W., “Commutative ordinary differential operators”, Proc. London Math. Society, Ser. 2, 21 (1923), 420–440 | DOI | Zbl

[3] Burchnall J. L., Chaundy I. W., “Commutative ordinary differential operators”, Proc. Royal Soc. London, Ser. A., 118 (1928), 557–583 | DOI | Zbl

[4] Burchnall J. L., Chaundy I. W., “Commutative ordinary differential operators”, Proc. Royal Soc. London, Ser. A., 134 (1931), 471–485 | DOI | Zbl

[5] Baker H. F., “Note on the foregoing paper “Commutative ordinary differential operators””, Proc. Royal Soc. London, Ser. A., 118 (1928), 584–593 | DOI | Zbl

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

[7] Krichever I. M., “Kommutativnye koltsa lineinykh obyknovennykh differentsialnykh operatorov”, Funkts. analiz i ego prilozh., 12:3 (1978), 20–31 | MR | Zbl

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

[9] Krichever I. M., Novikov S. P., “Golomorfnye rassloeniya i nelineinye uravneniya. Konechnozonnye resheniya ranga $2$”, Dokl. AN SSSR, 247:1 (1979), 33–36 | MR

[10] Krichever I. M., Novikov S. P., “Golomorfnye rassloeniya nad algebraicheskimi krivymi i nelineinye uravneniya”, UMN, 35:6 (1980), 47–68 | MR | Zbl

[11] Grinevich P. G., “Ratsionalnye resheniya uravnenii kommutatsii differentsialnykh operatorov”, Funkts. analiz i ego prilozh., 16:1 (1982), 19–24 | MR | Zbl

[12] Mokhov O. I., “Kommutiruyuschie obyknovennye differentsialnye operatory ranga $3$, otvechayuschie ellipticheskoi krivoi”, UMN, 37:4 (1982), 169–170 | MR | Zbl

[13] Mokhov O. I., Geometriya kommutiruyuschikh differentsialnykh operatorov ranga $3$ i gamiltonovykh potokov, Dis. $\dots$ kand. fiz.-matem. nauk, MGU, M., 1986

[14] Dehornoy P., “Operateurs differentiels et courbes elliptiques”, Compositio Mathematica, 43:1 (1981), 71–99 | MR | Zbl

[15] Diksme Zh., “Ob algebrakh Veilya”, Matematika, 13:4 (1969), 16–44

[16] Kon P., Svobodnye koltsa i ikh svyazi, Mir, M., 1975 | MR

[17] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1 (1976), 55–136 | MR | Zbl

[18] Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskii L. P., Teoriya solitonov: metod obratnoi zadachi, Nauka, M., 1980 | MR

[19] Novikov S. P., “Periodicheskaya zadacha Kortevega–de Friza, 1”, Funkts. analiz i ego prilozh., 8:3 (1974), 54–66 | MR | Zbl

[20] Krichever I. M., “Algebro-geometricheskoe postroenie uravnenii Zakharova–Shabata i ikh periodicheskikh reshenii”, Dokl. AN SSSR, 227:2 (1976), 291–294 | MR | Zbl

[21] Krichever I. M., “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, UMN, 32:6 (1977), 183–208 | MR | Zbl

[22] Krichever I. M., “Nelineinye uravneniya i ellipticheskie krivye”, Sovrem. problemy matem. (Itogi nauki i tekhn.), 23, 1983, 79–136 | MR | Zbl

[23] Svinolupov S. I., Sokolov V. V., “Ob evolyutsionnykh uravneniyakh s netrivialnymi zakonami sokhraneniya”, Funkts. analiz i ego prilozh., 16:4 (1982), 86–87 | MR | Zbl

[24] Bordag L. A., Matveev V. B., “Ob avtomodelnykh resheniyakh uravneniya Kortevega–de Vriza i potentsialakh s trivialnoi $S$-matritsei”, Teor. i matem. fizika, 34:3 (1978), 426–430 | MR | Zbl

[25] Drinfeld V. G., “O kommutativnykh podkoltsakh nekotorykh nekommutativnykh kolets”, Funkts. analiz i ego prilozh., 11:1 (1977), 15–31 | MR

[26] Novikov S. P., “Dvumernye operatory Shredingera v periodicheskikh polyakh”, Sovrem. problemy matem. (Itogi nauki i tekhn.), 23, 1983, 3–32

[27] Tyurin A. N., “Klassifikatsiya vektornykh rassloenii nad algebraicheskoi krivoi proizvolnogo roda”, Izv. AN SSSR, Ser. matem., 29:3 (1965), 657–688 | MR | Zbl

[28] Shafarevich I. R., Osnovy algebraicheskoi geometrii, Nauka, M., 1972 | MR | Zbl

[29] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR

[30] Van der Varden B. L., Algebra, Nauka, M., 1976 | MR

[31] Griffits F., Kharris Dzh., Printsipy algebraicheskoi geometrii, t. 1, Mir, M., 1982 | MR

[32] Akhiezer N. I., Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970 | MR | Zbl

[33] Sokolov V. V., “O gamiltonovosti uravneniya Krichevera–Novikova”, Dokl. AN SSSR, 277:1 (1984), 48–50 | MR

[34] Dorfman I. Ya., “Uravnenie Krichevera–Novikova i lokalnye simplekticheskie struktury”, Dokl. AN SSSR, 302:4 (1988), 792–795 | Zbl

[35] Dorfman I. Y., “Dirac structures of integrable evolution equations”, Phys. Lett. A., 125:5 (1987), 240–246 | DOI | MR

[36] Svinolupov S. I., Sokolov V. V., Yamilov R. I., “O preobrazovaniyakh Beklunda dlya integriruemykh evolyutsionnykh uravnenii”, Dokl. AN SSSR, 271:4 (1983), 802–805 | MR | Zbl

[37] Mikhailov A. V., Shabat A. B., “Usloviya integriruemosti sistem dvukh uravnenii vida $u_t=A(u)u_{xx}+F(u,u_x)$, I”, Teor. i matem. fiz., 62:2 (1985), 163–185 | MR

[38] Mikhailov A. V., Shabat A. B., “Usloviya integriruemosti sistem dvukh uravnenii vida $u_t=A(u)u_{xx}+F(u,u_x)$, II”, Teor. i matem. fiz., 66:1 (1986), 47–65 | MR

[39] Mikhailov A. V., Shabat A. B., Yamilov R. I., “O rasshirenii modulya obratimykh preobrazovanii”, Dokl. AN SSSR, 295:2 (1987), 288–291 | MR

[40] Mikhailov A. V., Shabat A. B., Yamilov R. I., “Simmetriinyi podkhod k klassifikatsii nelineinykh uravnenii. Polnye spiski integriruemykh sistem”, UMN, 42:4 (1987), 3–53 | MR