Loop groups and equations of KdV type
Publications Mathématiques de l'IHÉS, Tome 61 (1985), pp. 5-65. 
@article{PMIHES_1985__61__5_0,
     author = {Segal, Graeme and Wilson, George},
     title = {Loop groups and equations of {KdV} type},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {5--65},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {61},
     year = {1985},
     mrnumber = {783348},
     zbl = {0592.35112},
     language = {en},
     url = {http://archive.numdam.org/item/PMIHES_1985__61__5_0/}
}
TY  - JOUR
AU  - Segal, Graeme
AU  - Wilson, George
TI  - Loop groups and equations of KdV type
JO  - Publications Mathématiques de l'IHÉS
PY  - 1985
SP  - 5
EP  - 65
VL  - 61
PB  - Institut des Hautes Études Scientifiques
UR  - http://archive.numdam.org/item/PMIHES_1985__61__5_0/
LA  - en
ID  - PMIHES_1985__61__5_0
ER  - 
%0 Journal Article
%A Segal, Graeme
%A Wilson, George
%T Loop groups and equations of KdV type
%J Publications Mathématiques de l'IHÉS
%D 1985
%P 5-65
%V 61
%I Institut des Hautes Études Scientifiques
%U http://archive.numdam.org/item/PMIHES_1985__61__5_0/
%G en
%F PMIHES_1985__61__5_0
Segal, Graeme; Wilson, George. Loop groups and equations of KdV type. Publications Mathématiques de l'IHÉS, Tome 61 (1985), pp. 5-65. http://archive.numdam.org/item/PMIHES_1985__61__5_0/

[1] H. Airault, H. P. Mckean, J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure. Appl. Math. 30 (1977), 95-148. | MR | Zbl

[2] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Comm. Math. Phys. 61 (1978), 1-30. | MR | Zbl

[3] H. F. Baker, Note on the foregoing paper "Commutative ordinary differential operators", by J. L. BURCHNALL and T. W. CHAUNDY, Proc. Royal Soc. London (A) 118 (1928), 584-593. | JFM

[4] J. L. Burchnall, T. W. Chaundy, a) Commutative ordinary differential operators, Proc. London Math. Soc. 21 (1923), 420-440 ; b) Commutative ordinary differential operators, Proc. Royal Soc. London (A) 118 (1928), 557-583; c) Commutative ordinary differential operators II. The identity Pn = Qm, Proc. Royal Soc. London (A) 134 (1932), 471-485. | Zbl

[5] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations : I. Proc. Japan Acad. 57A (1981), 342-347 ; II. Ibid., 387-392 ; III. J. Phys. Soc. Japan 50 (1981), 3806-3812 ; IV. Physica 4D (1982), 343-365 ; V. Publ. RIMS, Kyoto Univ. 18 (1982), 1111-1119 ; VI. J. Phys. Soc. Japan 50 (1981), 3813-3818 ; VII. Publ. RIMS, Kyoto Univ. 18 (1982), 1077-1110. | Zbl

[6] V. G. Drinfel'D, V. V. Sokolov, Equations of Korteweg-de Vries type and simple Lie algebras, Dokl. Akad. Nauk SSSR 258 (I) (1981), 11-16 ; Soviet Math. Dokl. 23 (1981), 457-462. | MR | Zbl

[7] C. D'Souza, Compactification of generalized Jacobians, Proc. Ind. Acad. Sci. 88A (1979), 421-457. | MR | Zbl

[8] F. Ehlers, H. Knörrer, An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg-de Vries equation, Comment. Math. Helvetici 57 (1982), 1-10. | Zbl

[9] I. M. Gel'Fand, L. A. Dikii, Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (4) (1976), 13-29 (Russian), 259-273 (English). | MR | Zbl

[10] I. M. Krichever, Integration of non-linear equations by methods of algebraic geometry, Funct. Anal. Appl. 11 (1) (1977), 15-31 (Russian), 12-26 (English). | Zbl

[11] I. M. Krichever, Methods of algebraic geometry in the theory of non-linear equations, Uspekhi Mat. Nauk 32 (6) (1977), 183-208 ; Russian Math. Surveys 32 (6) (1977), 185-213. | Zbl

[12] B. A. Kupershmidt, G. Wilson, Modifying Lax equations and the second Hamiltonian structure, Inventiones Math. 62 (1981), 403-436. | MR | Zbl

[13] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979. | MR | Zbl

[14] I. Yu. Manin, Algebraic aspects of non-linear differential equations, Itogi Nauki i Tekhniki, ser. Sovremennye Problemi Matematiki 11 (1978), 5-152; J. Sov. Math. 11 (1) (1979), 1-122. | Zbl

[15] D. Mumford, Abelian varieties, Oxford University Press, 1974.

[16] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations, Proceedings of Symposium on Algebraic Geometry (M. NAGATA, ed.), Kinokuniya, Tokyo, 1978. | Zbl

[17] A. Pressley, G. Segal, Loop groups and their representations (Book in preparation ; Oxford University Press).

[18] G. Segal, Unitary representations of some infinite dimensional groups, Commun. Math. Phys. 80 (1981), 301-342. | MR | Zbl

[19] B. Simon, Notes on infinite determinants of Hilbert space operators, Adv. in Math. 24 (1977), 244-273. | MR | Zbl

[20] V. V. Sokolov, A. B. Shabat, (L, A)-pairs and a substitution of Riccati type, Funct. Anal. Appl. 14 (2) (1980), 79-80 (Russian), 148-150 (English). | MR | Zbl

[21] J.-L. Verdier, Equations différentielles algébriques, Séminaire Bourbaki (1977-1978), Exposé 512 = Lecture notes in Math. 710, 101-122. | Numdam | MR | Zbl

[22] G. Wilson, Commuting flows and conservation laws for Lax equations, Math. Proc. Camb. Phil. Soc. 86 (1979), 131-143. | MR | Zbl

[23] V. E. Zakharov, A. B. Shabat, Integration of the non-linear equations of mathematical physics by the inverse scattering method II, Funct. Anal. Appl. 13 (3) (1979), 13-22 (Russian), 166-174 (English). | MR | Zbl

[24] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques, in Modular functions of one variable, II (P. DELIGNE and W. KUYK, eds.), Lecture Notes in Math. 349, Springer, 1973. | MR | Zbl

[25] H. P. Mckean, E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143-226. | MR | Zbl

[26] M. Mulase, Geometry of soliton equations, MSRI preprint 035-83, Berkeley (1983). | MR | Zbl

[27] M. Mulase, Algebraic geometry of soliton equations I, MSRI preprint 040-83, Berkeley (1983). | MR | Zbl

[28] M. Mulase, Structure of the solution space of soliton equations, MSRI preprint 041-83, Berkeley (1983).

[29] M. Mulase, Complete integrability of the Kadomtsev-Petviashvili equation, MSRI preprint 053-83, Berkeley (1983).

[30] M. Mulase, Algebraic geometry of soliton equations, Proc. Japan Acad. 59, Ser. A (1983), 285-288. | MR | Zbl

[31] M. Mulase, Cohomological structure of solutions of soliton equations, isospectral deformation of ordinary differential operators and a characterization of Jacobian varieties, MSRI preprint 003-84-7, Berkeley (1984).

[32] M. Sato, Y. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Preprint, 13 pp. (date unknown). | Zbl

[33] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Preprint, 63 pp., Harvard University (1984). | Zbl

[34] C. J. Rego, The compactified Jacobian, Ann. Scient. Ec. Norm. Sup. 13 (1980), 211-223. | Numdam | MR | Zbl

[35] G. Wilson, Habillage et fonctions τ, C. R. Acad. Sc. Paris, 299, Sér. I, n° 13 (1984), 587-590. | MR | Zbl

[36] B. A. Dubrovin, Theta functions and non-linear equations, Uspekhi Mat. Nauk 36 (2) (1981), 11-80 ; Russian Math. Surveys 36 (2) (1981), 11-92. | MR | Zbl