Skip to main content
Log in

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Adler andJ. Moser, On a class of polynomials connected with the Korteweg-de Vries equation,Comm. Math. Phys.61 (1978), 1–30.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. F. Baker, Note on the foregoing paper “Commutative ordinary differential operators”, byJ. L. Burchnall andT. W. Chaundy,Proc. Royal Soc. London (A)118 (1928), 584–593.

    Article  Google Scholar 

  4. J. L. Burchnall, T. W. Chaundy, Commutative ordinary differential operators,Proc. London Math. Soc.21 (1923), 420–440;

    Article  Google Scholar 

  5. , Commutative ordinary differential operators,Proc. Royal Soc. London (A) 118 (1928), 557–583;

    Article  Google Scholar 

  6. , Commutative ordinary differential operators II. The identity Pn=Qm,Proc. Royal Soc. London (A) 134 (1932), 471–485.

    Article  Google Scholar 

  7. E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations: I.Proc. Japan Acad.57A (1981), 342–347;

    Google Scholar 

  8. Ibid., 387–392;

    Google Scholar 

  9. .J. Phys. Soc. Japan 50 (1981), 3806–3812;

    Article  MATH  MathSciNet  Google Scholar 

  10. .Physica 4D (1982), 343–365;

    MathSciNet  Google Scholar 

  11. .Publ. RIMS, Kyoto Univ. 18 (1982), 1111–1119;

    MATH  MathSciNet  Google Scholar 

  12. .J. Phys. Soc. Japan 50 (1981), 3813–3818;

    Article  MATH  MathSciNet  Google Scholar 

  13. .Publ. RIMS, Kyoto Univ. 18 (1982), 1077–1110.

    MATH  MathSciNet  Google Scholar 

  14. V. G. Drinfel’d, V. V. Sokolov, Equations of Korteweg-de Vries type and simple Lie algebras,Dokl. Akad. Nauk SSSR258 (1) (1981), 11–16;

    MathSciNet  Google Scholar 

  15. ,Soviet Math. Dokl. 23 (1981), 457–462.

    MATH  Google Scholar 

  16. C. D’Souza, Compactification of generalized Jacobians,Proc. Ind. Acad. Sci.88A (1979), 421–457.

    MathSciNet  Google Scholar 

  17. F. Ehlers, H. Knörrer, An algebro-geometric interpretation of the Bäcklund transformation for the Kortewegde Vries equation,Comment. Math. Helvetici57 (1982), 1–10.

    Article  MATH  Google Scholar 

  18. 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).

    MATH  MathSciNet  Google Scholar 

  19. 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).

    Article  MATH  Google Scholar 

  20. I. M. Krichever, Methods of algebraic geometry in the theory of non-linear equations,Uspekhi Mat. Nauk32 (6) (1977), 183–208;

    MATH  Google Scholar 

  21. ,Russian Math. Surveys 32 (6) (1977), 185–213.

    Article  MATH  Google Scholar 

  22. B. A. Kupershmidt, G. Wilson, Modifying Lax equations and the second Hamiltonian structure,Inventiones Math.62 (1981), 403–436.

    Article  MathSciNet  Google Scholar 

  23. I. G. Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1979.

  24. I. Yu. Manin, Algebraic aspects of non-linear differential equations,Itogi Nauki i Tekhniki, ser. Sovremennye Problemi Matematiki11 (1978), 5–152;

    MathSciNet  Google Scholar 

  25. ,J. Sov. Math. 11 (1) (1979), 1–122.

    Article  MATH  Google Scholar 

  26. D. Mumford,Abelian varieties, Oxford University Press, 1974.

  27. 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.

    Google Scholar 

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

  29. G. Segal, Unitary representations of some infinite dimensional groups,Commun. Math. Phys.80 (1981), 301–342.

    Article  MATH  Google Scholar 

  30. B. Simon, Notes on infinite determinants of Hilbert space operators,Adv. in Math.24 (1977), 244–273.

    MATH  Google Scholar 

  31. 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).

    Article  MATH  MathSciNet  Google Scholar 

  32. J.-L. Verdier, Equations différentielles algébriques,Séminaire Bourbaki (1977–1978), Exposé 512 =Lecture notes in Math.710, 101–122.

  33. G. Wilson, Commuting flows and conservation laws for Lax equations,Math. Proc. Camb. Phil. Soc.86 (1979), 131–143.

    Article  MATH  Google Scholar 

  34. 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).

    MathSciNet  Google Scholar 

  35. P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques, inModular functions of one variable, II (P. Deligne andW. Kuyk, eds.),Lecture Notes in Math.349, Springer, 1973.

  36. 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.

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Mulase, Geometry of soliton equations,MSRI preprint 035-83, Berkeley (1983).

  38. M. Mulase, Algebraic geometry of soliton equations I,MSRI preprint 040-83, Berkeley (1983).

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

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

  41. M. Mulase, Algebraic geometry of soliton equations,Proc. Japan Acad.59,Ser. A (1983), 285–288.

    Article  MATH  MathSciNet  Google Scholar 

  42. 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).

  43. M. Sato, Y. Sato,Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Preprint, 13 pp. (date unknown).

  44. T. Shiota,Characterization of Jacobian varieties in terms of soliton equations, Preprint, 63 pp., Harvard University (1984).

  45. C. J. Rego, The compactified Jacobian,Ann. Scient. Ec. Norm. Sup.13 (1980), 211–223.

    MATH  MathSciNet  Google Scholar 

  46. G. Wilson, Habillage et fonctionsτ, C. R. Acad. Sc. Paris,299, Sér. I, no 13 (1984), 587–590.

    MATH  Google Scholar 

  47. B. A. Dubrovin, Theta functions and non-linear equations,Uspekhi Mat. Nauk36 (2) (1981), 11–80;

    MathSciNet  Google Scholar 

  48. ,Russian Math. Surveys 36 (2) (1981), 11–92.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

About this article

Cite this article

Segal, G., Wilson, G. Loop groups and equations of KdV type. Publications Mathématiques de L’Institut des Hautes Scientifiques 61, 5–65 (1985). https://doi.org/10.1007/BF02698802

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02698802

Keywords

Navigation