Skip to main content

Solvability of the super KP equation and a generalization of the Birkhoff decomposition

Inventiones mathematicae volume 92, pages 1–46 (1988)Cite this article

Summary

The unique solvability of the initial value problem for the total hierarchy of the super Kadomtsev-Petviashvili system is established. To prove the existence we use a generalization of the Birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro- (i.e. pseudo-) differential operators. To show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space. The usual KP system with non-commutative coefficients is also studied. The KP system is obtained from the super KP system by reduction modulo odd variables. On the other hand, the first modified KP equation can be obtained from the super KP system by elimination of odd variables. Thus the super KP system is a natural unification of the KP system and the modified KP systems.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Atiyah, M.F.: Commentary on the article of Manin. In: Arbeitstagung Bonn 1984. Lect. Notes in Math., vol. 1111, pp. 103–109. Berlin Heidelberg New York Tokyo: Springer 1985

    Google Scholar 

  2. Connes, A.: Introduction to non commutative differential geometry. ibid.1111 3–16 (1985)

    Google Scholar 

  3. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Proc. Symp. on Non-linear Integrable Systems, RIMS, Kyoto Univ., ed. by Jimbo and Miwa, World Scientific, pp. 39–120 (1983)

  4. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math.21, 467–490 (1968)

    Google Scholar 

  5. Manin, Yu.I.: New dimensions in geometry. In: Arbeitstagung Bonn 1984. Lect. Notes in Math., vol. 1111, pp. 59–101. Berlin Heidelberg New York Tokyo: Springer 1985

    Google Scholar 

  6. Manin, Yu.I., Radul, A.O.: A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy. Commun. Math. Phys.98, 65–77 (1985)

    Google Scholar 

  7. Mulase, M.: Algebraic geometry of soliton equations. Proc. Japan Acad., Ser. A59, 285–288 (1983)

    Google Scholar 

  8. Mulase, M.: Cohomological structure in soliton equations and jacobian varieties. J. Differ. Geom.19, 403–430 (1984)

    Google Scholar 

  9. Mulase, M.: Complete integrability of the Kadomtsev-Petviashvili equation. Adv. Math.54, 57–66 (1984)

    Google Scholar 

  10. Pressley, A., Segal, G.B.: Loop groups. Oxford Univ. Press (1986)

  11. Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. In: Nonlinear PDE in Applied Science. U.S.-Japan Seminar, Tokyo, 1982. Lect. Notes in Num. Appl. Anal. vol. 5, pp. 259–271. Tokyo: Kinokuniya 1982

    Google Scholar 

  12. Segal, G.B., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. Inst. Hautes Etud. Sci.61, 5–65 (1985)

    Google Scholar 

  13. Ueno, K., Yamada, H.: A supersymmetric extension of nonlinear integrable systems. In: Hietarinta, J., Westerholm, J. (eds.). Topological and geometrical methods in field theory. World Scientific, pp. 59–72 (1986), see also: Supersymmetric extension of the Kadomtsev-Petviashvili hierarchy and the universal super Grassmann manifold. Adv. Stud. Pure Math., Kinokuniya (to appear 1987)

  14. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge University Press 1927

  15. Witten, E.: Physics and geometry (To appear in Proc. of the ICM-86, Berkeley)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    Motohico Mulase

Authors
  1. Motohico Mulase
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSF Grant No. DMS 86-03175

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mulase, M. Solvability of the super KP equation and a generalization of the Birkhoff decomposition. Invent Math 92, 1–46 (1988). https://doi.org/10.1007/BF01393991

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01393991

Keywords

  • Horizontal Section
  • Derivation Operator
  • Loop Variable
  • Unique Solvability
  • Loop Group