Skip to main content
SpringerLink
Log in

The Riemann-Hilbert Decomposition and the KP Hierarchy

  • Conference paper
Vertex Operators in Mathematics and Physics

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 3))

Abstract

By the Riemann-Hilbert (RH) problem, we mean the following problem. “For a given analytic curve C in ℙ1, and for a given matrix function H(ζ) (ζ ∈ ℙ1) which is holomorphic and invertible on C, find the decomposition of H(ζ) into a product of matrix functions, X±(ζ)

$$ {\rm H}\left( \zeta \right) = {\rm X}\_{\left( \zeta \right)^{{ - 1}}}{{\rm X}_{ + }}\left( \zeta \right) $$
((1))

where X+(ζ) (resp. X-(ζ)) is holomorphic and invertible in the inner (resp. outer) domain of C.”

Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. Ueno and K. Takasaki: Toda Lattice Hierarchy I. Proc. Japan Aca., 59A 167–170

    Google Scholar 

  2. K. Ueno and K. Takasaki: Toda Lattice Hierarchy II. Proc. Japan Aca., 59A 215–218

    Google Scholar 

  3. K. Ueno and K. Takasaki: Toda Lattice Hierarchy. RIMS preprint 425 (1983) (to appear in Advanced Study in Pure Math.)

    Google Scholar 

  4. K. Ueno and Y. Nakamure; Transformation Theory for Anti-Self-Dual Equation. Publ. RIMS, Kyoto Univ. 19, No. 2 (1983).

    Google Scholar 

  5. V.E. Zakharov and A.V. Mikhailov: Sov. Phys. JETP, 47, 1017 (1978).

    ADS  Google Scholar 

  6. I. Hauser and F.J. Ernst: J. Math. Phys., 21, 1126 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  7. M. Sato: Kokyoroku RIMS, Kyoto Univ., No. 439, 30–46 (1981).

    Google Scholar 

  8. E. Date, M. Jimbo, M. Kashiwara and T. Miwa: Transformation groups for soliton equations II. Proc. Japan Acad., 57A, 387–392 (1981)

    Google Scholar 

  9. E. Date, M. Jimbo, M. Kashiwara and T. Miwa: Transformation groups for soliton equations III. J Phys. Soc. Japan, 40, 3806–3812 (1981)

    ADS  MathSciNet  Google Scholar 

  10. E. Date, M. Jimbo, M. Kashiwara and T. Miwa: Transformation groups for soliton equations V. Physica 4 D, 343 (1982)

    ADS  MathSciNet  Google Scholar 

  11. E. Date, M. Jimbo, M. Kashiwara and T. Miwa: Transformation groups for soliton equations VI. J. Phys. Soc. Japan 50, 3813–3818 (1981).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. V.G. Kac and D.H. Peterson: Regular Functions on Certain Infinite-dimensional Groups. To appear in “Arithmetic and Geometry” edited by M. Artin and J. Tate Birkäuser, Boston, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yokohama City University, 22-2 Seto, Kanazawa-ku Yokohama 236, Japan

    Kimio Ueno

Authors
  1. Kimio Ueno
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    J. Lepowsky

  2. Department of Physics, University of California, Berkeley, CA, 94720, USA

    S. Mandelstam

  3. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    I. M. Singer

  4. Mathematical Sciences Research Institute, 2223 Fulton Street, Room 603, Berkeley, CA, 94720, USA

    I. M. Singer

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag New York Inc.

About this paper

Cite this paper

Ueno, K. (1985). The Riemann-Hilbert Decomposition and the KP Hierarchy. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds) Vertex Operators in Mathematics and Physics. Mathematical Sciences Research Institute Publications, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9550-8_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-9550-8_14

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9552-2

  • Online ISBN: 978-1-4613-9550-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation