Advertisement

On τ-Functions of Zakharov—Shabat and Other Matrix Hierarchies of Integrable Equations

  • L. A. Dickey
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 26)

Abstract

Matrix hierarchies are: multi-component KP, general Zakharov—Shabat (ZS) and its special cases, e.g., AKNS. The ZS comprises all integrable systems having a form of zero-curvature equations with rational dependence of matrices on a spectral parameter. The notion of a τ-function is introduced here in the most general case along with formulas linking τ-functions with wave Baker functions. The method originally invented by Sato et al. for the KP hierarchy is used. This method goes immediately from definitions and does not require any assumption about the character of a solution, being the most general. Applied to the matrix hierarchies, this involves considerable sophistication. The paper is self-contained and does not expect any special prerequisite from a reader.

Keywords

Soliton Equation Differential Polynomial Baker Function Bilinear Identity Hierarchy Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Sato, M., Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku 439 (1981), 30 - 46.Google Scholar
  2. [2]
    Date, E., Jimbo, M., Kashiwara, M., and Miwa, T., Transformation groups for soliton equations, in: Jimbo and Miwa (ed.) Non-linear integrable systems: classical theory and quantum theory, Proc. RIMS symposium, Singapore, 1983.Google Scholar
  3. [3]
    Date, E., Jimbo, M., Kashiwara, M., and Miwa, T., Operator approach to the Kadomtsev-Petviashvili equation - transformation groups for soliton equations III, Journ. Phys. Soc. Japan 50 (1981), 3806 - 3812.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Ueno, K., and Takasaki, K., Toda lattice hierarchy, in: Advanced Studies in Pure Mathematics 4, World Scientific, 1 - 95, 1984.Google Scholar
  5. [5]
    Dickey, L. A., On Segal-Wilson’s definition of the r-function and hierarchies AKNS-D and mcKP, in: Integrable systems, The Verdier Memorial Conference, Progress in Mathematics 115, 1993, Birkháuser, 147-162.Google Scholar
  6. [6]
    Dickey, L. A., On the r-function of matrix hierarchies of integrable equations, Journal Math. Physics 32 (1991), 2996 - 3002.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Dickey, L. A., Why the general Zakharov-Shabat equations form a hierarchy, Com. Math. Phys. 163 (1994), 509 - 521.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Vassilev, S., Tau functions of algebraic geometrical solutions to the general Zakharov-Shabat hierarchy, preprint of the University of Okla-homa, 1994.Google Scholar
  9. [9]
    Dickey, L. A., Soliton Equations and Hamiltonian Systems, Add Series in Mathematical Physics 12, World Scientific, 1991.Google Scholar

Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • L. A. Dickey
    • 1
  1. 1.University of OklahomaNormanUSA

Personalised recommendations