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On KP generators and the geometry of the HBDE

Authors
Journal
Journal of Geometry and Physics
0393-0440
Publisher
Elsevier
Publication Date
Volume
56
Issue
2
Identifiers
DOI: 10.1016/j.geomphys.2005.02.002
Keywords
  • Sato Grassmannian
  • Hirota Bilinear Difference Equation
  • Kp Hierarchy
  • Rank One Conditions
  • Infinite Dimensional Grassmannian
  • Tau-Functions
Disciplines
  • Mathematics

Abstract

Abstract Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the “shift” operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota bilinear difference equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a “rank one condition”, tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

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