On KP Generators and the Geometry of the HBDE

TL;DR

This paper explores the geometric origins of the Hirota Bilinear Difference Equation within the context of Sato theory, linking it to Grassmannian maps and infinitesimal generators for tau-functions in integrable systems.

Contribution

It rederives the HBDE as a statement about linear maps between Grassmannians and introduces a rank one condition for generating tau-functions using alternative infinitesimal operators.

Findings

HBDE derived from Grassmannian maps

Identifies a rank one condition for tau-function generation

Connects geometric interpretation to integrable system conditions

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.