Integrable Systems and Rank One Conditions for Rectangular Matrices

TL;DR

This paper introduces a determinantal formula for KP hierarchy tau-functions using rectangular matrices satisfying a rank one condition, unifying various previous results on integrable systems.

Contribution

It generalizes and unifies existing constructions of tau-functions from square matrices to rectangular matrices with a rank one condition.

Findings

Provides a new determinantal formula for tau-functions

Unifies multiple previous approaches under a common framework

Includes special cases like Wilson's formula and previous results with almost-intertwining matrices

Abstract

We provide a determinantal formula for tau-functions of the KP hierarchy in terms of rectangular, constant matrices $A$, $B$ and $C$ satisfying a rank one condition. This result is shown to generalize and unify many previous results of different authors on constructions of tau-functions for differential and difference integrable systems from square matrices satisfying rank one conditions. In particular, it contains as explicit special cases the formula of Wilson for tau-functions of rational KP solutions in terms of Calogero-Moser Lax matrices as well as our previous formula for KP tau functions in terms of almost-intertwining matrices.