On $\tau$-function of the quadrilateral lattice

TL;DR

This paper explores the $ au$-function of the quadrilateral lattice through the nonlocal $ard$-dressing method, revealing its equivalence to a Fredholm determinant of an associated integral equation.

Contribution

It demonstrates that the $ au$-function can be identified with a Fredholm determinant, linking integrable lattice theory with integral equation methods.

Findings

$ au$-function corresponds to a Fredholm determinant

Establishes a connection between $ard$-dressing method and integral equations

Provides a new perspective on quadrilateral lattice integrability

Abstract

We investigate the $\tau$-function of the quadrilateral lattice using the nonlocal $\bar\partial$-dressing method, and we show that it can be identified with the Fredholm determinant of the integral equation which naturally appears within that approach.