The tau function for ABS equations

TL;DR

This paper explains the tau-function for ABS quad-equations, showing how it linearizes the Bäcklund chain and unifies many equations through transformations, with illustrative examples.

Contribution

It introduces the tau-function for ABS equations, demonstrating its role in linearization and unification of different equations via transformations.

Findings

Tau-function systematically linearizes the Bäcklund chain.

Many ABS equations share the same tau function.

Transformations based on the tau-function unify different equations.

Abstract

The tau-function for quad-equations from the ABS classification is briefly explained. It is an auxiliary variable that systematically linearises the Backlund chain. Many equations have the same tau function and are unified by transformations constructed on that basis. Simple examples are included.