Ultradiscrete Painlev\'e VI with Parity Variables

TL;DR

This paper develops an ultradiscrete version of the Painlevé VI equation incorporating parity variables, demonstrating that Riccati-type solutions converge to solutions of this ultradiscrete system, and explores specific solutions.

Contribution

It introduces a novel ultradiscretization method with parity variables for Painlevé VI, extending the understanding of ultradiscrete integrable systems.

Findings

Ultradiscrete Riccati-type solutions satisfy the ultradiscrete Painlevé VI system.

The paper provides explicit solutions to the ultradiscrete Riccati-type and Painlevé VI equations.

Parity variables are effectively used to validate the ultradiscrete limit.

Abstract

We introduce a ultradiscretization with parity variables of the $q$-difference Painlev\'e VI system of equations. We show that ultradiscrete limit of Riccati-type solutions of $q$-Painlev\'e VI satisfies the ultradiscrete Painlev\'e VI system of equations with the parity variables, which is valid by using the parity variables. We study some solutions of the ultradiscrete Riccati-type equation and those of ultradiscrete Painlev\'e VI equation.