New Airy-type solutions of the ultradiscrete Painleve II equation with parity variables

TL;DR

This paper introduces new Airy-type solutions for the ultradiscrete Painleve II equation with parity variables, expanding the solution space and revealing richer structures than previously known solutions.

Contribution

It presents the first ultradiscrete limit of special solutions involving parity variables, resulting in novel solutions with more complex structures.

Findings

New Airy-type solutions with parity variables are derived.

Solutions exhibit richer structures compared to existing solutions.

Ultradiscretization process successfully applied to special solutions.

Abstract

The q-difference Painleve II equation admits special solutions written in terms of determinant whose entries are the general solution of the q-Airy equation. An ultradiscrete limit of the special solutions is studied by the procedure of ultradiscretization with parity varialbes. Then we obtain new Airy-type solutions of the ultradiscrete Painleve II equation with parity variables, and the solutions have richer structure than the known solutions.