Lattice equations arising from discrete Painlev\'e systems. II. $A_4^{(1)}$ case

TL;DR

This paper constructs lattices from tau functions of $A_4^{(1)}$-surface q-Painlevé equations, revealing ABS-type quad-equations and deriving Lax pairs through a hypercube structure, advancing understanding of integrable systems.

Contribution

It introduces a novel lattice construction from tau functions of $A_4^{(1)}$-surface q-Painlevé equations and derives associated Lax pairs using a hypercube approach.

Findings

Lattices constructed from tau functions exhibit ABS-type quad-equations.

Lax pairs for $A_4^{(1)}$-surface q-Painlevé equations are obtained.

Hypercube structure facilitates the derivation of integrable system properties.

Abstract

In this paper, we construct two lattices from the $\tau$ functions of $A_4^{(1)}$-surface $q$-Painlev\'e equations, on which quad-equations of ABS type appear. Moreover, using the reduced hypercube structure, we obtain the Lax pairs of the $A_4^{(1)}$-surface $q$-Painlev\'e equations.