Four-dimensional Painlev\'e systems of types $D_5^{(1)}$ and $B_4^{(1)}$

TL;DR

This paper introduces a new family of four-dimensional coupled Painlevé V systems with $D_5^{(1)}$ symmetry and describes their confluence to Painlevé III systems with $B_4^{(1)}$ symmetry, expanding understanding of these integrable systems.

Contribution

It constructs and analyzes a five-parameter family of Painlevé V systems with $D_5^{(1)}$ symmetry and explicitly describes their confluence to Painlevé III systems with $B_4^{(1)}$ symmetry.

Findings

Defined a five-parameter family of Painlevé V systems with $D_5^{(1)}$ symmetry.

Explicitly described the confluence process to Painlevé III systems with $B_4^{(1)}$ symmetry.

Established the affine Weyl group symmetries of the systems.

Abstract

We find and study a five-parameter family of four-dimensional coupled Painlev\'e V systems with affine Weyl group symmetry of type $D_5^{(1)}$. We then give an explicit description of a confluence from those systems to a four-parameter family of four-dimensional coupled Painlev\'e III systems with affine Weyl group symmetry of type $B_4^{(1)}$.