Coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$

TL;DR

This paper introduces four new four-dimensional coupled Painlevé VI systems with specific affine Weyl group symmetries, providing explicit transformations, holomorphy conditions, and a confluence process connecting different Painlevé types.

Contribution

It presents the first higher-order Painlevé systems of types B, D, with explicit polynomial Hamiltonians and symmetry properties, expanding the understanding of Painlevé equations.

Findings

Four families of coupled Painlevé VI systems with affine Weyl symmetries.

Explicit birational and symplectic transformations between systems.

Description of a confluence process from D6^{(1)} to A_5^{(1)}.

Abstract

We find four kinds of six-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order Painlev\'e equations of types $B_l^{(1)},D_l^{(1)}$ and $D_l^{(2)}$, respectively. Each system can be expressed as a polynomial Hamiltonian system. We show that these systems are equivalent by an explicit birational and symplectic transformation, respectively. By giving each holomorphy condition, we can recover each system. These symmetries, holomorphy conditions and invariant divisors are new. We also give an explicit description of a confluence process from the system of type $D_6^{(1)}$ to the system of type $A_5^{(1)}$ by taking the coupling confluence process from the Painlev\'e VI system to the Painlev\'e V system.