Four-dimensional Painlev\'e-type difference equations

TL;DR

This paper investigates four-dimensional Painlevé-type difference equations derived from Fuchsian equations with specific parameters, exploring their degeneration schemes and providing explicit examples of discrete isomonodromic deformations.

Contribution

It introduces a degeneration scheme for Fuchsian equations with four accessory parameters and computes explicit examples of related discrete isomonodromic deformation equations.

Findings

Degeneration scheme of Fuchsian equations established

Explicit example of discrete isomonodromic deformation provided

Insights into four-dimensional phase space dynamics

Abstract

We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete isomodromic deformation equations with four-dimensional phase space. We compute an example of discrete isomonodromic deformation equations of a certain Fuchsian equation.