A $q$-analogue of the matrix fifth Painlev\'e system

TL;DR

This paper introduces a degeneration of the $q$-matrix sixth Painlevé system, leading to a new non-linear $q$-difference system that deforms a non-Fuchsian linear $q$-difference system, and connects it to the matrix fifth Painlevé system through a continuous limit.

Contribution

It defines the spectral type for non-Fuchsian $q$-difference systems and characterizes the associated linear problem, extending the understanding of $q$-Painlevé systems.

Findings

Derived a non-linear $q$-difference system from the degeneration.

Established the spectral type for non-Fuchsian $q$-difference systems.

Connected the $q$-difference system to the matrix fifth Painlevé system via a continuous limit.

Abstract

We consider a degeneration of the $q$-matrix sixth Painlev\'e system. As a result, we obtain a system of non-linear $q$-difference equations, which describes a deformation of a certain non-Fuchsian linear $q$-difference system. We define the spectral type for non-Fuchsian $q$-difference systems and characterize the associated linear problem in terms of the spectral type. We also consider a continuous limit of the non-linear $q$-difference system and show that the resulting system of non-linear differential equations coincides with the matrix fifth Painlev\'e system.