q-Difference equations of KdV type and "Chazy-type" second-degree difference equations

TL;DR

This paper derives a hierarchy of second-degree q-difference equations from integrable KdV-type equations, introducing a q-analogue of Chazy equations and exploring their relation to Painleve-type equations.

Contribution

It introduces a new hierarchy of second-degree q-difference equations derived from KdV-type equations and connects them to Chazy and Painleve equations.

Findings

Derived a hierarchy of second-degree q-difference equations.

Established isomonodromic deformation problems for these equations.

Explored relations to Painleve-type equations.

Abstract

By imposing special compatible similarity constraints on a class of integrable partial $q$-difference equations of KdV-type we derive a hierarchy of second-degree ordinary $q$-difference equations. The lowest (non-trivial) member of this hierarchy is a second-order second-degree equation which can be considered as an analogue of equations in the class studied by Chazy. We present corresponding isomonodromic deformation problems and discuss the relation between this class of difference equations and other equations of Painleve type.