Singularity confinement in delay-differential Painlev\'e equations

TL;DR

This paper investigates singularity confinement in delay-differential Painlevé equations, providing a geometric interpretation and demonstrating confinement in three specific examples, advancing understanding of their integrability properties.

Contribution

It introduces a geometric framework for analyzing singularity confinement in delay-differential Painlevé equations and confirms confinement in three studied cases.

Findings

All singularities in the three examples are confined.

A geometric interpretation of singularity confinement is established.

The results enhance understanding of integrability in delay-differential equations.

Abstract

We study singularity confinement phenomena in examples of delay-differential Painlev\'e equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in terms of mappings between jet spaces, defining certain singularities analogous to those of interest in the singularity analysis of discrete systems, and what it means for them to be confined. For three previously studied examples of delay-differential Painlev\'e equations, we describe all such singularities and show they are confined in the sense of our geometric description.