Integrable systems without the Painlev\'e property

TL;DR

This paper investigates the relationship between integrability and the Painlevé property in nonlinear differential equations, showing that some integrable systems lack the Painlevé property, especially in discrete analogues.

Contribution

It demonstrates that the Painlevé property is not a necessary condition for integrability in certain linearisable systems, including their discrete versions.

Findings

Linearisable systems can be integrable without having the Painlevé property.

Discrete analogues of these systems may have nonconfined singularities.

The Painlevé property is not essential for integrability in all cases.

Abstract

We examine whether the Painlev\'e property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlev\'e continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities.