Do All Integrable Evolution Equations Have the Painlev\'e Property?

TL;DR

This paper investigates the relationship between integrability and the Painleve property in PDEs and lattice equations, revealing that some integrable equations do not exhibit the Painleve or singularity confinement properties.

Contribution

It demonstrates that certain integrable PDEs and lattice equations can be linearisable without possessing the Painleve or singularity confinement properties.

Findings

Existence of linearisable PDEs without Painleve property

Existence of linearisable lattice equations without singularity confinement

Analogy between continuous and discrete integrability properties

Abstract

We examine whether the Painleve property is necessary for the integrability of partial differential equations (PDEs). We show that in analogy to what happens in the case of ordinary differential equations (ODEs) there exists a class of PDEs, integrable through linearisation, which do not possess the Painleve property. The same question is addressed in a discrete setting where we show that there exist linearisable lattice equations which do not possess the singularity confinement property (again in analogy to the one-dimensional case).