Painleve tests, singularity structure and integrability

TL;DR

This paper reviews Painlevé tests and singularity analysis methods to assess the integrability of differential equations, highlighting their connection to the equations' singularity structures.

Contribution

It provides a concise overview of Painlevé tests and explores their relationship with integrability in both ordinary and partial differential equations.

Findings

Painlevé tests effectively identify integrable equations.

Singularity structures are closely linked to integrability.

The paper demonstrates applications to various differential equations.

Abstract

After a brief introduction to the Painlev\'{e} property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlev\'{e} tests. The tests are applied to several different examples, and we discuss the connection between singularity structure and integrability for ordinary and partial differential equations.