A Lagrangian Description of the Higher-Order Painlev\'e Equations

TL;DR

This paper derives Lagrangian formulations for higher-order Painlevé equations using Jacobi's last multiplier, highlighting their integrability properties and satisfying specific mathematical conditions.

Contribution

It introduces a method to obtain Lagrangians for complex Painlevé equations, expanding understanding of their mathematical structure.

Findings

Higher-order Painlevé equations can be described by Lagrangians.

Some equations pass the Painlevé test and meet Jurás's conditions.

The approach reveals new integrability features of these equations.

Abstract

We derive the Lagrangians of the higher-order Painlev\'e equations using Jacobi's last multiplier technique. Some of these higher-order differential equations display certain remarkable properties like passing the Painlev\'e test and satisfy the conditions stated by Jur\'a$\check{s}$, (Acta Appl. Math. 66 (2001) 25--39), thus allowing for a Lagrangian description.