On an equation arising by reduction of the Drinfeld-Sokolov hierarchy

TL;DR

This paper demonstrates that a specific seventh-order ODE from the Drinfeld-Sokolov hierarchy is equivalent to a similarity reduction of the Sawada-Kotera hierarchy and connects it to a higher order Painlevé function.

Contribution

It establishes a novel link between a seventh-order ODE from integrable hierarchies and a higher order Painlevé equation, expanding understanding of their interrelations.

Findings

Identifies the seventh-order ODE as a similarity reduction of the Sawada-Kotera hierarchy

Links the ODE to a particular higher order Painlevé function

Provides insights into the structure of integrable hierarchies and special functions

Abstract

A seventh order ordinary differential equation (ODE) arising by reduction of the Drinfeld-Sokolov hierarchyis shown to be identical to a similarity reduction of an equationin the hierarchy of Sawada-Kotera.We also exhibit its link with a particular F-VI,a fourth order ODE isolated by Cosgrove which is likely to define a higher order Painlev\'e function.