Rational solutions of the fourth and fifth Painlev\'{e} hierarchies

TL;DR

This paper investigates rational solutions of higher-order Painleve hierarchies, establishing conditions for such solutions and revealing that certain P4 hierarchies universally admit the solution 1/x, similar to the original P4.

Contribution

It provides necessary and sufficient conditions for rational solutions in four Painleve hierarchies and shows that specific P4 hierarchies always admit the solution 1/x.

Findings

Conditions for rational solutions in P4 and P5 hierarchies

Existence of solutions in the form of 1/x for certain P4 hierarchies

All considered P4 hierarchies admit the solution 1/x

Abstract

We will consider four hierarchies of higher order analogues of the fourth (P4) and fifth (P5) Painleve equations. The necessary and sufficient conditions for having rational solutions will be presented. Further we well consider two more hierarchies of the (P4). We well show that both of them admit the solution in the form of $1/x$. The last result shows that all the considered P4-hierarchies have the common property: they assume the solution $1/x$ as also the original P4 does.