Rational solutions of Painleve systems

TL;DR

This paper explores rational solutions of Painlevé systems, using symmetric Riccati-like equations, Maya diagrams, and orthogonal polynomials, with explicit constructions for Hermite-type solutions in several Painlevé equations.

Contribution

It provides a new explicit determinantal representation of rational solutions for Painlevé systems via combinatorial and orthogonal polynomial methods.

Findings

Explicit Hermite-type solutions for PIV and PV equations

Determinantal formulas for rational solutions using Maya diagrams

Connection between Painlevé solutions and classical orthogonal polynomials

Abstract

Although the solutions of Painlev\'e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very successful approach in the study of rational solutions to Painlev\'e equations involves the reformulation of these scalar equations into a symmetric system of coupled, Riccati-like equations known as dressing chains. Periodic dressing chains are known to be equivalent to the $A_N$-Painlev\'e system, first described by Noumi and Yamada. The Noumi-Yamada system, in turn, can be linearized as using bilinear equations and $\tau$-functions; the corresponding rational solutions can then be given as specializations of rational solutions of the KP hierarchy. The classification of rational solutions to Painlev\'e equations and systems may now be reduced to an analysis of combinatorial objects known as Maya diagrams. The upshot of this analysis is a an explicit determinental representation for rational solutions in terms of classical orthogonal polynomials. In this paper we illustrate this approach by describing Hermite-type rational solutions of Painlev\'e of the Noumi-Yamada system in terms of cyclic Maya diagrams. By way of example we explicitly construct Hermite-type solutions for the PIV, PV equations and the $A_4$ Painlev\'e system.