Rational solutions of Painleve systems

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.