Rational solutions of Painlevé systems.

Although the solutions of Painlevé 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 Painlevé 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 -Painlevé system, first described by Noumi and Yamada. The Noumi-Yamada system, in turn, can be linearized as using bilinear equations and τ-functions; the corresponding rational solutions can then be given as specializations of rational solutions of the KP hierarchy.

The classification of rational solutions to Painlevé equations and systems may now be reduced to an analysis of combinatorial objects known as Maya diagrams. The upshot of this analysis is an explicit determinantal representation for rational solutions in terms of classical orthogonal polynomials. We illustrate this approach by describing Hermite-type rational solutions of Painlevé of the Noumi-Yamada system in terms of cyclic Maya diagrams. By way of example we explicitly construct Hermite-type solutions for the P_IV, P_V equations and the A ₄ Painlevé system.