An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Rational solutions and associated special polynomials associated for the Painlev\'e equations

21st September 2006

Author: Clarkson, P (Kent)

Abstract

In this talk I shall discuss rational solutions and associated polynomials for the second, third, fourth and fifth Painlev\'e equations(PII--PV). The Painlev\'e equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past twenty-five years, which have arisen in a variety of physical applications and may be thought of as nonlinear special functions.Rational solutions of the Painlev\'e equations are expressible as the logarithmic derivative of special polynomials. For PII these special polynomials are known as the {\it Yablonskii-Vorob'ev polynomials\/}. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials for PIII, PIV and PV are derived and I shall show that the roots of these special polynomials also have a highly regular structure.