The Painlevé equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. Rational solutions of the Painlevé equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the second Painlevé equation (PII) these polynomials are known as the Yablonskii–Vorob’ev polynomials, first derived in the 1960’s by Yablonskii and Vorob’ev. 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 associated with rational solutions of the fourth Painlevé equation (PIV), which are known as the generalized Hermite polynomials and generalized Okamoto polynomials, are described and it is shown that their roots also have a highly regular structure. The Yablonskii–Vorob’ev polynomials arise in string theory and the generalized Hermite polynomials in the theories of random matrices and orthogonal polynomials.

It is well known that soliton equations have symmetry reductions which reduce them to the Painlevé equations, e.g. scaling reductions of the Korteweg-de Vries and modified Korteweg-de Vries equations are expressible in terms of PII and scaling reductions of the Boussinesq and nonlinear Schrödinger equations are expressible in terms of PIV. Hence rational solutions of these soliton equations can be expressed in terms of the Yablonskii and Vorob’ev, generalized Hermite and generalized Okamoto polynomials. Furthermore the motion of the poles of the rational solutions of the Korteweg-de Vries equation is described by a constrained Calogero-Moser system, as shown by Airault, McKean, and Moser in 1977.

The motion of the poles of more general rational solutions of equations in the Korteweg-de Vries and the modified Korteweg-de Vries hierarchies as well as rational solutions Boussinesq equation, the classical Boussinesq system and variants will be discussed. Further for the nonlinear Schrödinger equation, rational solutions and rational-oscillatory solutions, which are new solutions, will also be described.

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