Peter Clarkson

Rational solutions of Painleve equations

The six Painleve equations, whose solutions are called the Painleve transcendents, were derived by Painleve and his colleagues in the late 19th and early 20th centuries in a classification of second order, nonlinear ordinary differential equations. In the 18th and 19th centuries, the classical special functions such as Bessel, Airy, Legendre and hypergeometric functions, were recognized and developed in response to the problems of the day in electromagnetism, acoustics, hydrodynamics, elasticity and many other areas.
Around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of functions, the Painleve functions, started to appear in applications. The list of problems now known to be described by the Painleve equations is large, varied and expanding rapidly, ranging from the scattering of neutrons off heavy nuclei to the distribution of the zeros of the Riemann-zeta function.

In this talk I shall discuss rational solutions of Painleve equations. Although the general solutions of the six Painleve equations are transcendental, all except the first Painleve equation possess rational solutions for certain values of the parameters. These solutions are usually expressed in terms of special polynomials that are determinants, often of classical orthogonal polynomials such as Hermite and Laguerre polynomials. It is also known that the roots of these special polynomials have highly symmetric patterns in the complex plane. The polynomials arise in applications such as rogue waves, random matrix theory, vortex dynamics, in supersymmetric quantum mechanics, as coefficients of recurrence relations for semi-classical orthogonal polynomials and are examples of exceptional orthogonal polynomials.