# Painleve Equations - Nonlinear Special Functions I

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CATW01 - The complex analysis toolbox: new techniques and perspectives

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 ordinary differential equations whose solutions have no movable critical points.

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. The list includes, at one end, the scattering of neutrons off heavy nuclei, and at the other, the distribution of the zeros of the Riemann-zeta function on the critical line Re(z) =1/2. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar solutions of integrable equations, combinatorial problems such as the longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are comparable and large, and also random growth problems.

The Painleve equations possess a plethora of interesting properties including a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. Solutions of the Painleve equations have some interesting asymptotics which are useful in applications. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations.

In these lectures I shall first review many of the remarkable properties which the Painleve equations possess. In particular I will 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 logarithmic derivatives of special polynomials that are Wronskians, often of classical orthogonal polynomials such as Hermite and Laguerre. It is also known that the roots of these special polynomials are highly symmetric in the complex plane. The polynomials arise in applications such as 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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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