Ahmad Barhoumi

Special Function Solutions of Painlevé Equations via Orthogonal Polynomials

The six Painlevé equations are fundamental families of differential equations in mathematical physics with a wide range of applications. While their generic solutions are transcendental, it is known that all but Painlevé-I possess families of special–function solutions: solutions written in terms of elementary and/or classical special functions. These are often generated by iterating Bäcklund transformations which map one solution of a Painlevé equation to another of the same equation with possibly modified parameters. This produces a family of solutions indexed by a natural number n counting the number of iterations.

In this talk, I will describe an approach to studying the large-n behavior of these solutions by connecting them with families of orthogonal polynomials. Two particular examples will be highlighted: Airy solutions of Painlevé-II and parabolic cylinder function solutions of Painlevé-IV. Parts of the presentation are based on joint work with Pavel Bleher, Alfredo Deaño, and Maxim Yattselev.