Ahmad Barhoumi: Painlevé-III D6 -> D8 Confluence via Bäcklund Transformations

Abstract.

Among the six Painlevé equations, Painlevé-III (P-III) is the simplest equation with two non-movable singularities, one at infinity and one at the origin. While its generic solutions are transcendental, it is known to 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 a solution of P-III to another solution of P-III with possibly modified parameters. This produces a family of solutions indexed by a natural number n counting the number of iterations. It is in this way that many families of solutions of interest (e.g. rational solutions and Bessel-function solutions) are constructed.  

In this talk, I will describe an approach to studying the large-n behavior near the origin of solutions to P-III constructed in this way with a generic choice of seed function. One of the main findings is that the limiting function, under appropriate scaling) solves the "double-degenerate" version of P-III, known as P-III(D8). This talk is based on the preprint arXiv:2307.11217, joint with Oleg Lisovyy, Peter Miller, and Andrei Prokhorov.