Mathematics

We study the distribution, in the complex plane, of the singularities of rational solutions of the Painleve fourth equation (P IV). Rational solutions are classified by two integer parameters 'm,n'. First we derive a variety of theoretical results (including the number of real singularities), then we we compute the asymptotic distribution of such singularities, for large value of the parameter 'm'. It turns out that in such limit the singularities asymptotically lie on the disjoint union of n curves, and, on such curves, their asymptotic density satisfies the Wigner's semicircle law.

The seminar is based on a joint paper with Pieter Roffelsen: Poles of Painlevé IV Rationals and their Distribution, SIGMA 14 (2018).