One of the main open analytic problems concerning Painlevé equations is to determine the distribution of singularities (poles, zeros etc.) of their solutions. We consider this problem for Painlevé IV and discuss how singularities can be classified in terms of inverse monodromy problems of a certain class of anharmonic oscillators.
We showcase how this classification can be made effective to analyse the distribution of singularities by considering a particular example, namely one of the three families of Hermite-type rational solutions. The corresponding anharmonic oscillators are quasi-exactly solvable and can be studied using Nevanlinna theory and WKB methods, yielding both theoretical and asymptotic results concerning the distribution of singularities.
The seminar is based on a joint paper with Davide Masoero: Poles of Painlevé IV Rationals and their Distribution, SIGMA 14 (2018).