Thursday 31 August 2017, 16:00-17:00 in HG03.085
Pieter Roffelsen (University of Sydney)
Singularities of Painlevé IV Rationals and their Distribution
The fourth Painlevé equation can be realised as the nonlinear differential equation governing the isomonodromic deformation of certain rank two linear systems. Given a solution and point z in the complex plane, corresponding inverse monodromy problem furnishes the value of the solution at z, except when z is a zero or pole, in which case the inverse monodromy problem does not have any solution. In this talk I will discuss how the zeros and poles can in fact be characterised as the solutions of a simpler inverse monodromy problem concerning an anharmonic oscillator of degree two. Upon specialising to rational solutions of the fourth Painlevé equation, we find that their zeros and poles are classified by the monodromy representation of a class of meromorphic functions with a finite number of branch points introduced by Nevanlinna. In particular this allows us to compute the asymptotic distribution of zeros and poles for Hermite type rationals, as their degree becomes large.