Classification of Real Solutions of the Fourth Painlevé Equation

Abstract

Painlevé transcendents are usually considered as complex functions of a complex variable, but, in applications, it is often the real cases that are of interest. We propose a classification of the real solutions of Painlevé IV (or, more precisely, symmetric Painlevé IV) using a sequence of symbols describing their asymptotic behavior and singularities. We give rules for the sequences that are allowed (depending on the parameter values). We look in detail at the existence of globally nonsingular real solutions and determine the dimensions of the spaces of such solutions with different asymptotic behaviors. We show that for a generic choice of the parameters, there exists a unique finite sequence of singularities for which Painlevé IV has a two-parameter family of solutions with this singularity sequence. There also exist solutions with singly infinite and doubly infinite sequences of singularities, and we identify which such sequences are possible. We look at the singularity sequences (and asymptotics) of special solutions of Painlevé IV. We also show two other results concerning special solutions: rational solutions can be obtained as the solution of a set of polynomial equations, and other special solutions can be obtained as the solution of a first-order differential equation.