Solutions of mixed Painlevé P III—V model

Abstract

We review the construction of the mixed Painlevé P III –V system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of a hybrid differential equation that for special limits of its parameters reduces to either Painlevé P III or P V . The aim of this paper is to describe solutions of the P III – V model. In particular, we determine and classify rational, power series and transcendental solutions of the P III – V model. A class of power series solutions is shown to be convergent in accordance with the Briot–Bouquet theorem. Moreover, the P III – V equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of a P III – V equation is quantized by integer n ∈ Z.