Tau-functions and special solutions in a coupled Painlev\'e system

Abstract

For a pair of coupled Painlev\'e equations obtained as a similarity reduction of the Hirota-Satsuma systems we describe special parameter-families of solutions given in terms of mixtures of rational and Airy functions, and in terms of a second Painlev\'e transcendent. The tau-functions associated to some of these solutions are also given explicitly.