On Cauchy Problems for Higher Order Painleve' Equations

Abstract

A rigorous methodology for studying the initial value problems associated with certain integrable nonlinear ordinary differential equations was introduced by Fokas and Zhou, and was used to investigate Painlev´e second and fourth Painlev´e equations. The same methodology has been applied to the other Painlev´e equations and to the second member of a fourth Painlev´e hierarchy. We studied the application of this methodology to the second members of fourth and second Painlev´e hierarchies. We show that the Cauchy problems for these equations admit, in general, global solutions, meromorphic in x. Furthermore, for special relations among the monodromy data, the associated Riemann-Hilbert problems can be reduced to a set of scalar Riemann-Hilbert problems. By solving these RiemannHilbert problems, we obtain special solutions of the second members of fourth and second Painlev´e hierarchies which can be written in terms of special functions.