Isomonodromy problems and Painleve Hierarchies

Abstract

In this thesis we show that the expansion of linear problems of the Painlev´e equation in powers of the spectral variable can be used to derive hierarchies of ordinary differential equations. We study the derivation of the first and second Painlev´e hierarchy from scalar isomonodromy problems. Moreover, we will give some special solutions of the second Painlev´e hierarchy. After this, we study the derivation of the first, second, third and fourth Painlev´e hierarchy from matrix isomonodromy problems. Moreover, we will study some special solutions of the hierarchies of the third and fourth painlev´e equations.