A method to obtain the Schlesinger transformations for Painlevi VI equation is given. The procedure involves formulating a Riemann-Hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same.

In this paper, we show that the expansion of linear problems of the Painlevé equation in powers of the spectral variable can be used to derive hierarchies of ordinary differential equations. We applied this approach to linear problems of the first, second, third and fourth Painlevé equations. We derived a new hierarchy of the third Painlevé equation and rederived known hierarchies of the other equations. Moreover some special solutions of the hierarchies of the second, third and fourth Painlevé equations are also given.

The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painleve equations is used to obtain a one-to-one correspondence´ between the Painleve I, II and III equations and certain second-order second degree equations´ of Painleve type.´

No abstract

One-to-one correspondence between the Painlevé I-VI equations and certain second-order second-degree equations of Painlevé type is investigated. The transformation between the Painlevé equations and second-order second-degree equations is the one involving the Fuchsian-type equation.