On the Riemann--Hilbert--Birkhoff inverse monodromy problem and the Painlevé equations

• Published in: St. Petersburg mathematical journal 2005-02, Vol.16 (1), p.105-143
• Main Authors: Bolibruch, A. A., Its, A. R., Kapaev, A. A.
• Format: Article
• Language: English
• ISSN: 1061-0022; 1547-7371
• DOI: 10.1090/S1061-0022-04-00845-3

A generic 2\times2 system of first order linear ordinary differential equations with second degree polynomial coefficients is considered. The problem of finding such a system with the property that its Stokes multipliers coincide with a given set of relevant 2\times2-matrices constitutes the first nontrivial case of the Riemann--Hilbert--Birkhoff inverse monodromy problem. The meromorphic (with respect to the deformation parameter) solvability of this problem is proved. The approach is based on Malgrange's generalization of the classical Birkhoff--Grothendieck theorem to the case with the parameter. As a corollary, a new proof of meromorphicity of the second Painlevé transcendent is obtained. An elementary proof of a particular case of Malgrange's theorem, needed for our goals, is also presented (following an earlier work of the first author).