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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

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


Authors: A. A. Bolibruch, A. R. Its and A. A. Kapaev
Original publication: Algebra i Analiz, tom 16 (2004), nomer 1.
Journal: St. Petersburg Math. J. 16 (2005), 105-142
MSC (2000): Primary 34M55, 34M50, 34M40
Published electronically: December 14, 2004
MathSciNet review: 2069003
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Abstract: 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).


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Additional Information

A. A. Bolibruch
Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 117966, GSP-1, Russia

A. R. Its
Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, Indianapolis, IN 46202-3216, USA
Email: itsa@math.iupui.edu

A. A. Kapaev
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00845-3
PII: S 1061-0022(04)00845-3
Keywords: Riemann--Hilbert problem, Painlev\'e equations
Received by editor(s): October 6, 2003
Published electronically: December 14, 2004
Additional Notes: The first author was supported in part by RFBR (grant no. 99-01-00157) and by INTAS (grant no. 97-1644). The second author was supported in part by NSF (grants nos. DMS-9801608 and DMS-0099812). The third author was supported in part by RFBR (grant no. 99-01-00687). The second author also thanks the Département de Mathématiques, Université de Strasbourg, where part of this work was done, for hospitality during his visit.
Dedicated: This paper is dedicated to Professor M. Sh. Birman on the occasion of his 75th birthday
Article copyright: © Copyright 2004 American Mathematical Society