On the Riemann–Hilbert–Birkhoff inverse monodromy problem and the Painlevé equations
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- by A. A. Bolibruch, A. R. Its and A. A. Kapaev
- St. Petersburg Math. J. 16 (2005), 105-142
- DOI: https://doi.org/10.1090/S1061-0022-04-00845-3
- Published electronically: December 14, 2004
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Correction: St. Petersburg Math. J. 18 (2007), 679-680.
Abstract:
A generic $2\times 2$ 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 $2\times 2$-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).References
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Bibliographic 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
- 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.
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 16 (2005), 105-142
- MSC (2000): Primary 34M55, 34M50, 34M40
- DOI: https://doi.org/10.1090/S1061-0022-04-00845-3
- MathSciNet review: 2069003
Dedicated: This paper is dedicated to Professor M. Sh. Birman on the occasion of his 75th birthday