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Riemann Monodromy Problem, Isomonodromy Deformation Equations and Completely Integrable Systems

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Bifurcation Phenomena in Mathematical Physics and Related Topics

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 54))

Abstract

These lectures serve as an introduction into the Riemann-Hilbert monodromy problem. Our main aim is to relate known completely integrable systems with isomonodromy deformation. We describe Schlesinger isomonodromy deformation equations for Fuchsian linear differential equations and their connection with Painlevé transcendents. Moreover it is shown that all the one-dimensional classical completely integrable systems, connected with commuting matrix differential operators, can be represented as simplified Schlesinger systems. We generalize isomonodromy deformation equations for the case of two-dimensional systems. This way arise Painlevé type equations in one space and one time situation. The last part of the lectures is devoted to Riemann boundary value problem. It is explained, how using boundary problem for analytic function on a Riemann surface of finite genus one can solve classical multidimensional isospectral and isomonodromy deformation equations. Examples of so called Bakes’s functions are given.

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Authors and Affiliations

  1. DPh.T. CEN-Saclay, 91190, Gif-sur-Yvette, France

    D. V. Chudnovsky

  2. Dept. of Mathematics, Columbia University, New-York, NY, 10027, USA

    D. V. Chudnovsky

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  1. D. V. Chudnovsky
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Editors and Affiliations

  1. Départment de Mathematiques, Université de Paris-Nord, Villetaneuse, France

    Claude Bardos

  2. CEN, CEA-Saclay, Gif-sur-Yvette, France

    Daniel Bessis

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Chudnovsky, D.V. (1980). Riemann Monodromy Problem, Isomonodromy Deformation Equations and Completely Integrable Systems. In: Bardos, C., Bessis, D. (eds) Bifurcation Phenomena in Mathematical Physics and Related Topics. NATO Advanced Study Institutes Series, vol 54. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9004-3_20

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  • DOI: https://doi.org/10.1007/978-94-009-9004-3_20

  • Publisher Name: Springer, Dordrecht

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  • Online ISBN: 978-94-009-9004-3

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