Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence

Abstract

It is well known that the sixth Painlevé equation $\PVI$ admits a group of Bäcklund transformations which is isomorphic to the affine Weyl group of type ${\mathrm{D}}_4^{(1)}$. Although various aspects of this unexpectedly large symmetry have been discussed by many authors, there still remains a basic problem yet to be considered, that is, the problem of characterizing the Bäcklund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the Bäcklund transformations are just the pull-back of very simple transformations on the moduli of monodromy representations by the Riemann-Hilbert correspondence. This result gives a natural and clear picture of the Bäcklund transformations. Key words: Bäcklund transformation, the sixth Painlevé equation, Riemann-Hilbert correspondence, isomonodromic deformation, affine Weyl group of type ${\mathrm{D}}_4^{(1)}$.

24pages, 4 figures, 3 eps files

References in corpus (3)

• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• First degree birational transformations of the Painlevé equations and their contiguity relations
• Nodal curves and Riccati solutions of Painlevé equations

Cited by in corpus (13)

• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• Some explicit solutions to the Riemann-Hilbert problem
• Dynamics of the Sixth Painlevé Equation
• Six results on Painleve VI
• Regge and Okamoto symmetries
• Holomorphic dynamics, Painlevé VI equation and Character Varieties
• Integral transformation of Heun's equation and some applications
• Cubic and quartic transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence
• Finite branch solutions to Painleve VI around a fixed singular point
• Moduli spaces of d-connections and difference Painleve equations
• Okamoto's symmetry on the representation space of the sixth Painlevé equation
• The fifty-two icosahedral solutions to Painleve VI
• From Klein to Painleve via Fourier, Laplace and Jimbo