Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
英文摘要：
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 $\mathrm{D}_4^{(1)}$.
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