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B\"acklund Transformations of the Sixth Painlev\'e Equation in Terms of Riemann-Hilbert Correspondence

Michi-aki Inaba·Katsunori Iwasaki·Masa-Hiko Saito
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Abstract

It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of B\"acklund 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 B\"acklund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the B\"acklund 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 B\"acklund transformations. Key words: B\"acklund transformation, the sixth Painlev\'e equation, Riemann-Hilbert correspondence, isomonodromic deformation, affine Weyl group of type $\mathrm{D}_4^{(1)}$.

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Michi-aki Inaba, Katsunori Iwasaki, Masa-Hiko Saito. 2003-09-20. B\"acklund Transformations of the Sixth Painlev\'e Equation in Terms of Riemann-Hilbert Correspondence. https://doi.org/10.1155/s1073792804131310

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