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29 March 2024
 
  » arxiv » math.AG/0309341

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Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
Michi-aki Inaba ; Katsunori Iwasaki ; Masa-Hiko Saito ;
Date 20 Sep 2003
Journal Internat. Math. Res. Notices, 2004, no. 1, 1-30
Subject Algebraic Geometry; Analysis of PDEs; Exactly Solvable and Integrable Systems MSC-class: 33E17; 37K20, 37J35 | math.AG math.AP nlin.SI
AffiliationKyushu), Katsunori Iwasaki (Kyushu), Masa-Hiko Saito (Kobe
AbstractIt 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 $mathrm{D}_4^{(1)}$.
Source arXiv, math.AG/0309341
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