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arXiv · 1207.4335

A Riemann--Hilbert approach to Painlev\'e IV

Marius van der Put·Jaap Top
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Abstract

This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlev\'e equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto--Painlev\'e varieties and the Painlev\'e property follows. For an explicit computation of the full group of B\"acklund transformations, rank three connections on $\mathbb{P}^1$ are introduced, inspired by the symmetric form for ${\rm PIV}$ as was studied by M. Noumi and Y. Yamada.

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Marius van der Put, Jaap Top. 2012-07-18. A Riemann--Hilbert approach to Painlev\'e IV. https://arxiv.org/abs/1207.4335

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