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arXiv · 1610.01299

Unitary monodromy implies the smoothness along the real axis for some Painlev\'{e} VI equation, I

Zhijie Chen·Ting-Jung Kuo·Chang-Shou Lin
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Abstract

In this paper, we study the Painlev\'{e} VI equation with parameter $(\frac {9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. We prove (i) An explicit formula to count the number of poles of an algebraic solution with the monodromy group $D_{N}$, where $D_{N}$ is the dihedral group of order $2N$. (ii) There are only four solutions without poles in $\mathbb{C}\backslash \left \{ 0,1\right \} $. (iii) If the monodromy group of the associated linear ODE of a solution $\lambda \left( t\right) $ is unitary, then $\lambda ( t) $ has no poles in $\mathbb{R}% \backslash \{ 0,1\} $.

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Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin. 2016-10-05. Unitary monodromy implies the smoothness along the real axis for some Painlev\'{e} VI equation, I. https://doi.org/10.1016/j.geomphys.2017.01.016

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