Monodromy dependence and connection formulae for isomonodromic tau functions
Monodromy dependence and connection formulae for isomonodromic tau functions
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DOI：
10.1215/00127094-2017-0055
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发表时间：
2016-04
影响因子：
2.5
通讯作者：
A. Its;O. Lisovyy;A. Prokhorov
中科院分区：
文献类型：
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作者：
A. Its;O. Lisovyy;A. Prokhorov
我们讨论了 Jimbo-Miwa-Ueno 微分 1-形式到具有有理系数的线性常微分方程组的扩展单向数据的整个空间上封闭的形式的扩展。此扩展基于 M. Bertola 的结果，概括了 B. Malgrange 先前的构造。我们展示了如何使用这种 1-形式来解决等单向 tau 函数的连接公式的评估这一长期存在的问题，其中包括相关常数因子的显式计算。我们解释了该方案如何适用于 Fuchsian 系统，特别是计算通用 Painleve VI tau 函数的连接常数。结果证明了 Iorgov、Lisovyy 和 Tykhyy 提出的该常数的猜想公式。我们还将该方法应用于非 Fuchsian 系统，并评估 Painleve II tau 函数渐进中的常数因子。
We discuss an extension of the Jimbo–Miwa–Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for the generic Painleve VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painleve II tau function.