A note on the R. Fuchs's problem for the Painlevé equations

Details A note on the R. Fuchs's problem for the Painlevé equations A note on the R. Fuchs's problem for the Painlevé equations

2012-04-01

arxiv.org/abs/1204.0157v1

Mathematics

Classical Analysis and ODEs

16 pages

Scientific paper

In this article we consider a first-order completely integrable system of partial differential equations $\partial \Fi/partial x=A(x, t) \Fi, \partial \Fi/partial t=B(x, t) \Fi$ with $\Fi=(\fi_1, \fi_2)^{\tau}$ where $A(x, t)$ and $B(x, t)$ are 2 by 2 holomorphic matrices functions. Under some assumptions we find a variable change by which the system $\partial \Fi/\partial x=A(x, t) \Fi$ is reduced to an equation independent on the variable $t$. As an application we show that the R. Fuchs's conjecture for the Painlev\'e equations is true for some algebraic solutions.

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