An Isomonodromy Cluster of Two Regular Singularities

Details An Isomonodromy Cluster of Two Regular Singularities An Isomonodromy Cluster of Two Regular Singularities

2006-06-22

arxiv.org/abs/math/0606562v1

Mathematics

Classical Analysis and ODEs

44 pages, 8 figures

Scientific paper

10.1088/0305-4470/39/39/S03

We consider a linear $2\times2$ matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations. In our main example the latter limit reads as $P_6\to P_5$, where $P_n$ is the $n$-th Painlev\'e equation. We also discuss some general problems which arise while studying the above-mentioned limits for the Painlev\'e equations.

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