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arXiv · 1307.8383

Confluence of singularities of non-linear differential equations via Borel--Laplace transformations

Martin Klimes
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Abstract

Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel--Laplace transformation. This article shows how to generalize the Borel--Laplace transformation in order to investigate bounded solutions of parameter dependent non-linear differential systems with two simple (regular) singular points unfolding a double (irregular) singularity. We construct parametric solutions on domains attached to both singularities, that converge locally uniformly to the sectoral Borel sums. Our approach provides a unified treatment for all values of the complex parameter.

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Martin Klimes. 2015-11-03. Confluence of singularities of non-linear differential equations via Borel--Laplace transformations. https://doi.org/10.1007/s10883-015-9290-7

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