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

TL;DR

This paper extends Borel--Laplace transformations to analyze bounded solutions of parameter-dependent nonlinear differential systems with singularities, providing a unified approach for all parameter values.

Contribution

It generalizes Borel--Laplace transformations to study solutions near multiple singular points in nonlinear differential systems with parameters.

Findings

Constructed parametric solutions converging to Borel sums.

Unified treatment for all complex parameter values.

Analyzed solutions near regular and irregular singularities.

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.