Confluence of Singularities of Nonlinear Differential Equations via Borel–Laplace Transformations

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.Keywords: Ordinary differential equations irregular singularity unfolding confluence center manifold of a saddle-node singularity Borel summation

On Boundary Layer Expansions for a Singularly Perturbed Problem with Confluent Fuchsian Singularities

“…with Fuchsian singularities at x = ± 1/2 that coalesce to the origin as tends to 0. This paper [7] can be seen as a continuation of the contribution by B. Sternin and V. Shatalov [8] who focused on linear scalar ODEs with holomorphic coefficients of the form…”

“…In the study [5], Hideshi Yamane constructs solutions to nonlinear wave equations that blow up along prescribed noncharacteristic hypersurfaces using the so-called Fuchsian reduction method introduced by S. Kichenassamy which transforms the initial problem into a Fuchsian PDE which, in general, contains logarithmic terms, see the excellent textbook [6] for a reference. The idea of considering such special types of confluence (3) stems from a work by M. Klimes, see [7], where nonlinear differential systems with irregular singularity at x = 0…”

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

“…We will now define "sectoral" domains in the -and x-space on which the system can be normalized. They are of the same type as those introduced in [10,13,15,16,25].…”

“…The reason for introducing the two Banach algebras the above way lies in the following theorem adapted from [13] which is the other essential tool in our proof of Theorem 17.…”

Zero Level Perturbation of a Certain Third-Order Linear Solvable ODE with an Irregular Singularity at the Origin of Poincaré Rank 1

“…On the other hand, following [4,5,8,12,16,17], we can always regard the irregular singularity at the origin as a result of confluence of two Fuchsian singularities. Namely, Date: 06.05.2018.…”