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

Abstract

We study an irregular singularity of Poincaré rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter ε ∈ (R + , 0) (ε < 1), which causes the splitting of the irregular singularity into two finite Fuchsian singularities. We show that when the solutions of the perturbed equation contain logarithmic terms, the Stokes matrices of the initial equation are limits of the part of the monodromy matrices around the finite resonant Fuchsian singularities of the perturbed equation.

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

“…Assume that the list of constraints (10), (12), (15), (21), (47), (51) hold. We consider a good covering E in = {E in p } 0≤p≤η−1 in C * and an admissible set of sectors…”

“…This result has been extended to general linear systems of ODEs in two seminal papers [10,11] by Alexey Glutsyuk that describe the convergence of monodromy data of well-chosen fundamental solutions to the Stokes matrices in the confluence process. More recently in [12], Tsvetana Stoyanova has studied particular cases of [10,11] and obtained explicit formulas for the solutions of an unfolding of a third order linear scalar ODE with irregular singularity at the origin. Confluence under the additional constraint of isomonodromic deformation has been investigated for linear systems of ODEs for Fuchsian singularities by Andrey Bolibrukh, ref.…”

Stokes matrices for a class of reducible equations

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This paper is a continuation of our previous work [15] where we have studied the Stokes phenomenon for a particular family of equation (1.1) with (1.2)-(1.3) from a perturbative point of view. Here we focis on the explicit computation of the Stokes matrices at the non-resonant irregular singularity for a more general situation.

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“…Global, formal, actual fundamental matrices and Stokes matrices. In [15] we proved that the equation (1.1) possesses a global fundamental matrix whose off-diagonal elements are expressed in terms of iterated integrals. In the present paper we are going to use the same fundamental matrix with respect to which we compute the Stokes matrices.…”

“…This equation, in general, has over CP 1 two singular points -an irregular singular point at the origin of Poincaré rank 1 and a regular singular point at x = ∞. In [15] we considered a particular family of equation (1.1) for α 1 = 0, α 2 = ν − 2, α 3 = ν − 4, β 1 = 1, β 2 = 2, β 3 = 0, where ν ∈ C is an arbitrary parameter. There we Date: 24.06.2019. investigated the Stokes phenomenon applying a perturbative approach.…”