Zero level perturbation of a certain third-order linear solvable ODE with an irregular singularity at the origin of Poincar\'e rank 1

TL;DR

This paper investigates how a small perturbation in a third-order linear differential equation with an irregular singularity causes the singularity to split, revealing a relationship between Stokes matrices and monodromy matrices in the limit.

Contribution

It introduces a perturbation method for a third-order ODE with an irregular singularity, linking Stokes matrices to monodromy matrices through limiting behavior.

Findings

Irregular singularity splits into two Fuchsian singularities under perturbation.

Logarithmic solutions relate Stokes matrices to monodromy matrices.

Limits of monodromy matrices approximate original Stokes matrices.

Abstract

We study an irregular singularity of Poincar\'e rank 1 at the origin of a certain third-order linear solvable homogeneous ODE. We perturb the equation by introducing a small parameter $\varepsilon\in (\mathbb{R}_+, 0)\,(\varepsilon < 1)$ 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.