On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

Abstract

We study a family of singularly perturbed linear partial differential equations with irregular type $ϵt^2{\partial }_t{\partial }_z^S{X}_i(t,z,ϵ)+(ϵt+1){\partial }_z^S{X}_i(t,z,ϵ)=\sum (s,k_0,k_1)\in 𝒮b_{s,k_0,k_1}(z,ϵ)t^s{\partial }_t^{k_0}{\partial }_z^{k_1}{X}_i(t,z,ϵ)$ in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter $ϵ$ converges near the origin in $ℂ$ and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say ${\kappa }_i\in [0,2\pi )$, $0\le i\le \nu -1$ for some integer $\nu \ge 2$. The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors $\{ϵ\in {ℂ}^{\ast }/\text{arg}(ϵ)\in ({\kappa }_i,{\kappa }_{i+1})\}$, where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.