On the parametric Stokes phenomenon for solutions of singularly perturbed linear partial differential equations


We study a family of singularly perturbed linear partial differential equations with irregular type (1) in the complex domain. In a previous work [31], we have given sufficient conditions under which the Borel transform of a formal solution to (1) with respect to the perturbation parameter ɛ converges near the origin in C and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say κi ∈ [0, 2π), 0 ≤ i ≤ ν − 1 for some integer ν ≥ 2. The proof rests on the construction of neighboring sectorial holomorphic solutions to (1) whose difference 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 {ɛ ∈ C ∗ /arg(ɛ) ∈ (κi, κi+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 A. Fruchard and R. Schäfke in [19] and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above

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