On the parametric Stokes phenomenon for solutions of singularly perturbed linear partial differential equations (2012)
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@MISC{Malek12onthe,
author = {Stéphane Malek},
title = {On the parametric Stokes phenomenon for solutions of singularly perturbed linear partial differential equations},
year = {2012}
}
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Abstract
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.
Keyphrases
linear partial differential equation borel transform parametric stokes phenomenon analytic continuation stokes phenomenon singular perturbation partial differential equation formal solution sch fke perturbation parameter converges formal power series logarithmic type isolated singularity sectorial solution finite number sufficient condition sector arg new condition previous work perturbation parameter unbounded sector accurate description classical ramis-sibuya theorem small opening asymptotic expansion half lattice borel-laplace transform complex domain bisecting direction irregular type cauchy problem integro-differential equation sectorial holomorphic solution key word small bound
