12-73 S. Malek

On the parametric Stokes phenomenon for solutions of singularly perturbed linear partial differential equations (761K, pdf) Jul 9, 12

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Abstract. We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work we have given sufficient conditions under which the Borel transform of a formal solution with respect to the perturbation parameter converges near the origin in the complex domain and can be extended on a finite number of unbounded sectors with small opening. The proof rests on the construction of neighboring sectorial holomorphic solutions 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 larger sectors 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. Schaefke in a recent paper and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.

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