Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems

Remy, Pascal

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 645-685.

Dans cet article, nous considérons un système différentiel linéaire méromorphe à multiples niveaux. Nous démontrons que les transformées de Borel de ses solutions formelles réduites de plus haut niveau sont résurgentes-sommables et nous donnons la forme générale de toutes leurs singularités. Celle-ci est ensuite précisée pour certaines configurations géométriques des points singuliers. Pour ces mêmes configurations, nous énonçons également des formules exactes permettant d’exprimer les multiplicateurs de Stokes de plus haut niveau du système initial à l’aide de constantes de connexion dans le plan de Borel, généralisant ainsi les formules déjà données par M. Loday-Richaud et l’auteur pour les systèmes de niveau unique. Ces formules sont illustrées par un exemple.

In this article, we consider a linear meromorphic differential system with several levels. We prove that the Borel transforms of its highest level’s reduced formal solutions are summable-resurgent and we give the general form of all their singularities. This one is then precised in restriction to some convenient hypotheses on the geometric configuration of singular points. Next, under the same hypotheses, we state exact formulæ to express some highest level’s Stokes multipliers of the initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already displayed by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one example.

Classification : 34M03, 34M25, 34M30, 34M35, 34M40
Mots clés : Linear differential system, multisummability, Stokes phenomenon, Stokes multipliers, resurgence, singularities, connection constants

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