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

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1548
Classification : 34M03, 34M25, 34M30, 34M35, 34M40
Mots clés : Linear differential system, multisummability, Stokes phenomenon, Stokes multipliers, resurgence, singularities, connection constants
Pascal Remy 1

1 Lycée Les Pierres Vives, 1 rue des Alouettes, 78420 Carrières-sur-Seine, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AFST_2017_6_26_3_645_0,
     author = {Pascal Remy},
     title = {Resurgence and highest level{\textquoteright}s {connection-to-Stokes} formul{\ae} for some linear meromorphic differential systems},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {645--685},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {3},
     year = {2017},
     doi = {10.5802/afst.1548},
     zbl = {1384.34092},
     mrnumber = {3669968},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1548/}
}
TY  - JOUR
AU  - Pascal Remy
TI  - Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2017
SP  - 645
EP  - 685
VL  - 26
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1548/
DO  - 10.5802/afst.1548
LA  - en
ID  - AFST_2017_6_26_3_645_0
ER  - 
%0 Journal Article
%A Pascal Remy
%T Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2017
%P 645-685
%V 26
%N 3
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1548/
%R 10.5802/afst.1548
%G en
%F AFST_2017_6_26_3_645_0
Pascal Remy. Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 645-685. doi : 10.5802/afst.1548. https://afst.centre-mersenne.org/articles/10.5802/afst.1548/

[1] Werner Balser Summation of formal power series through iterated Laplace integrals, Math. Scand., Volume 70 (1992) no. 2, pp. 161-171 | DOI | MR | Zbl

[2] Werner Balser Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer, New-York, 2000, xviii+299 pages | Zbl

[3] Werner Balser; Boele L. J. Braaksma; Jean-Pierre Ramis; Yasutaka Sibuya Multisummability of formal power series solutions of linear ordinary differential equations, Asymptot. Anal., Volume 5 (1991) no. 1, pp. 27-45 | DOI | MR | Zbl

[4] Werner Balser; Wolfgang B. Jurkat; Donald A. Lutz Birkhoff invariants and Stokes multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., Volume 71 (1979), pp. 48-94 | DOI | MR | Zbl

[5] Werner Balser; Wolfgang B. Jurkat; Donald A. Lutz A general theory of invariants for meromorphic differential equations; Part I, formal invariants, Funkcial. Ekvac., Volume 22 (1979), pp. 197-221 | Zbl

[6] Werner Balser; Wolfgang B. Jurkat; Donald A. Lutz On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities I, SIAM J. Math. Anal., Volume 12 (1981) no. 5, pp. 691-721 | DOI | MR | Zbl

[7] Werner Balser; Wolfgang B. Jurkat; Donald A. Lutz Transfer of connection problems for meromorphic differential equation of rank r2 and representations of solutions, J. Math. Anal. Appl., Volume 85 (1982), pp. 488-542 | DOI

[8] Werner Balser; Wolfgang B. Jurkat; Donald A. Lutz Transfer of connection problems for first level solutions of meromorphic differential equations and associated Laplace transforms, J. Reine Angew. Math., Volume 344 (1983), pp. 149-170 | MR | Zbl

[9] Werner Balser; Wolfgang B. Jurkat; Donald A. Lutz On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities II, SIAM J. Math. Anal., Volume 19 (1988) no. 2, pp. 398-443 | DOI | MR | Zbl

[10] George David Birkhoff Singular points of ordinary linear differential equations, Trans. Am. Math. Soc., Volume 10 (1909), pp. 436-470 | DOI | MR

[11] Boele L. J. Braaksma Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Differ. Equations, Volume 92 (1991), pp. 45-75 | DOI | MR | Zbl

[12] Boele L. J. Braaksma; Geertrui Klara Immink; Yasutaka Sibuya The Stokes phenomenon in exact asymptotics, Pac. J. Math., Volume 187 (1999) no. 1, pp. 13-50 | DOI | MR

[13] Pierre Deligne Equations différentielles à points singuliers réguliers, Lecture Notes in Math., 163, Springer, Berlin, 1970, 133 pages | Zbl

[14] J. Écalle Les fonctions résurgentes. III : l’équation du pont et la classification analytique des objets locaux, Publ. Math. Orsay, 85-05, Université Paris-Sud, 1985, 585 pages | Zbl

[15] Michèle Loday-Richaud Calcul des invariants de Birkhoff des systèmes d’ordre deux, Funkcial. Ekvac., Volume 33 (1990) no. 2, pp. 161-225 | MR | Zbl

[16] Michèle Loday-Richaud Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier, Volume 44 (1994) no. 3, pp. 849-906 | DOI | Numdam | MR | Zbl

[17] Michèle Loday-Richaud Rank reduction, normal forms and Stokes matrices, Expo. Math., Volume 19 (2001) no. 3, pp. 229-250 | DOI | MR | Zbl

[18] Michèle Loday-Richaud Divergent series, summability and resurgence II. Simple and multiple summability., Lecture Notes in Math., 2154, Springer, 2016, xxiii+272 pages | MR | Zbl

[19] Michèle Loday-Richaud; Pascal Remy Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems, J. Differ. Equations, Volume 250 (2011) no. 3, pp. 1591-1630 | DOI | MR | Zbl

[20] Donald A. Lutz; Reinhard Schäfke Calculating connection coefficients for meromorphic differential equations, Complex Var. Theory Appl., Volume 34 (1997) no. 1-2, pp. 145-170 | MR | Zbl

[21] Bernard Malgrange Introduction aux travaux de J. Écalle, Enseign. Math., Volume 31 (1985), pp. 261-282 | Zbl

[22] Bernard Malgrange Fourier transform and differential equations, Recent Developments in Quantum Mechanics (Mathematical Physics Studies), Volume 12 (1991), pp. 33-48 | MR

[23] Bernard Malgrange; Jean-Pierre Ramis Fonctions multisommables, Ann. Inst. Fourier, Volume 42 (1992) no. 1-2, pp. 353-368 | DOI | Numdam | MR | Zbl

[24] Jean Martinet; Jean-Pierre Ramis Théorie de Galois différentielle et resommation, Computer algebra and differential equations (1989), pp. 117-214 | Zbl

[25] Jean Martinet; Jean-Pierre Ramis Elementary acceleration and multisummability, Ann. Inst. H. Poincaré Phys. Théor., Volume 54 (1991) no. 4, pp. 331-401 | Numdam | MR | Zbl

[26] Juan J. Morales-Ruiz; Jean-Pierre Ramis Galoisian obstructions to integrability of Hamiltonian systems I & II, Methods Appl. Anal., Volume 8 (2001) no. 1, pp. 33-111 | MR | Zbl

[27] Juan J. Morales-Ruiz; Jean-Pierre Ramis; Carles Simó Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. Éc. Norm. Supér., Volume 40 (2007) no. 6, pp. 845-884 | DOI | Numdam | MR | Zbl

[28] Jean-Pierre Ramis Théorèmes d’indices Gevrey pour les équations différentielles ordinaires, Mem. Am. Math. Soc., Volume 286 (1984), viii+95 pages | Zbl

[29] Jean-Pierre Ramis Phénomène de Stokes et resommation, C. R. Acad. Sci. Paris, Volume 301 (1985) no. 4, pp. 99-102 | MR | Zbl

[30] Jean-Pierre Ramis Filtration de Gevrey sur le groupe de Picard-Vessiot d’une équation différentielle irrégulière (juin 1985), P. Deligne, B. Malgrange, J.-P. Ramis, Singularités irrégulières (Documents Mathématiques (Paris) (Mathematical Documents (Paris))), Volume 5, Société Mathématique de France, Paris, 2007

[31] Pascal Remy First level’s connection-to-Stokes formulae for meromorphic linear differential systems, Adv. Dyn. Syst. Appl., Volume 7 (2012) no. 2, pp. 249-285 | MR

[32] Pascal Remy Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique, Ann. Fac. Sci. Toulouse, Volume 21 (2012) no. 1, pp. 93-150 | DOI | Numdam | MR | Zbl

[33] David Sauzin Resurgent functions and splitting problems, RIMS Kôkyûroku, Volume 1493 (2005), pp. 48-117

[34] Yasutaka Sibuya Linear differential equations in the complex domain: problems of analytic continuation, Translation of Mathematical Monographs, 82, Amer. Math. Soc., 1990 | MR | Zbl

Cité par Sources :