A new proof of multisummability of formal solutions of non linear meromorphic differential equations
Annales de l'Institut Fourier, Tome 44 (1994) no. 3, p. 811-848
Nous donnons une nouvelle preuve de la multisommabilité des solutions séries formelles des équations différentielles méromorphes non linéaires. Nous utilisons une définition récente de la multisommabilité due à B. Malgrange et J.-P. Ramis. La première démonstration du résultat central est due à B. Braaksma. Notre méthode est très différente : Braaksma utilisait la définition de J. Écalle de la multisommabilité et la transformation de Laplace. Partant d’une forme normale préliminairexdydx=G0(x)+λ(x)+A0y+xμG(x,y),l’idée de notre démonstration est de représenter une solution série formelle par une cochaîne holomorphe, dont le cobord est exponentiellement petit d’un certain ordre. Ensuite on augmente cet ordre en un nombre fini d’étapes. (Pour cela on utilise la connaissance des pentes d’un polygone de Newton.) Le lemme clé est basé sur des réductions à des formes normales résonnantes et sur l’analyse détaillée de phénomènes de Stokes non linéaires.
We give a new proof of multisummability of formal power series solutions of a non linear meromorphic differential equation. We use the recent Malgrange-Ramis definition of multisummability. The first proof of the main result is due to B. Braaksma. Our method of proof is very different: Braaksma used Écalle definition of multisummability and Laplace transform. Starting from a preliminary normal form of the differential equationxdydx=G0(x)+λ(x)+A0y+xμG(x,y),the idea of our proof is to interpret a formal power series solution as a holomorphic cochain, whose coboundary is exponentially small of some order. Then we increase this order in a finite number of steps. (In this process we use the knowledge of the slopes of a Newton polygon.) The key lemma is based on reductions to some resonant normal forms and on a precise description of some non linear Stokes phenomena.
@article{AIF_1994__44_3_811_0,
     author = {Ramis, Jean-Pierre and Sibuya, Yasutaka},
     title = {A new proof of multisummability of formal solutions of non linear meromorphic differential equations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {3},
     year = {1994},
     pages = {811-848},
     doi = {10.5802/aif.1418},
     zbl = {0812.34005},
     mrnumber = {95h:34012},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_3_811_0}
}
Ramis, Jean-Pierre; Sibuya, Yasutaka. A new proof of multisummability of formal solutions of non linear meromorphic differential equations. Annales de l'Institut Fourier, Tome 44 (1994) no. 3, pp. 811-848. doi : 10.5802/aif.1418. http://www.numdam.org/item/AIF_1994__44_3_811_0/

[1] W. Balser, B.L.J. Braaksma, J.-P. Ramis and Y. Sibuya, Multisummability of formal power series solutions of linear ordinary differential equations, Asymptotic Analysis, 5 (1991), 27-45. | MR 93f:34011 | Zbl 0754.34057

[2] B.L.J. Braaksma, Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Differential Equations, 92 (1991), 45-75. | MR 93c:34010 | Zbl 0729.34005

[3] B.L.J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier, Grenoble, 42-3 (1992), 517-540. | Numdam | MR 93j:34006 | Zbl 0759.34003

[4] J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Act. Math., Ed. Hermann, Paris (1993).

[5] M. Hukuhara, Intégration formelle d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier, Ann. di Mat. Pura Appl., 19 (1940), 34-44. | JFM 66.0397.02 | MR 2,49c | Zbl 0023.22802

[6] M. Iwano, Intégration analytique d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier, I, II, Ann. di Mat. Pura Appl., 44 (1957), 261-292 et 47 (1959), 91-150. | Zbl 0089.29102

[7] B. Malgrange and J.-P. Ramis, Fonctions multisommables, Ann. Inst. Fourier, Grenoble, 42, 1-2 (1992), 353-368. | Numdam | MR 93e:40007 | Zbl 0759.34007

[8] J. Martinet and J.-P. Ramis, Elementary acceleration and multisummability, Ann. Inst. Henri Poincaré, Physique Théorique, 54 (1991), 331-401. | Numdam | MR 93a:32036 | Zbl 0748.12005

[9] J.-P. Ramis, Les séries k-sommables et leurs applications, Analysis, Microlocal Calculus and Relativistic Quantum Theory, Proc. “Les Houches” 1979, Springer Lecture Notes in Physics, 126 (1980), 178-199.

[10] J.-P. Ramis and Y. Sibuya, Hukuhara's domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type, Asymptotic Analysis, 2 (1989), 39-94. | MR 90k:58209 | Zbl 0699.34058

[11] Y. Sibuya, Normal forms and Stokes multipliers of non linear meromorphic differential equations, Computer Algebra and Differential Equations, 3 (1994), Academic Press.

[12] Y. Sibuya, Linear Differential Equations in the Complex Domain. Problems of Analytic Continuation, Transl. of Math. Monographs, Vol. 82, A. M. S., (1990). | Zbl 00048899