# A new proof of multisummability of formal solutions of non linear meromorphic differential equations

• Volume: 44, Issue: 3, page 811-848
• ISSN: 0373-0956

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## Abstract

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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 equation$x\frac{d\stackrel{\to }{y}}{dx}={\stackrel{\to }{G}}_{0}\left(x\right)+\left[\lambda \left(x\right)+{A}_{0}\right]\stackrel{\to }{y}+{x}^{\mu }\stackrel{\to }{G}\left(x,\stackrel{\to }{y}\right),$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.

## How to cite

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Ramis, Jean-Pierre, and Sibuya, Yasutaka. "A new proof of multisummability of formal solutions of non linear meromorphic differential equations." Annales de l'institut Fourier 44.3 (1994): 811-848. <http://eudml.org/doc/75082>.

@article{Ramis1994,
abstract = {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 equation\begin\{\} x\{d\vec\{y\}\over dx\}=\vec\{G\}\_\{0\}(x) +\left[ \lambda (x) + A\_\{0\} \right]\vec\{y\} + x^\{\mu \} \vec\{G\}(x,\vec\{y\}) , \end\{\}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.},
author = {Ramis, Jean-Pierre, Sibuya, Yasutaka},
journal = {Annales de l'institut Fourier},
keywords = {nonlinear meromorphic differential equations; multisummability; formal power series solutions; nonlinear Stokes phenomenon; normal forms; resonances; exponential decay},
language = {eng},
number = {3},
pages = {811-848},
publisher = {Association des Annales de l'Institut Fourier},
title = {A new proof of multisummability of formal solutions of non linear meromorphic differential equations},
url = {http://eudml.org/doc/75082},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Ramis, Jean-Pierre
TI - A new proof of multisummability of formal solutions of non linear meromorphic differential equations
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 3
SP - 811
EP - 848
AB - 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 equation\begin{} x{d\vec{y}\over dx}=\vec{G}_{0}(x) +\left[ \lambda (x) + A_{0} \right]\vec{y} + x^{\mu } \vec{G}(x,\vec{y}) , \end{}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.
LA - eng
KW - nonlinear meromorphic differential equations; multisummability; formal power series solutions; nonlinear Stokes phenomenon; normal forms; resonances; exponential decay
UR - http://eudml.org/doc/75082
ER -

## References

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1. [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. Zbl0754.34057MR93f:34011
2. [2] B.L.J. BRAAKSMA, Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Differential Equations, 92 (1991), 45-75. Zbl0729.34005MR93c:34010
3. [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. Zbl0759.34003MR93j:34006
4. [4] J. ECALLE, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Act. Math., Ed. Hermann, Paris (1993).
5. [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. Zbl0023.22802MR2,49cJFM66.0397.02
6. [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. Zbl0089.29102
7. [7] B. MALGRANGE and J.-P. RAMIS, Fonctions multisommables, Ann. Inst. Fourier, Grenoble, 42, 1-2 (1992), 353-368. Zbl0759.34007MR93e:40007
8. [8] J. MARTINET and J.-P. RAMIS, Elementary acceleration and multisummability, Ann. Inst. Henri Poincaré, Physique Théorique, 54 (1991), 331-401. Zbl0748.12005MR93a:32036
9. [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. [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. Zbl0699.34058MR90k:58209
11. [11] Y. SIBUYA, Normal forms and Stokes multipliers of non linear meromorphic differential equations, Computer Algebra and Differential Equations, 3 (1994), Academic Press.
12. [12] Y. SIBUYA, Linear Differential Equations in the Complex Domain. Problems of Analytic Continuation, Transl. of Math. Monographs, Vol. 82, A. M. S., (1990). Zbl1145.34378

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