Partial differential equations/Complex analysis
On the summability of divergent power series satisfying singular PDEs
Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 258-262.

The aim of this note is to apply the Borel–Laplace summation method studied by H. Chen, Z. Luo and C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) to the divergent power series solutions to two families of nonlinear PDEs. The first one contains particularly a two-dimensional version of the so-called Euler equation (ODE), while the second is called totally characteristic type PDE by H. Chen and H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).

Le but de cette Note est d'appliquer la méthode de sommation de Borel–Laplace étudiée par H. Chen, Z. Luo et C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) aux solutions séries entières de deux familles d'EDP non linéaires. La première contient particulièrement une version bidimensionnelle de ce qu'on appelle équation d'Éuler, alors que la seconde famille d'EDP est dite de type totalement caractéristique par H. Chen et H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.02.008
Chen, Hua 1, 2; Luo, Zhuangchu 1, 2; Zhang, Changgui 1, 2

1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2 Laboratoire P. Painlevé (UMR – CNRS 8524) et CEMPI, Dep. Math., FST, Université de Lille, Cité scientifique, 59655, Villeneuve d'Ascq cedex, France
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Chen, Hua; Luo, Zhuangchu; Zhang, Changgui. On the summability of divergent power series satisfying singular PDEs. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 258-262. doi : 10.1016/j.crma.2019.02.008. http://www.numdam.org/articles/10.1016/j.crma.2019.02.008/

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The research supported partially by the NSFC Grants (No. 11171261, 11371282 and 11631011).