Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity
Annales de l'Institut Fourier, Volume 51 (2001) no. 6, pp. 1599-1620.

In this paper, we calculate the formal Gevrey index of the formal solution of a class of nonlinear first order totally characteristic type partial differential equations with irregular singularity in the space variable. We also prove that our index is the best possible one in a generic case.

Dans cet article, nous calculons l'indice Gevrey des solutions formelles (avec des conditions initiales données) d'une certaine classe d'équations aux dérivées partielles non linéaires du premier ordre, du type totalement caractéristique et ayant une singularité irrégulière en la variable spatiale. Nous montrons également que l'indice obtenu est génériquement optimal.

DOI: 10.5802/aif.1867
Classification: 35A07, 35A10, 35A20
Keywords: formal solution, totally characteristic PDF, Gevrey index
Mot clés : solution formelle, PDF totalement caractéristique, indice Gevrey
Chen, Hua 1; Luo, Zhuangchu 2; Tahara, Hidetoshi 

1 Wuhan University, Institute of Mathematics, Wuhan (Rép. Pop. Chine)
2 Sophia University, Department of Mathematics, Tokyo (Japon)
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     title = {Formal solutions of nonlinear first order totally characteristic type {PDE} with irregular singularity},
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Chen, Hua; Luo, Zhuangchu; Tahara, Hidetoshi. Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity. Annales de l'Institut Fourier, Volume 51 (2001) no. 6, pp. 1599-1620. doi : 10.5802/aif.1867. http://archive.numdam.org/articles/10.5802/aif.1867/

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math., Volume 115 (1982) | MR | Zbl

[1] H. Chen; H. Tahara On the holomorphic solution of non-linear totally characteristic equations (To appear in Mathematische Nachrichten, Germany) | Zbl

[2] H. Chen; H. Tahara On totally characteristic type non-linear partial differential equations in the complex domain, Publ. RIMS, Kyoto Univ., Volume 26 (1999), pp. 621-636 | DOI | MR | Zbl

[3] H. Chen; Z. Luo On the holomorphic solution of nonlinear totally characteristic equations with several space variables (Preprint) | Zbl

[4] R. Gérard; H. Tahara Nonlinear singular first order partial differential equations of Briot-Bouquet type, Proc. Japan Acad., Volume 66 (1990), pp. 72-74 | DOI | MR | Zbl

[5] R. Gérard; H. Tahara Holomorphic and singular solution of nonlinear singular first order partial differential equations, Publ. RIMS, Kyoto Univ., Volume 26 (1990), pp. 979-1000 | DOI | MR | Zbl

[6] R. Gérard; H. Tahara Singular nonlinear partial differential equations, Aspects of Mathematics, E 28, Vieweg, 1996 | MR | Zbl

[7] R. Gérard; H. Tahara Formal power series solutions of nonlinear first order partial differential equations, Funkcial. Ekvac., Volume 41 (1998), pp. 133-166 | MR | Zbl

[8] S. Ouchi Formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, Volume 1 (1994), pp. 205-237 | MR | Zbl

[9] A. Shirai Maillet type theorems for nonlinear partial differential equations and the Newton polygons (Preprint) | MR | Zbl

[10] E. T. Whittaker; G. N. Watson A course of modern analysis, Cambridge Univ. Press, 1958 | MR

[11] H. Yamazawa Newton polyhedrons and a formal Gevrey space of double indices for linear partial differential operators, Funkcial. Ekvac., Volume 41 (1998), pp. 337-345 | MR | Zbl

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