Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain

• Autorzy: Masatake Miyake , Akira Shirai
• Języki publikacji: EN

Abstrakty

EN

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations
   (SE) f(x,u,D_x u) = 0 with u(0)=0.
Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,ξ^{0}) ∈ ℂ^{n}_{x} × ℂ_{u} × ℂ^{n}_{ξ} (ξ^{0} = D_{x}u(0))$ and $f(0,0,ξ^{0}) = 0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $(ξ ∈ ℂ^{n})$. The criterion of convergence of a formal solution $u(x) = ∑_{|α| ≥ 1} u_{α}x^{α}$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal solution diverges a precise rate of divergence or the formal Gevrey order is specified which can be interpreted in terms of the Newton polygon as in the case of linear equations but for nonlinear equations it depends on the individual formal solution.

Słowa kluczowe

EN

singular equation; formal solution

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 74
• Numer: 1
• Strony: 215-228

Daty

wydano

2000

otrzymano

1999-05-31

poprawiono

1999-12-10

Twórcy

autor

Masatake Miyake

• Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

autor

Akira Shirai

• Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

Bibliografia

• [G-T 1] R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations in Complex Domain, Vieweg, 1996.
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• [H 2] M. Hibino, Divergence property of formal solutions for singular first order linear partial differential equations, Publ. RIMS Kyoto Univ. 35 (1999), 893-919.
• [M] M. Miyake, Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations, J. Math. Soc. Japan 43 (1991), 305-330.
• [M-H] M. Miyake and Y. Hashimoto, Newton polygons and Gevrey indices for linear partial differential operators, Nagoya Math. J. 128 (1992), 15-47.
• [O] T. Oshima, On the theorem of Cauchy-Kowalevski for first order linear differential equations with degenerate principal symbols, Proc. Japan Acad. 49 (1973), 83-87.
• [R] J. P. Ramis, Théorèmes d'indices Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc. 48 (1984).
• [S 1] A. Shirai, Convergence of formal solutions to nonlinear first order singular partial differential equations, Master Thesis, Grad. School of Math., Nagoya Univ., 1998 (in Japanese).
• [S 2] A. Shirai, Maillet type theorem for nonlinear partial differential equations and the Newton polygons, J. Math. Soc. Japan, submitted.