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

• Volume: 74, Issue: 1, page 215-228
• ISSN: 0066-2216

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## Abstract

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We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $\left(0,0,{\xi }^{0}\right)\in {ℂ}_{x}^{n}×{ℂ}_{u}×{ℂ}_{\xi }^{n}\left({\xi }^{0}={D}_{x}u\left(0\right)\right)$ and $f\left(0,0,{\xi }^{0}\right)=0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $\left(\xi \in {ℂ}^{n}\right)$. The criterion of convergence of a formal solution $u\left(x\right)={\sum }_{|\alpha |\ge 1}{u}_{\alpha }{x}^{\alpha }$ 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.

## How to cite

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Miyake, Masatake, and Shirai, Akira. "Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain." Annales Polonici Mathematici 74.1 (2000): 215-228. <http://eudml.org/doc/208367>.

@article{Miyake2000,
abstract = {We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,Dx 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.},
author = {Miyake, Masatake, Shirai, Akira},
journal = {Annales Polonici Mathematici},
keywords = {singular equation; formal solution; formal power series solutions; precise rate of divergence; Gevrey order; Newton polygon; convergence},
language = {eng},
number = {1},
pages = {215-228},
title = {Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain},
url = {http://eudml.org/doc/208367},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Miyake, Masatake
AU - Shirai, Akira
TI - Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 215
EP - 228
AB - We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,Dx 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.
LA - eng
KW - singular equation; formal solution; formal power series solutions; precise rate of divergence; Gevrey order; Newton polygon; convergence
UR - http://eudml.org/doc/208367
ER -

## References

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1. [G-T 1] R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations in Complex Domain, Vieweg, 1996. Zbl0874.35001
2. [G-T 2] R. Gérard and H. Tahara, Formal power series solutions of nonlinear first order partial differential equations, Funkcial. Ekvac. 41 (1998), 133-166. Zbl1142.35310
3. [H 1] M. Hibino, Gevrey asymptotic expansion for singular first order linear partial differential equations of nilpotent type, Master Thesis, Grad. School of Math., Nagoya Univ., 1998 (in Japanese).
4. [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. Zbl0956.35024
5. [M] M. Miyake, Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations, J. Math. Soc. Japan 43 (1991), 305-330. Zbl0743.35015
6. [M-H] M. Miyake and Y. Hashimoto, Newton polygons and Gevrey indices for linear partial differential operators, Nagoya Math. J. 128 (1992), 15-47. Zbl0815.35007
7. [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. Zbl0283.35002
8. [R] J. P. Ramis, Théorèmes d'indices Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc. 48 (1984). Zbl0555.47020
9. [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).
10. [S 2] A. Shirai, Maillet type theorem for nonlinear partial differential equations and the Newton polygons, J. Math. Soc. Japan, submitted. Zbl0995.35002

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