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

Annales Polonici Mathematici (2000)

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

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topMiyake, 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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- [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
- [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
- [R] J. P. Ramis, Théorèmes d'indices Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc. 48 (1984). Zbl0555.47020
- [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. Zbl0995.35002

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