MR2154381
Miyake, Masatake and Shirai, Akira
Masatake MIYAKE and Akira SHIRAI
Structure of Formal Solutions of Nonlinear First Order Singular Partial Differential Equations in Complex Domain
Funkcialaj Ekvacioj. Serio Internacia
48
2005
113--136
http://fe.math.kobe-u.ac.jp/FE/FullPapers/48-1/48_113.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2154381
This paper is a continuation of our early paper [MS]. In this paper, we study the structure of formal power series solutions of a first order nonlinear partial differential equation which is defined and holomorphic in a neighborhood of the origin of several complex variables. Our main theorem (Theorem 1.1) characterizes the convergence or the divergence of a given formal power series solution a priori. Especially, in the case of divergence, we give the rate of divergence in terms of Gevrey index which is known as Maillet type theorem (Theorem 1.1, (ii)). It should be mentioned that the notion of singularity or singular point for nonlinear equations depends on each solution, and the coexistence of a convergent solution and a divergent solution for a nonlinear equation is possible.
Singular equation, Formal solution, Convergence, Maillet type theorem, Gevrey index.
Primary 35F20, Secondary 35A20, 35C10.
48-113
2005
Structure of Formal Solutions of Nonlinear First Order Singular Partial Differential Equations in Complex Domain
MIYAKE, Masatake and SHIRAI, Akira
Masatake MIYAKE and Akira SHIRAI
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