Structure of Formal Solutions of Nonlinear First Order Singular Partial Differential Equations in Complex Domain

Abstract

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.