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 di¤erential 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.

No abstract

Abstract. In this paper, we study the following nonlinear first order partial differential equation:f (t, x, u, ∂tu, ∂xu) = 0 with u(0, x) ≡ 0. Especially the last-mentioned paper is regarded as part I of this paper.

We shall study the first order singular nonlinear

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations (SE) f (x, u, Dxu) = 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 × Cu × C n ξ (ξ 0 = Dxu(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 uαx 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.