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

Abstract

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.

Convergence or Divergence of Formal Solutions of First Order Singular Nonlinear Ordinary Differential Equations in Complex Domain

“…On the other hand, M. Miyake and A. Shirai [2,3] In this paper we shall extend a part of results of [2] in the case of ordinary di¤erential equations. Precisely, in [2] for the criterion of convergence or divergence of formal solution the Taylor coe‰cients of first order of formal solution and equation are used, but in this paper the Taylor coe‰cients of higher order are used.…”

Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type. Part II

“…To this equation, a lot of Maillet type theorems have been studied by many mathematicians. For example, Gérard-Tahara, and Miyake-Shirai study the nonlinear case, which is found in the book or papers [4,6,7] and [8]. They obtained the Maillet type theorem, which include the convergent case.…”

Existence of Classical Solutions near Characteristic Points of First Order Nonlinear Partial Differential Equations

“…As for existence of analytic solutions in [7], [8], [9], they firstly constructed a solutions of formal power series and next showed the convergence under a condition called Poincaré's condition, which is essential to show the convergence. In this paper F ðx; u; pÞ is real valued, so Poincaré's condition is as follows: Definition 1.4.…”

“…However, if Under the condition that F ðx; u; pÞ is an analytic function, there are several reseaches about F ðx; u; u x Þ ¼ 0 with (0.2). For example, its solutions of formal power series, existence of asymptotic or analytic solutions were studied by using analytical functions methods ( [1], [4], [6], [7], [8], [9] and [10] etc. )…”