Formal Gevrey theory for singular first order semi-linear partial differential equations

Abstract

This paper is concerned with the existence, the uniqueness, convergence and divergence of formal power series solutions of singular first order quasi-linear partial differential equations. Firstly we give the condition under which the formal solution exists uniquely. However, this formal power series solution does not necessarily converge. So we characterize the rate of divergence of the formal solution via the Gevrey order of formal power series. The Gevrey orders of formal solutions are determined by the Newton polyhedrons of nonlinear partial differential operators.