Maillet type theorem for nonlinear totally characteristic partial differential equations

Abstract

The paper discusses a holomorphic nonlinear singular partial differential equation (t∂t)mu=F(t,x,{(t∂t)j∂αxu}j+α≤m,j<m)(t∂t)mu=F(t,x,{(t∂t)j∂xαu}j+α≤m,j<m) under the assumption that the equation is of nonlinear totally characteristic type. By using the Newton polygon at x=0x=0, the notion of the irregularity at x=0x=0 of the equation is defined. In the case where the irregularity is greater than one, it is proved that every formal power series solution belongs to a suitable formal Gevrey class. The precise bound of the order of the formal Gevrey class is given, and the optimality of this bound is also proved in a generic case.