Maillet type theorem for nonlinear totally characteristic partial differential equations

Abstract

The paper discusses a holomorphic nonlinear singular partial differential equation $(t{\partial }_t{)}^{m}u=F(t,x,\{(t{\partial }_t{)}^j{\partial }_x^{\alpha }u{\}}_{j+\alpha \le m,j<m})$ under the assumption that the equation is of nonlinear totally characteristic type. By using the Newton Polygon at $x=0$, the notion of the irregularity at $x=0$ 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.