Maillet type theorem for nonlinear totally characteristic partial differential equations

TL;DR

This paper extends Maillet's theorem to certain nonlinear totally characteristic PDEs, establishing conditions under which formal solutions are Gevrey and providing bounds on their regularity.

Contribution

It introduces a new framework using the Newton Polygon to analyze irregularity and Gevrey class bounds for nonlinear PDEs of totally characteristic type.

Findings

Formal solutions are Gevrey when irregularity exceeds one.

Bounds on the Gevrey order are explicitly given.

Optimality of bounds is proved in generic cases.

Abstract

The paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^{\alpha}u \}_{j+\alpha \leq m, j<m})$ 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.