Generalized Poincaré Condition and Convergence of Formal Solutions of Some Nonlinear Totally Characteristic Equations

Abstract

This paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^{\alpha}u \}_{j+\alpha \leq m, j<m})$ that is of nonlinear totally characteristic type. The Newton Polygon at $x=0$ of the equation is defined, and by means of this polygon we define a generalized Poincaré condition (GP) and a condition (R) that the equation has a regular singularity at $x=0$. Under these conditions, (GP) and (R), it is proved that every formal power series solution is convergent in a neighborhood of the origin.