Summability of the formal power series solutions of a certain class of inhomogeneous nonlinear partial differential equations with a single level

Abstract

In this article, we investigate the summability of the formal power series solutions in time of a class of inhomogeneous nonlinear partial differential equations in two variables, whose the attached Newton polygon admits a unique positive slope k, the latter being determined by the highest spatial-derivative order of the initial equation. We give in particular a necessary and sufficient condition for the k-summability of the solutions in a given direction, and we illustrate this result by some examples. This condition generalizes the ones already given by the author in the linear case [1,2] and, more recently, in the semilinear case [3,4]. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof of our main result.