Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

Abstract

We study a singularly perturbed PDE with cubic nonlinearity depending on a complex perturbation parameter $\epsilon$. This is the continuation of a precedent work by the first author. We construct two families of sectorial meromorphic solutions obtained as a small perturbation in $\epsilon$ of two branches of an algebraic slow curve of the equation in time scale. We show that the nonsingular part of the solutions of each family shares a common formal power series in $\epsilon$ as Gevrey asymptotic expansion which might be different one to each other, in general.