Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

TL;DR

This paper investigates singular solutions of a perturbed PDE with cubic nonlinearity, constructing sectorial meromorphic solutions that share a common Gevrey asymptotic expansion in the perturbation parameter.

Contribution

It introduces two families of solutions with Gevrey asymptotics for a complex-perturbed PDE, extending previous work and analyzing their shared formal expansions.

Findings

Construction of sectorial meromorphic solutions

Identification of common Gevrey asymptotic expansions

Analysis of solution branches in the complex perturbation parameter

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.