On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

Abstract

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter $\mathrm{ϵ}$. The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in $\mathbb{C}$. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in $\mathrm{ϵ}$ of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in $\mathrm{ϵ}$ as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.