Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

• Main Authors: Alberto Lastra, Stephane Malek
• Format: Article
• Language: English
• Published: Texas State University 2018-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: Asymptotic expansion; Borel-Laplace transform; Fourier transform; Cauchy problem; formal power series; nonlinear integro-differential equation; nonlinear partial differential equation; singular perturbation
• Online Access: http://ejde.math.txstate.edu/Volumes/2018/46/abstr.html

We study a singularly perturbed PDE with cubic nonlinearity depending on a complex perturbation parameter $\epsilon$. This is a continuation of the precedent work [22] 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.