Electron. J. Differential Equations, Vol. 2018 (2018), No. 46, pp. 1-89.

Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

Alberto Lastra, Stephane Malek

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.

Submitted July 6, 2017. Published February 13, 2018.
Math Subject Classifications: 35C10, 35C20.
Key Words: Asymptotic expansion; Borel-Laplace transform; Fourier transform; Cauchy problem; formal power series; nonlinear integro-differential equation; nonlinear partial differential equation; singular perturbation.

Show me the PDF file (720 KB), TEX file for this article.

Alberto Lastra
Dpto. de Física y Matemáticas
Universidad de Alcalá, Ap. Correos 20
E-28871 Alcalá de Henares, Madrid, Spain
email: alberto.lastra@uah.es
Stephane Malek
University of Lille 1, Laboratoire Paul Painlevé
59655 Villeneuve d'Ascq cedex, France
email: Stephane.Malek@math.univ-lille1.fr

Return to the EJDE web page