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Abstract and Applied Analysis
Volume 2017, Article ID 9405298, 32 pages
https://doi.org/10.1155/2017/9405298
Research Article

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

Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Correspondence should be addressed to Stéphane Malek; rf.1ellil-vinu.htam@kelam.enahpets

Received 5 May 2017; Accepted 1 August 2017; Published 13 September 2017

Academic Editor: Sining Zheng

Copyright © 2017 Stéphane Malek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter . 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 . We construct a family of sectorial meromorphic solutions obtained as a small perturbation in 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 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.