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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 153169, 29 pages
http://dx.doi.org/10.1155/2014/153169
Research Article

On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

1Departamento de Física y Matemáticas, University of Alcalá, Apartado de Correos 20, 28871 Alcalá de Henares, Spain
2Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Received 18 March 2014; Accepted 18 June 2014; Published 22 December 2014

Academic Editor: Graziano Crasta

Copyright © 2014 A. Lastra and S. 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 investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.