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

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 sho...

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Main Authors: A. Lastra, S. Malek
Format: Article
Language:English
Published: Wiley 2014-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2014/153169
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author A. Lastra
S. Malek
author_facet A. Lastra
S. Malek
author_sort A. Lastra
collection DOAJ
description 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.
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institution Kabale University
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series Abstract and Applied Analysis
spelling doaj-art-9f11259bf2d34ea6afdc6d9cfb2581cd2025-02-03T01:10:39ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/153169153169On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function SpacesA. Lastra0S. Malek1Departamento de Física y Matemáticas, University of Alcalá, Apartado de Correos 20, 28871 Alcalá de Henares, SpainLaboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d’Ascq Cedex, FranceWe 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.http://dx.doi.org/10.1155/2014/153169
spellingShingle A. Lastra
S. Malek
On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces
Abstract and Applied Analysis
title On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces
title_full On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces
title_fullStr On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces
title_full_unstemmed On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces
title_short On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces
title_sort on parametric gevrey asymptotics for some cauchy problems in quasiperiodic function spaces
url http://dx.doi.org/10.1155/2014/153169
work_keys_str_mv AT alastra onparametricgevreyasymptoticsforsomecauchyproblemsinquasiperiodicfunctionspaces
AT smalek onparametricgevreyasymptoticsforsomecauchyproblemsinquasiperiodicfunctionspaces