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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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| 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. |
| format | Article |
| id | doaj-art-9f11259bf2d34ea6afdc6d9cfb2581cd |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| 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 |