On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity
We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. 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 C. We constr...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2017/9405298 |
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| _version_ | 1832549227575640064 |
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| author | Stéphane Malek |
| author_facet | Stéphane Malek |
| author_sort | Stéphane Malek |
| collection | DOAJ |
| description | We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. 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 C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ 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 ϵ 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. |
| format | Article |
| id | doaj-art-43021a360e384f91abf3a87913b7979f |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-43021a360e384f91abf3a87913b7979f2025-02-03T06:11:47ZengWileyAbstract and Applied Analysis1085-33751687-04092017-01-01201710.1155/2017/94052989405298On Singular Solutions to PDEs with Turning Point Involving a Quadratic NonlinearityStéphane Malek0Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d’Ascq Cedex, FranceWe study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter ϵ. 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 C. We construct a family of sectorial meromorphic solutions obtained as a small perturbation in ϵ 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 ϵ 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.http://dx.doi.org/10.1155/2017/9405298 |
| spellingShingle | Stéphane Malek On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity Abstract and Applied Analysis |
| title | On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity |
| title_full | On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity |
| title_fullStr | On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity |
| title_full_unstemmed | On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity |
| title_short | On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity |
| title_sort | on singular solutions to pdes with turning point involving a quadratic nonlinearity |
| url | http://dx.doi.org/10.1155/2017/9405298 |
| work_keys_str_mv | AT stephanemalek onsingularsolutionstopdeswithturningpointinvolvingaquadraticnonlinearity |