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| HAL: hal-00622240, version 1 |
| arXiv: 1109.2407 |
| Detailed view |  Export this paper |
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| Available versions: | v1 (2011-09-12) | v2 (2012-10-11) |
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| Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus |
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| Erwan Faou 1, 2Ludwig Gauckler 3 |
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| (2011-09-12) |
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| It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time. |
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| 1: | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne |
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| 2: | IPSO (INRIA - IRMAR) |
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CNRS : UMR6074 – INRIA – Université de Rennes 1 |
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| 3: | Institut für Mathematik [Berlin] |
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Technische Universität Berlin |
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| 4: | Mathematisches Institut [Tubigen] |
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Universität Tübingen |
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| Subject | : | Mathematics/Analysis of PDEs |
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| Nonlinear Schrödinger equation – Normal form – Modulated Fourier expansion |
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| hal-00622240, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00622240 | |
| oai:hal.archives-ouvertes.fr:hal-00622240 | |
| From: Marie-Annick Guillemer | |
| Submitted on: Monday, 12 September 2011 11:07:25 | |
| Updated on: Monday, 12 September 2011 11:09:49 | |
