hal-00646217, version 1
Resonant dynamics for the quintic non linear Schrödinger equation
(2011-11-29)
Abstract: We consider the quintic nonlinear Schrödinger equation (NLS) on the circle. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomena does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomena can not occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.
- 1:
- CNRS : UMR6629 – Université de Nantes – École Centrale de Nantes
- Domain : Mathematics/Analysis of PDEs
- Keywords : Nonlinear Schrödinger equation – Resonant normal form – energy exchange.
- Comment : 32 pages – 4 figures
- hal-00646217, version 1
- http://hal.archives-ouvertes.fr/hal-00646217
- oai:hal.archives-ouvertes.fr:hal-00646217
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- Submitted on: Tuesday, 29 November 2011 14:32:34
- Updated on: Tuesday, 29 November 2011 15:38:54



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