Resonant dynamics for the quintic nonlinear Schrödinger equation

Grébert, Benoît; Thomann, Laurent

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 455-477.

Nous considérons lʼéquation de Schrödinger non linéaire (NLS) quintique sur le cercle

$$i{\partial }_{t}u+{\partial }_x^{2}u=\pm \nu {\left|u\right|}^{4}u,\nu \ll 1,x\in {𝕊}^1,t\in ℝ.$$

Nous montrons quʼil existe des solutions issues dʼune condition initiale construite sur quatre modes de Fourier formant un ensemble résonant (voir définition 1.1) ont une dynamique non triviale mettant en jeu des échanges périodiques dʼénergie entre ces quatre modes initialement excités. Il est remarquable que ce phénomène non linéaire soit indépendant du choix de lʼensemble résonant.Le résultat dynamique est obtenu en mettant dʼabord sous forme normale résonante jusquʼà lʼordre 10 lʼHamiltonien de NLS quintique puis en isolant un terme effectif dʼordre 6. Il est à noter que ce phénomène ne peut pas se produire pour NLS cubique pour lequel les amplitudes des modes de Fourier sont des presque-actions et donc ne varient quasiment pas au cours du temps.

We consider the quintic nonlinear Schrödinger equation (NLS) on the circle

$$i{\partial }_{t}u+{\partial }_x^{2}u=\pm \nu {\left|u\right|}^{4}u,\nu \ll 1,x\in {𝕊}^1,t\in ℝ.$$

We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set (see Definition 1.1), which have a nontrivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomenon 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 phenomenon cannot occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.

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