Resonant dynamics for the quintic non linear Schrödinger equation

Benoît Grébert (1) , Laurent Thomann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: energy exchange.; energy exchange; Nonlinear Schrödinger equation; Resonant normal form

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]

Benoît Grébert, Laurent Thomann. Resonant dynamics for the quintic non linear Schrödinger equation. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2012, 29 (3), pp.455--477. ⟨hal-00646217⟩