Resonant dynamics for the quintic nonlinear Schrödinger equation

Abstract

We consider the quintic nonlinear Schrödinger equation (NLS) on the circleWe 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. © 2012 RésuméNous considérons l'équation de Schrödinger non linéaire (NLS) quintique sur le cercleNous montrons qu'il existe des solutions issues d'une 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 d'énergie entre ces quatre modes initialement excités. Il est remarquable que ce phénomène non linéaire soit indépendant du choix de l'ensemble résonant. Le résultat dynamique est obtenu en mettant d'abord sous forme normale résonante jusqu'à l'ordre 10 l'Hamiltonien de NLS quintique puis en isolant un terme effectif d'ordre 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.

Long time behavior of the NLS-Szegő equation

“…In the first term of the preceding formula, there is a resonance set k 2 1 − k 2 2 + k 2 3 − k 2 4 + k 2 5 − k 2 6 = 0 that cannot be cancelled by the other two terms. We can see Grébert-Thomann [20] and Haus-Procesi [21] for instance to analyse the resonant set for 6 indices for the quintic NLS equation.…”

On nonlinear equation of Schrödinger type

“…The equation of such type were studied in many articles under different conditions on the function f in the dynamic case (see for example [2,4,6,7,8,9,14,17,18,20,22,24,31,34] and the references therein) and in the steady-state case (see, for example, [1,3,5,10,11,12,13,14,15,16,18,19,23,24,25,26,28,31,32] and references therein). It is known that in this case the equation (1) in the steady-state case (i.e.…”

“…NLS) (see, [1,2,3,10,13] and references therein). Considerable attention has been paid in recent years to the problem (1) for small ε > 0 as the coeffi-cients of the linear part since the solutions are known as in the semiclassical states, which can be used to describe the transition from quantum to classical mechanics (see, [3,10,11,12,13,14,23,24,25,31,32,33,34] and references therein).…”

A KAM algorithm for the resonant non-linear Schrödinger equation

“…This is due to the fact that the NLS equation after one step of Birkhoff normal form is integrable and non-degenerate. Unfortunately this very strong property holds only for the cubic NLS in dimension one, indeed for d > 1 the non-integrability of the NLS normal form has been exploited (see for instance [6] and [14]) to construct diffusive orbits. In order to overcome this problem Bourgain proposed the idea of choosing the initial data wisely.…”

“…We thus assume that the parameters ε, r, s satisfy (14). Formula (12) puts in action angle variables (y; x) = (y 1 , .…”