In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the L 2 critical and super-critical NLS (without harmonic potential).

Abstract. We study the continuous resonant (CR) equation which was derived in [7] as the large-box limit of the cubic nonlinear Schrödinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schrödinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the striking structure of the equation. We study in particular the dynamics on a few invariant subspaces: eigenspaces of the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space. We focus on stationary waves and their stability.

Abstract. -In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.

We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in . Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod convergence theorem. To begin with, we consider the non linear Schrödinger equation (NLS) on the tri-dimensional sphere. Then we focus on the Benjamin-Ono equation and on the derivative nonlinear Schrödinger equation on the circle. Next, we construct a Gibbs measure and global solutions to the so-called periodic half-wave equation. Finally, we consider the cubic 2d defocusing NLS on an arbitrary spatial domain and we construct global solutions on the support of the associated Gibbs measure.

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.

We study dynamical properties of the cubic lowest Landau level equation, which is used in the modeling of fast rotating Bose-Einstein condensates. We obtain bounds on the decay of general stationary solutions. We then provide a classification of stationary waves with a finite number of zeros. Finally, we are able to establish which of these stationary waves are stable, through a variational analysis.2000 Mathematics Subject Classification. 35Q55 ; 37K05 ; 35C07 ; 35B08. 1 2 PATRICK GÉRARD, PIERRE GERMAIN, AND LAURENT THOMANN 7.2. Minimizers of G µ = 8πH + µP for M fixed 30 7.3. Minimizers of P for H and M fixed 33 7.4. Stability of stationary waves with finite mass and a finite number of zeros 37 Appendix A. Some explicit M -stationary waves 38 Appendix B. The dictionary 40 Appendix C. Sobolev spaces 41 References 42

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