On reducibility of Quantum Harmonic Oscillator on Rd\mathbb{R}^d with quasiperiodic in time potential

We prove that a linear d-dimensional Schr{\"o}dinger equation on $\mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\partial\_t u -- \Delta u + |x|^2 u + \epsilon V (t\omega, x)u = 0, x \in \mathbb{R}^d$ reduces to an autonomous system for most values of the frequency vector $\omega \in \mathbb{R}^n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms

KAM for the Klein Gordon equation on Sd\mathbb S^d

Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the structure of the resonant sets is quite simple. In the present paper, we consider an important physical example that do not fit in this context: the Klein Gordon equation on $\mathbb S^d$. Our abstract KAM theorem also allow to prove the reducibility of the corresponding linear operator with time quasiperiodic potentials.Comment: arXiv admin note: substantial text overlap with arXiv:1410.808

Modified scattering for the cubic Schr{\"o}dinger equation on product spaces: the nonresonant case

We consider the cubic nonlinear Schr{\"o}dinger equation on the spatial domain $\mathbb{R}\times \mathbb{T}^d$, and we perturb it with a convolution potential. Using recent techniques of Hani-Pausader-Tzvetkov-Visciglia, we prove a modified scattering result and construct modified wave operators, under generic assumptions on the potential. In particular, this enables us to prove that the Sobolev norms of small solutions of this nonresonant cubic NLS are asymptotically constant

Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

We prove a Lieb-Thirring type inequality for potentials such that the associated Schr\"{o}dinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg inequality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated

NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS

International audienceWe consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d,\ t\in \R where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least.\\ We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$.\\ As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I: Finite dimensional discretization.

International audienceWe consider {\em discretized} Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite dimensional Birkhoff normal form result, we show the almost preservation of the {\em actions} of the numerical solution associated with the splitting method over arbitrary long time, provided the Sobolev norms of the initial data is small enough, and for asymptotically large level of space approximation. This result holds under {\em generic} non resonance conditions on the frequencies of the linear operator and on the step size. We apply this results to nonlinear Schrödinger equations as well as the nonlinear wave equation.

Manifestations de la sécheresse en Afrique de l'Ouest non sahélienne : cas de la Côte d'Ivoire, du Togo et du Bénin

La sécheresse qui sévit depuis une vingtaine d'années dans les régions sahéliennes d'Afrique de l'Ouest semble avoir des manifestations également plus au sud dans les pays riverains du golfe de Guinée. Une double analyse, ponctuelle et spatialisée, concernant les précipitations annuelles de la Côte d'Ivoire, du Togo et du Bénin permet de mettre ce fait en évidence. Les séries chronologiques d'indices pluviométriques confirment la chute brutale de la pluviométrie à la fin des années 60. La représentation cartographique des résultats montre le net glissement des courbes isohyètes vers le sud et permet de prendre en compte la dimension régionale du phénomène. (Résumé d'auteur