Dynamics on resonant clusters for the quintic non linear Schr\"odinger equation

We construct solutions to the quintic nonlinear Schr\"odinger equation on the circle with initial conditions supported on arbitrarily many different resonant clusters. This is a sequel of a work of Beno\^it Gr\'ebert and the second author.Comment: 11 page

Growth of Sobolev norms for the quintic NLS on T2\mathbb T^2

We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with arXiv:0808.1742 by other author

Exact controllability for quasi-linear perturbations of KdV

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inequality and HUM method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash-Moser implicit function theorems by H\"ormander.Comment: 39 page

Controllability of quasi-linear Hamiltonian NLS equations

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash-Moser-H\"ormander implicit function theorem as a black box

Asymptotic Behavior of an Elastic Satellite with Internal Friction

We study the dynamics of an elastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. The main result is that, if all the deformations of the satellite dissipate some energy, then under a suitable nondegeneracy condition there are only three possible outcomes for the dynamics: (i) the orbit of the satellite is unbounded, (ii) the satellite falls on the planet, (iii) the satellite is captured in synchronous resonance i.e. its orbit is asymptotic to a motion in which the barycenter moves on a circular orbit, and the satellite moves rigidly, always showing the same face to the planet. The result is obtained by making use of LaSalle\u2019s invariance principle and by a careful kinematic analysis showing that energy stops dissipating only on synchronous orbits. We also use in quite an extensive way the fact that conservative elastodynamics is a Hamiltonian system invariant under the action of the rotation group