Emanuele Haus

Growth of Sobolev norms for the quintic NLS on T 2

We study the quintic Non Linear Schrodinger 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 discussed in [15] for the case of the cubic NLS. This requires an accurate combinatorial analysis.

Asymptotic stability of synchronous orbits for a gravitating viscoelastic sphere

We study the dynamics of a viscoelastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. We work in the quadrupole approximation. We consider the solution in which the center of mass of the body moves on a circular orbit, and the body rotates in a synchronous way about its axis, so that it always shows the same face to the planet as the Moon does with the Earth. We prove that if any internal deformation of the body dissipates some energy, then such an orbit is locally asymptotically stable. The proof is based on the construction of a suitable system of coordinates and on the use of LaSalle’s principle. A large part of the paper is devoted to the analysis of the kinematics of an elastic body interacting with a gravitational field. We think this could have some interest in itself.

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions—namely periodic and even in the space variable x—of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the fully nonlinear nature of the gravity water waves equations—the highest order x-derivative appears in the nonlinear term but not in the linearization at the origin—and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators, obtained at each approximate quasi-periodic solution along a Nash–Moser iterative scheme, to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions which lose derivatives both in time and space. Despite the fact that the depth parameter moves the linear frequencies by just exponentially small quantities, we are able to verify such non-resonance conditions for most values of the depth, extending degenerate KAM theory.

A KAM Result on Compact Lie Groups

We describe some recent results on existence of quasi-periodic solutions of Hamiltonian PDEs on compact manifolds. We prove a linear stability result for the non-linear Schrödinger equation in the case of SU(2) and SO(3).

KAM for Beating Solutions of the Quintic NLS

We consider the nonlinear Schrödinger equation of degree five on the circle

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

{\mathbb{T}= \mathbb{R} / 2\pi}

Dynamics on resonant clusters for the quintic non linear Schrodinger equation

T=R/2π. We prove the existence of quasi-periodic solutions with four frequencies which bifurcate from “resonant” solutions [studied in Grébert and Thomann (Ann Inst Henri Poincaré Anal Non Linéaire 29(3):455–477, 2012)] of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.

Growth of Sobolev norms for the analytic NLS on T-2

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’s 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.