On weak interaction between a ground state and a non-trapping potential

We show that ground states of the NLS moving at nonzero speed are asymptotically stable if they either stay far from the potential, or the potential is small, or the ground state has large speed

On orbital instability of spectrally stable vortices of the NLS in the plane

We explain how spectrally stable vortices of the Nonlinear Schr\"odinger Equation in the plane can be orbitally unstable. This relates to the nonlinear Fermi golden rule, a mechanism which exploits the nonlinear interaction between discrete and continuous modes of the NLS.Comment: Revised versio

On weak interaction between a ground state and a trapping potential

We study the interaction of a ground state with a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.Comment: 34 page

On small energy stabilization in the NLS with a trapping potential

We describe the asymptotic behavior of small energy solutions of an NLS with a trapping potential. In particular we generalize work of Soffer and Weinstein, and of Tsai et. al. The novelty is that we allow generic spectra associated to the potential. This is yet a new application of the idea to interpret the nonlinear Fermi Golden Rule as a consequence of the Hamiltonian structure.Comment: Revised versio

A note on virial method for decay estimates (Nonlinear and Random Waves)

In this note, we show how to prove decay estimates for Schrodinger equations by virial methods. The virial method used in this note are based on the series of work by Kowalczyk, Martel, Munoz and Van Den Bosch [10, 11, 12, 13, 14]

Stability of bound states of Hamiltonian PDEs in the degenerate cases

We consider a Hamiltonian systems which is invariant under a one-parameter unitary group. We give a criterion for the stability and instability of bound states for the degenerate case. We apply our theorem to the single power nonlinear Klein-Gordon equation and the double power nonlinear Schr\"odinger equation.Comment: 16 page

On small energy stabilization in the NLKG with a trapping potential

We consider a nonlinear Klein Gordon equation (NLKG) with short range potential with eigenvalues and show that in the contest of complex valued solutions the small standing waves are attractors for small solutions of the NLKG. This extends the results already known for the nonlinear Schr\"odinger equation and for the nonlinear Dirac equation. In addition, this extends a result of Bambusi and Cuccagna (which in turn was an extension of a result by Soffer and Weinstein) which considered only real valued solutions of the NLKG