abstract: It is well known that in finite dimensional Hamiltonian systems asymptotic stability is impossible. On the contrary it is a typical phenomenon in infinite dimensional Hamiltonian systems in infinite domains. This is due to the presence of continuous spectrum which typically introduces some dispersive behaviour. Furthermore the generic existence of resonances between discrete and continuous spectrum (FGR) often implies the non existence of localized states. A typical example is the wave equation with a potential (which will be discussed).
On the contrary, when some symmetries exist, some localized objects exist and they are typically asymptotically stable. An example is the ground state of the NLS (on which I will say a few words).
A further interesting situation where some localized objects exist is that of lattices in which, due to the fact that the continuous spectrum is bounded, one can prove existence of breathers, which, due to the presence of continuous spectrum, turn out to be asymptotically stable.
The techniques used to prove the above kind of results are a combination of techniques from finite dimensional Hamiltonian systems (normal form, Darboux coordinates) and techniques from dispersive equations (Strichartz estimates).