  We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian
and are perturbations of linear dispersive equations. The unperturbed dynamical
system has a bound state, a spatially localized and time periodic solution. We
show that, for generic nonlinear Hamiltonian perturbations, all small amplitude
solutions decay to zero as time tends to infinity at an anomalously slow rate.
  In particular, spatially localized and time-periodic solutions of the linear
problem are destroyed by generic nonlinear Hamiltonian perturbations via slow
radiation of energy to infinity. These solutions can therefore be thought of as
metastable states.
  The main mechanism is a nonlinear resonant interaction of bound states
(eigenfunctions) and radiation (continuous spectral modes), leading to energy
transfer from the discrete to continuum modes.
  This is in contrast to the KAM theory in which appropriate nonresonance
conditions imply the persistence of invariant tori. A hypothesis ensuring that
such a resonance takes place is a nonlinear analogue of the Fermi golden rule,
arising in the theory of resonances in quantum mechanics. The techniques used
involve: (i) a time-dependent method developed by the authors for the treatment
of the quantum resonance problem and perturbations of embedded eigenvalues,
(ii) a generalization of the Hamiltonian normal form appropriate for infinite
dimensional dispersive systems and (iii) ideas from scattering theory. The
arguments are quite general and we expect them to apply to a large class of
systems which can be viewed as the interaction of finite dimensional and
infinite dimensional dispersive dynamical systems, or as a system of particles
coupled to a field.
