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 timeperiodic 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. Section 3: Existence theory Section 4: Isolation of the key resonant terms and formulation as a coupled finite and infinite dimensional dynamical system Section 5: Dispersive Hamiltonian normal form Section 6: Asymptotic behavior of solutions of perturbed normal form equations Section 7: Asymptotic behavior of solutions of the nonlinear Klein Gordon equation Section 8: Summary and discussionIt is interesting to contrast our results with those known for Hamiltonian partial differential equations for a function u(x, t), where x varies over a compact spatial domain, e.g. periodic or Dirichlet boundary conditions [4], [17], [37]. For nonlinear wave equations of the form, (1.1), with periodic boundary conditions in x, KAM type results have been proved; invariant tori, associated with a nonresonance condition persist under small perturbations. The nonresonance hypotheses of such results fail in the current context , a consequence of the continuous spectrum associated with unbounded spatial domains. In our situation, non-vanishing resonant coupling (condition (1.8)) provides the mechanism for the radiative decay and therefore nonpersistence of localized periodic solutions.Remarks:

We consider a class of nonlinear Schr6dinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t ~ + oo) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result of asymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different for t ~ + ov from that+for t- .-or. Moreover the solution acquires an extra constant asymptotic phase e '~- .

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We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multi-soliton systems.

We develop an approach to scattering theory for generalized N -body systems. In particular we consider a general class of three quasi-particle systems, for which we prove Asymptotic Completeness.