 98501 Soffer, A., Weinstein, M.I.
 Resonances, Radiation Damping and Instability in Hamiltonian
Nonlinear Wave Equations
(543K, LATeX 2e)
Jul 12, 98

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Abstract. We consider a class of nonlinear KleinGordon 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 {\it 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
timedependent 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.
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