1998/06/29 by Avy Soffer, A. Soffer, Michael I. Weinstein +1 · 348 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Bound state #Classical mechanics #Eigenfunction #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Hamiltonian system #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Nonlinear resonance #Nonlinear system #Physics #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #chao-dyn #gr-qc #math-ph #math.MP #nlin.CD
paper · pdf · doi:10.1007/s002220050303
published in Inventiones mathematicae 136(1), 9-74 (Springer Science+Business Media) · To appear in Inventiones Mathematicae
arxiv created 1998/06/29 · openalex publication_date 1999/03/18 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.