Avy Soffer

Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations

Abstract. 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.

SELECTION OF THE GROUND STATE FOR NONLINEAR SCHRÖDINGER EQUATIONS

We prove for a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

Phase Space Analysis on some Black Hole Manifolds

Abstract The Schwarzschild and Reissner–Nordstrom solutions to Einsteins equations describe space–times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space–time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L 6 norm in space decays like t − 1 3 . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an ϵ loss of angular derivatives.

MINIMAL ESCAPE VELOCITIES

We give a new derivation of the minimal velocity estimates (SiSo1) for unitary evolutions. LetH andA be selfadjoint operators on a Hilbert space H. The starting point is Mourres inequality i(H,A) ≥ � > 0, which is supposed to hold in form sense on the spectral subspace Hof H for some interval � ⊂ R. The second assumption is that the multiple commutators ad (k) A (H) are well- behaved fork = 1...n (n ≥ 2) . Then we show that, for a dense set of s in Hand allm < n−1, t = exp(−iHt) is contained in the spectral subspace A ≥ �t as t → ∞, up to an error of order t −m in norm. We apply this general result to the case where H is a Schrodinger operator on R n and A the dilation generator, proving that t(x) is asymptotically supported in the set |x| ≥ t √ � up to an error of order t −m in norm.

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Prices Law states that linear perturbations of a Schwarzschild black hole fall off as

Resonance Theory for Schrödinger Operators

t^{-2ell-3}

A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation

for

N-BODY HAMILTONIANS WITH HARD-CORE INTERACTIONS

t to infty

The nonlinear Schrödinger equation with a random potential: results and puzzles

provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be

Perturbation theory for the nonlinear Schrödinger equation with a random potential

t^{-2ell-4}