Resonances, radiation damping and instabilitym 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 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:

Asymptotic dynamics of nonlinear Schrödinger equations: Resonance‐dominated and dispersion‐dominated solutions

“…The resonance solutions for nonlinear Klein-Gordon equations were first proved in an important paper [12] by Soffer and Weinstein (see also [2]). They consider real solutions to the nonlinear Klein-Gordon equation…”

“…We first remark that the proof in [12] has only established the upper bound t −1/4 . Furthermore, an universal lower bound of the form t −1/4 is in fact incorrect.…”

“…Notice that all solutions in [12] decay as a function of t. Therefore, we can view [12] as a study of asymptotic dynamics around vacuum. Although most works concerning asymptotic dynamics of nonlinear evolution equations have been concentrated on cases with vacuum as the unique profile at t = ∞, more interesting and relevant cases are asymptotic dynamics around solitons (such as the Hartree equations [4], see also [2] Recall that we need to approximate the wave function ψ t by nonlinear ground states for all t. Since we aim to show that the error between them decay like t −1/2 , we have to track the nonlinear ground states with accuracy at least like t −1/2 .…”

Global Attraction to Solitary Waves in Models Based on the Klein-Gordon Equation

“…This establishes asymptotics of type (2.4) but only for solutions close to the solitary waves, proving the existence of a local attractor. This was first done by Soffer and Weinstein and by Buslaev and Perelman in [7,8,50,51], and then developed in [9,12,13,14,40,52] and other papers.…”

Nonlinear Dynamics of a System of Particle-Like Wavepackets

“…Since we want to establish a general property of a wide class of systems, we apply a general enough dynamical approach. There is a number of general approaches developed for the studies of high-dimensional and infinite-dimensional nonlinear evolutionary systems of hyperbolic type, [12], [15], [21], [24], [31], [37], [42], [47], [49], [51], [53]) and references therein. The approach we develop here is based on the introduction of a wavepacket interaction system.…”