Classical and quantum-mechanical phase locking transition in a nonlinear oscillator driven by a chirped frequency perturbation is discussed. Different limits are analyzed in terms of the dimensionless parameters P1 = ε/ √ 2m ω0α and P2 = (3 β)/(4m √ α) (ε, α, β and ω0 being the driving amplitude, the frequency chirp rate, the nonlinearity parameter and the linear frequency of the oscillator). It is shown that for P2 ≪ P1 + 1, the passage through the linear resonance for P1 above a threshold yields classical autoresonance (AR) in the system, even when starting in a quantum ground state. In contrast, for P2 ≫ P1 + 1, the transition involves quantum-mechanical energy ladder climbing (LC). The threshold for the phase-locking transition and its width in P1 in both AR and LC limits are calculated. The theoretical results are tested by solving the Schrodinger equation in the energy basis and illustrated via the Wigner function in phase space.
A method for adiabatic excitation and control of multiphase ( N -band) waves of the periodic nonlinear Schrödinger (NLS) equation is developed. The approach is based on capturing the system into successive resonances with external, small amplitude plane waves having slowly varying frequencies. The excitation proceeds from zero and develops in stages, as an (N+1) -band (N=0,1,2,...) , growing amplitude wave is formed in the (N+1) th stage from an N -band solution excited in the preceding stage. The method is illustrated in simulations, where the excited multiphase waves are analyzed via the spectral approach of the inverse scattering transform method. The theory of excitation of 0- and 1-band NLS solutions by capture into resonances is developed on the basis of a weakly nonlinear version of Whitham's averaged variational principle. The phenomenon of thresholds on the driving amplitudes for capture into successive resonances and the stability of driven, phase-locked solutions in these cases are discussed.
Electron phase-space holes are formed and controlled in a plasma by adiabatic nonlinear phase locking (autoresonance) with a chirped frequency driving wave. The process has a threshold on the driving amplitude and involves dragging a void region in phase space into the bulk of the distribution via persistent Cherenkov-type resonance.
It is shown that stable, large amplitude, spatially coherent solutions of the nonlinear Schrödinger equation can be excited by a weak forcing composed of an oscillation and a standing wave with a slowly varying frequency. The excitation involves autoresonant transition from a growing amplitude, uniform state to spatially modulated solution approaching the soliton, as the frequency increases in time.[S0031-9007 (98)07585-1] PACS numbers: 42.65.Tg, 03.40.Kf, 52.35.MwThe ac-driven, damped nonlinear Schrödinger equation (NLSE) ic t 1 c xx 1 jcj 2 c 2iGc 1 f, where f͑t͒ ´exp͑iLt͒ was proposed originally [1] as a model describing dipolar excitations in one-dimensional condensates. Later this equation was used in other applications, including ferromagnets in rotating magnetic fields [2], long Josephson junctions in ac fields [3], and rf-driven plasmas [4]. Among a variety of solutions of driven NLSE, the simplest are stationary, phase-locked states c a͑x͒ exp͑iLt͒, where a͑x͒ satisfies a xx 2 La 1 jaj 2 a 2iGa 1´. The existence and stability of these solutions were addressed previously [1,5,6]. In the present work, we suggest a simple way of adiabatic excitation (from zero) of the phase-locked states by slowly varying a single parameter, i.e., by chirping the frequency of the forcing. The proposed scheme is based on imposing the periodic boundary condition c͑x, t͒ c͑x 1 L, t͒ and spatially modulating the amplitude of the driving force. In particular, we shall study the case f͑x, t͒ ͑x͒ exp͓iw͑t͔͒, where L͑t͒ w t is a slow function of time,´͑x͒ ϵ´0 1´1 cos͑k 0 x͒, and k 0 2p͞L.Our approach to controlling the nonlinear wave excitation process is based on the autoresonance effect, which reflects a natural tendency of the nonlinear system to preserve, under certain conditions, the resonance with external perturbations despite variation of the system's parameters. Applications of this idea to Korteweg-de Vries, sine-Gordon, and other nonlinear wave systems exist in the literature [7,8]. Recently, the autoresonance was also studied in the context of generating spatially modulated NLSE solutions [9]. However, the proposed scheme required a special choice of initial and boundary conditions, as well as space and time variation of parameters of the driver. Instead, in the present work, we formulate an initial value problem, vary the frequency of the driver only, and include a weak dissipation in our analysis.We shall use vanishing initial conditions and view the driver as a perturbation. In this case, the initial evolution of our solution comprises a linear wave excitation problem. Since the driver is a combination of an oscillation and a standing wave with slowly varying frequencies, one expects this linear excitation to be effective at times when one of the driver components passes the resonance with a linear NLSE wave. The linear dispersion relation of the dissipationless NLSE is v 1 k 2 0, and, therefore, the resonances with´0 or´1 components of the driver are expected when L ഠ 0 or L ഠ 2k 2 0 , respectively. Suppose L is inc...
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