Excitation of Solitons by Adiabatic Multiresonant Forcing

Abstract

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...

Migration Timescale Thresholds for Resonant Capture in the Plutino Problem

“…The term autoresonance is used in describing the continuing phase locking in driven nonlinear systems with slow parameters. The theory of autoresonance was extended recently to applications in fluid dynamics (Friedland 1999), plasmas (Fajans, Gilson, & Friedland 1999), and nonlinear waves (Friedland & Shagalov 1998). One of the main predictions of this theory is the existence of a sharp threshold on the driving frequency sweep rate (the rate of variation of the angular frequency of Neptune in the Plutino problem) for capture into resonance.…”

Parametric autoresonance

“…It was extensively studied in the context of relativistic particle acceleration: in the 40-ies by McMillan [5], Veksler [6] and Bohm and Foldy [7,8], and more recently [9][10][11][12]. Additional applications include a quasiclassical scheme of excitation of atoms [13] and molecules [14], excitation of nonlinear waves [15,16], solitons [17,18], vortices [19,20] and other collective modes [21] in fluids and plasmas, an autoresonant mechanism of transition to chaos in Hamiltonian systems [22,23], etc.…”

Control of integrable Hamiltonian systems and degenerate bifurcations

“…In Ref. [21], Friedland and Shagalov demonstrate that the plane wave state of the NLS can be autoresonantly excited, and that as the amplitude reaches a certain threshold, a spatially modulated form arises and eventually becomes a shape not unlike a soliton. In Ref.…”