Time-periodic Modulation Research Articles

Long-wave Marangoni Convection in Two-layer Films Under the Action of Gravity Modulation

The influence of the gravity on the long-wave Marangoni patterns in two-layer films is considered. The numerical analysis is carried out in the lubrication approximation. Periodic boundary conditions have been applied on the boundaries of the computational region. The development of instabilities is investigated by means of nonlinear simulations. The cases of a constant gravity and a time-periodic gravity modulation are considered. In the case of a constant gravity, non-stationary three-dimensional and two-dimensional structures have been found. It is shown that the periodic modulations of the gravity force lead to the development of new three-dimensional spatially-periodic patterns. Outside the region of parameters, corresponding to three-dimensional structures, dynamical regime of two-dimensional traveling waves have been observed.

Molecular wires acting as quantum heat ratchets

We explore heat transfer in molecular junctions between two leads in the absence of a finite net thermal bias. The application of an unbiased time-periodic temperature modulation of the leads entails a dynamical breaking of reflection symmetry, such that a directed heat current may emerge (ratchet effect). In particular, we consider two cases of adiabatically slow driving, namely, (i) periodic temperature modulation of only one lead and (ii) temperature modulation of both leads with an ac driving that contains a second harmonic, thus, generating harmonic mixing. Both scenarios yield sizable directed heat currents, which should be detectable with present techniques. Adding a static thermal bias allows one to compute the heat current-thermal load characteristics, which includes the ratchet effect of negative thermal bias with positive-valued heat flow against the thermal bias, up to the thermal stop load. The ratchet heat flow in turn generates also an electric current. An applied electric stop voltage, yielding effective zero electric current flow, then mimics a solely heat-ratchet-induced thermopower ("ratchet Seebeck effect"), although no net thermal bias is acting. Moreover, we find that the relative phase between the two harmonics in scenario (ii) enables steering the net heat current into a direction of choice.

Stabilization and destabilization of second-order solitons against perturbations in the nonlinear Schrödinger equation

We consider splitting and stabilization of second-order solitons (2-soliton breathers) in a model based on the nonlinear Schrödinger equation, which includes a small quintic term, and weak resonant nonlinearity management (NLM), i.e., time-periodic modulation of the cubic coefficient, at the frequency close to that of shape oscillations of the 2-soliton. The model applies to the light propagation in media with cubic-quintic optical nonlinearities and periodic alternation of linear loss and gain and to Bose–Einstein condensates, with the self-focusing quintic term accounting for the weak deviation of the dynamics from one dimensionality, while the NLM can be induced by means of the Feshbach resonance. We propose an explanation to the effect of the resonant splitting of the 2-soliton under the action of the NLM. Then, using systematic simulations and an analytical approach, we conclude that the weak quintic nonlinearity with the self-focusing sign stabilizes the 2-soliton, while the self-defocusing quintic nonlinearity accelerates its splitting. It is also shown that the quintic term with the self-defocusing/focusing sign makes the resonant response of the 2-soliton to the NLM essentially broader in terms of the frequency.

Effect of Thermal Modulation on the Onset of Electrothermoconvection in a Dielectric Fluid Saturated Porous Medium

An electroconvection in a horizontal dielectric fluid saturated with a densely packed porous layer is investigated under the simultaneous action of vertical electric field and vertical temperature gradient when the walls of the layer are subjected to time periodic temperature modulation. The dielectric constant is assumed to be a linear function of temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and wave number. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation, electric Rayleigh number, Prandtl number, and Darcy number, and their effects on the critical Rayleigh number are discussed. The situations which are favorable for the design of artificial organs without the side effect of hemolysis are explained.

Application of the Feshbach-resonance management to a tightly confined Bose-Einstein condensate

We study suppression of the collapse and stabilization of matter-wave solitons by means of time-periodic modulation of the effective nonlinearity, using the nonpolynomial Schroedinger equation (NPSE) for BEC trapped in a tight cigar-shaped potential. By means of systematic simulations, a stability region is identified in the plane of the modulation amplitude and frequency. In the low-frequency regime, solitons feature chaotic evolution, although they remain robust objects.

Model for modulated and chaotic waves in zero-Prandtl-number rotating convection

The effects of time-periodic forcing in a few-mode model for zero-Prandtl-number convection with rigid body rotation is investigated. The time-periodic modulation of the rotation rate about the vertical axis and gravity modulation are considered separately. In the presence of periodic variation of the rotation rate, the model shows modulated waves with a band of frequencies. The increase in the external forcing amplitude widens the frequency band of the modulated waves, which ultimately leads to temporally chaotic waves. The gravity modulation, on the other hand, with small frequencies, destroys the quasiperiodic waves at the onset and leads to chaos through intermittency. The spectral power density shows more power to a band of frequencies in the case of periodic modulation of the rotation rate. In the case of externally imposed vertical vibration, the spectral density has more power at lower frequencies. The two types of forcing show different routes to chaos.

Internal Variability of the Winter Stratosphere. Part II: Time-Dependent Forcing

Abstract This paper considers the effect of time-dependent lower boundary wave forcing on the internal variability found to appear spontaneously in a stratosphere-only model when the forcing is perfectly steady. While the time-dependent forcing is found to modulate the internal variability, leading in some cases to frequency locking of the upper-stratospheric response to the forcing, the temporal and spatial structure of the variability remains similar to the case when the forcing is time independent. Experiments with a time-periodic modulation of the forcing amplitude indicate that the wave flux through the lower boundary is only partially related to the instantaneous forcing, but is more significantly influenced by the condition of the polar vortex itself. In cases of purely random wave forcing with zero time mean, the stratospheric response is similar to that obtained with steady forcing of magnitude equal to the root-mean-square of the time-varying forcing.

Dressed matter waves

We suggest to view ultracold atoms in a time-periodically shifted optical lattice as a ‘dressed matter wave’, analogous to a dressed atom in an electromagnetic field. A possible effect lending support to this concept is a transition of ultracold bosonic atoms from a superfluid to a Mott-insulating state in response to appropriate "dressing" achieved through time-periodic lattice modulation. In order to observe this effect in a laboratory experiment, one has to identify conditions allowing for effectively adiabatic motion of a many-body Floquet state.

Nonlinear mode coupling and resonant excitations in two-component Bose-Einstein condensates

Nonlinear excitations in two-component Bose-Einstein condensates (BECs) described by two coupled Gross-Pitaevskii equations are investigated analytically and numerically. The beating phenomenon, the higher-harmonic generation, and the mixing of the excited modes are revealed by both variational approximation and numerical method. The strong excitations induced by the parametric resonance are also studied by time-periodic modulation for the intercomponent interaction. The resonance conditions in terms of the modulation frequency and the strength of intercomponent interaction are obtained analytically, which are confirmed by numerical method. Direct numerical simulations show that, when the resonance takes place, periodic phase separation and multisoliton configurations (including soliton trains, soliton pairs, and multidomain walls) can be excited. In particular, we demonstrate a method for formation of multisoliton configurations through parametric resonance in two-component BECs.

Effect of thermal modulation on the onset of convection in a rotating fluid layer

The stability of a rotating horizontal fluid layer heated from below is examined when, the walls of the layer are subjected to time-periodic temperature modulation. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and wavenumber. The shift in critical Rayleigh number is calculated as a function of frequency of modulation, Taylor number and Prandtl number. It is established that the instability can be enhanced by the rotation at low frequency symmetric modulation and with moderate to high frequency lower wall temperature modulation, whereas the stability can be enhanced by the rotation in case of asymmetric modulation. The effect of Taylor number and Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number and Prandtl number it is possible to advance or delay the onset of convection.

Rabi oscillations and quantum beats in a qubit in a distorted magnetic field

The time-periodic modulation of the magnetic field stabilizing the magnetic resonance position in a two-level system is investigated. It is shown that the fundamental resonance is stable against a harmonized variation of the longitudinal and transverse magnetic fields. The time dependence of the Rabi oscillations and quantum beats of the spin-flip probability is investigated numerically in different parameter regimes, taking into account dissipation and decoherence in the Lindblad form. The present study may be useful in the analysis of interference experiments and for manipulation of quantum bits.

Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer

The stability of a fluid-saturated horizontal rotating porous layer subjected to time-periodic temperature modulation is investigated when the condition for the principle of exchange of stabilities is valid. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Darcy–Rayleigh number and wavenumber. The shift in critical Darcy–Rayleigh number is calculated as a function of frequency of modulation, Taylor number, and Darcy–Prandtl number. It is established that the convection can be advanced by the low frequency in-phase and lower-wall temperature modulation, where as delayed by the out-of-phase modulation. The effect of Taylor number and Darcy–Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number, and Darcy–Prandtl number it is possible to advance or delay the onset of convection.

Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice

We investigate stability of gap solitons (GSs) in the first two band gaps in the framework of the one-dimensional Gross-Pitaevskii equation, combining the repulsive nonlinearity and a moderately strong optical lattice (OL), which is subjected to ``management,'' in the form of time-periodic modulation of its depth. The analysis is performed for parameters relevant to the experiment, characteristic values of the modulation frequency being $\ensuremath{\omega}\ensuremath{\sim}2\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}20\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$. First, we present several GS species in the two band gaps in the absence of the management. These include fundamental solitons and their bound states, as well as a subfundamental soliton in the second gap, featuring two peaks of opposite signs in a single well of the periodic potential. This soliton is always unstable, and quickly transforms into a fundamental GS, losing a considerable part of its norm. In the first band gap, (stable) bound states of two fundamental GSs are possible solely with opposite signs, if they are separated by an empty site. Under the periodic modulation of the OL depth, we identify stability regions for various GS species, in terms of $\ensuremath{\omega}$ and modulation amplitude, at fixed values of the soliton's norm, $N$. In either band gap, the GS species with smallest $N$ has a largest stability area; in the first and second gaps, they are, respectively, the fundamental GS proper, or the one spontaneously generated from the subfundamental soliton. However, with the increase of $N$, the stability region of every species expands in the first gap, and shrinks in the second one. The outcome of the instability development is also different in the two band gaps: it is destruction of the GS in the first gap, and generation of extra side lobes by unstable GSs in the second one.

Spatial nonlinear dynamics near principal parametric resonance for a fluid-conveying cantilever pipe

Abstract Flow-induced vibrations of a fluid-conveying cantilever pipe are examined theoretically under the condition that the fluid velocity has a small harmonic pulsatile component. More specifically, the case of principal parametric resonance is considered for the pipe free to undergo three-dimensional motions. The mean flow is considered to be near the critical flow rate at which the tube undergoes a Hopf bifurcation into self-excited oscillations. When the governing equations of motion for the tube with steady flow are reduced to those on the center manifold in the neighborhood of Hopf bifurcation, the normal form equations are O(2)-equivariant. The weak harmonic fluctuations due to pulsatile flow result in symmetry-breaking terms in the normal form. The eigenvalues of an O(2)-equivariant system undergoing a symmetry-breaking Hopf bifurcation have multiplicity two. When an additive linear term, arising from time-periodic modulations of the original dynamic system, is introduced into the normal form, the symmetry-breaking bifurcation structure for the trivial solution splits into three categories: a steady-state bifurcation giving rise to standing wave fixed-point solutions, a Hopf bifurcation giving rise to two-frequency solutions, and an O(2)-Takens–Bogdanov bifurcation. The resulting dynamics in each case are studied along with secondary and tertiary bifurcations. The dynamics of the tube system are studied as a function of the mean flow rate and the frequency of flow fluctuations. Amplitude response diagrams constructed for a specific example tube system using the continuation and bifurcation analysis software package AUTO illustrate the variety of possible behaviors.

Many-particle tunnelling in a driven Bosonic Josephson junction

We study the quantum many-body dynamics of a Bose–Einstein condensate in a double well potential subjected to a time-periodic modulation. For modulation frequencies moderately higher than the single-particle tunnelling frequency, this system displays an interplay of self-trapping and coherent tunnelling destruction, allowing one to effectively shut off the tunnelling contact in extended parameter regimes. For modulation frequencies roughly equalling the tunnelling frequency, one encounters chaotic dynamics, and finds sudden population jumps in the regime of low-frequency driving. A driven Bosonic Josephson junction thus constitutes a promising tool for the coherent control of mesoscopic matter waves. Our predictions can be verified under presently accessible experimental conditions.

Wave-form sampling using a driven electron ratchet in a two-dimensional electron system

We utilize a time-periodic ratchetlike potential modulation imposed onto a two-dimensional electron system inside a GaAs∕AlxGa1−xAs heterostructure to evoke a net pumped direct current. The modulation is induced by two sets of interdigitated gates, interlacing off center, which can be independently addressed. When the transducers are driven by two identical but phase-shifted periodic signals, a lateral pumped direct current I(φ) results, which strongly depends on both, the phase shift φ and the wave form V(t) of the imposed gate voltages. We find that for different periodic signals, the phase dependence I(φ) closely resembles V(t). A simple linear model of pumping in two-dimensional electron systems is presented, which reproduces well our experimental findings.

Double parametric resonance for matter-wave solitons in a time-modulated trap

We analyze the motion of solitons in a self-attractive Bose-Einstein condensate, loaded into a quasi-one-dimensional parabolic potential trap, which is subjected to time-periodic modulation with an amplitude epsilon and frequency Omega. First, we apply the variational approximation, which gives rise to decoupled equations of motion for the center-of-mass coordinate of the soliton, xi (t), and its width a (t). The equation for xi (t) is the ordinary Mathieu equation (ME) (it is an exact equation that does not depend on the adopted ansatz), the equation for a (t) being a nonlinear generalization of the ME. Both equations give rise to the same map of instability zones in the (epsilon,Omega) plane, generated by the parametric resonances (PRs), if the instability is defined as the onset of growth of the amplitude of the parametrically driven oscillations. In this sense, the double PR is predicted. Direct simulations of the underlying Gross-Pitaevskii equation give rise to a qualitatively similar but quantitatively different stability map for oscillations of the soliton's width a (t). In the direct simulations, we identify the soliton dynamics as unstable if the instability (again, realized as indefinite growth of the amplitude of oscillations) can be detected during a time comparable with, or smaller than, the lifetime of the condensate (therefore accessible to experimental detection). Two-soliton configurations are also investigated. It is concluded that multiple collisions between solitons are elastic, and they do not affect the instability borders.

High-Frequency Asymptotic for the Forced Periodical Flows of an Inviscid Fluid through a Given Domain

In this paper, we employ the method of multiple scales to study high-frequency time-periodic solutions for a boundary value problem for Euler equations in a domain with permeable boundary. Here the permeability means that an inflow and an outflow of fluid through some parts of the boundary are allowed. The boundary data given at the flow inlet and at the flow outlet are subjected to a high-frequency time-periodic modulation while the fluid is supposed to be free of mass forces. The relevant example is a flow through a finite duct or pipe affected by the modulation of the inflow and of the outflow of fluid. The keynote of our study is that the fluid passing through a domain is a non-conservative system even if the fluid is inviscid and incompressible. Indeed, the pumping and dissipation of energy occur when the fluid particles are entering the domain and, correspondingly, leaving it. In particular, this means that the flow should react on the time-periodic forcing like a non-conservative system (e.g. a viscous fluid). But since we deal with an inviscid fluid, the high-frequency modulation should produce short fast waves that propagate from the inlet downstream. They contribute into the leading term of asymptotic in spite of small amplitude they have in the case under consideration. Moreover, this contribution admits a simple explicit expression which makes the asymptotic and stability analysis for the time-periodic regimes readily accessible.

Rayleigh-Benard convection subject to time dependent wall temperature/gravity in a fluid-saturated anisotropic porous medium

The effect of time-periodic temperature/gravity modulation at the onset of convection in a Boussinesq fluid-saturated anisotropic porous medium is investigated by making a linear stability analysis. Brinkman flow model with effective viscosity larger than the viscosity of the fluid is considered to give a more general theoretical result. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature/gravity modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of the modulation, viscosity ratio, anisotropy parameter and porous parameter. We have shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature and to advance convection by gravity modulation. It is also shown that the small anisotropy parameter has a strong influence on the stability of the system. The effect of viscosity ratio, anisotropy parameter, the porous parameter and the Prandtl number is discussed.

Dynamics of spiral waves in excitable media with local time-periodic modulation

The propagation of spiral waves in excitable media with locally inhomogeneous time-periodic modulations is studied numerically, and it is shown that the size of the local inhomogeneity and the amplitude and frequency of the periodic forcing play a major role in determining the spiral wave dynamics, suppression and breakup. It is also shown that, in the absence of breakup, spiral waves flatten and thicken as the size of the inhomogeneity is increased, and that spiral waves may penetrate into the modulation region if the forcing frequency is sufficiently small. At high frequencies, however, it is shown that the inhomogeneity behaves as an obstacle. It is also reported that, for sufficiently large modulation regions, the spiral wave may be annihilated and become an almost cylindrical wave with initially high concentration of the activator in the center of the inhomogeneity.