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Front Dynamics with Time Delays in a Bistable System of the Reaction-Diffusion Type: Role of the Symmetry of the Rate Function

100%

Bakanas R., Ivanauskas F., Jasaitis V.

Acta Physica Polonica A

2011

vol. 119

issue 3

282-297

The retardation effects in dynamics of the ac driven "bistable" fronts joining two states of the different stability in a bistable system of the reaction-diffusion type are investigated by use of the macroscopic kinetic equation of the reaction kinetics. We approximate the rate (reaction) function in the governing equation of "bistable" fronts by the piecewise linear dependence of the flexible symmetry, encompassing both cases of the symmetrical and asymmetrical rate functions. By numerically simulating the drift motion of the ac driven front being subjected to the time-dependent step-like (rectangular) forcing we investigate the lag time between the ac force and the instantaneous velocity of the ac driven front. We find that the time lags derivable by the symmetrical and asymmetrical rate functions notably differ, namely, we show that (a) the lag time is a function of the outer slope coefficients of the rate function and is not sensitive to the inner, (b) it has only weak dependence on the strength of the applied forcing, (c) the retardation effects (time lags) in the front dynamics are describable adequately enough by use of the perturbation theory. Another aspect of the front dynamics discussed in this report is the influence of the retardation effects on the ratchet-like transport of the ac driven fronts being described by the asymmetrical rate functions of the "low" symmetry. By considering the response of "bistable" front to the single-harmonic ac force we find that the occurrence of the time lags in the oscillatory motion of the ac driven front shrink the spurious drift of the front; the spurious drift practically disappears if the frequency of the oscillatory force significantly exceeds the characteristic relaxation rate of the system. Furthermore, the occurrence of the time lags in the front dynamics leads to the vanishing of the reversals in the directed net motion of the ac driven fronts, being always inherent in the case of the slow (quasi-stationary) ac drive, i.e., the possibilities of controlling the directed net motion of the self-ordered fronts by the low- and high-frequency zero-mean ac forces radically differ.

Self-Ordered Front under Temporally Irregular Forcing: Ratchet-Like Transport of the Quasi-Periodically Forced Front

100%

Bakanas R., Ivanauskas F., Jasaitis V.

Acta Physica Polonica A

2011

vol. 119

issue 6

731-739

Ratchet-like transport of the quasi-periodically forced "bistable" front joining two states of the different stability in the reaction-diffusion system is considered by use of the piecewise linear rate (reaction) function of the reaction kinetics. We approximate the oscillatory force acting on the front in the system by the bi-harmonic forcing functions being a superposition of the single-harmonic components (the Fourier modes) of the different frequencies, either commensurate or incommensurable ones. By considering the response of the self-ordered front to the oscillatory forces used we analyze the effect of the temporally irregular oscillations of the ac forcing on the ratchet-like shuttling of the ac driven front. By comparing the average characteristics of the spurious drift derivable in both cases of the periodically and quasi-periodically forced fronts we show that the temporally irregular fluctuations of the oscillatory force shrink the spurious drift of the front. More specifically, we find the performance of the ratchet-like shuttling of the self-ordered fronts is much lesser pronounced with the quasi-periodic, temporally irregular ac forcing if compared to that derivable by the rigorously periodic forcing, in both cases of the symmetrical and asymmetrical rate functions satisfying the different symmetry. The average characteristics of the spurious drift, that describe the dependence of the mean drift velocity of the ac driven front versus both the amplitude (strength) and the frequency of the oscillatory forces used are presented.

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2 Acta Physica Polonica A

2 2011

2 Bakanas R.

2 Ivanauskas F.

2 Jasaitis V.

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Front Dynamics with Time Delays in a Bistable System of the Reaction-Diffusion Type: Role of the Symmetry of the Rate Function

Self-Ordered Front under Temporally Irregular Forcing: Ratchet-Like Transport of the Quasi-Periodically Forced Front