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

摘要

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.