Retarding accelerations and ratchet-like transport of the self-ordered front in a bistable system of reaction-diffusion type

Abstract

Front dynamics with time delays in a reaction-diffusion equation with the piecewise linear rate function is studied by considering the response of the self-ordered front to the rapidly varying force. More specifically, the retarding accelerations of the bistable front (BF) joining two states of different stability in the system are examined. Two cases of differently shaped forcing functions are studied. In examining the ‘size’ of the retardation, namely, the lag time between the propagation velocity of the front and the driving force we approximate the forcing function by the step-like dependence of the flexible steepness. The lag time is found to be sensitive to both the rate (steepness) of the driving force and the characteristic parameters of the rate function. A significant retardation effect occurred when the steepness of the force exceeded the characteristic relaxation rate of the system; the lag time monotonically increases with increasing steepness and approaches the maximal value depending on the relaxation time of the system, when the driving force becomes extremely steep. The influence of delays on the ratchet-like transport of the self-ordered front is studied by considering the response of BF to the periodically oscillating square-pulse force. The average characteristics of the unforced dc motion of BFs are presented. The occurrence of delays, as shown, significantly reduces the unforced dc motion of BF. The ratchet-like transport is highly suppressed if the period of the ac force becomes low compared to the characteristic relaxation time of the system.