Retarded Accelerations of the Self-organized Front: Propagation of the Bistable Front under Step-like Force

Retarded accelerations of the self-organized front that separates two steady states of a continuous bistable system are studied by considering the response of the “bistable” front (BF) to the step-like force of flexible steepness. The driven system is described by a nonlinear differential equation of reaction–diffusion type, with the rate function approximated by the linear pieces. The retarded response of BF is examined by considering the lag (delay) time between the driving force and the propagation velocity of the driven front. The considered delay time is shown to be sensitive both to the rate (steepness) of the driving force and to the characteristic relaxation time of the system that describes the rate of transient processes within it. At low rates of driving force the response of BF is almost instantaneous. The delay time monotonically increases with increasing the rate (steepness) of the driving force and approaches some fixed value that does not exceed the characteristic relaxation time of the system. The dependence of delay time on the strength of the driving force is weak, insignificant. The derived results evidently show that the ratchet-like transport of BF, previously discussed in [4], should be significantly suppressed in the case of fast driving, when the frequency of applied zero-mean force exceeds the characteristic relaxation rates in the system.