Rectified oscillatory motion of the self-ordered front under zero-mean ac force: Role of symmetry of the rate function

The rectified oscillatory motion of the “bistable” fronts (BFs) joining two states of the different stability in a spatially extended system with two stable equilibria is studied by use of the macroscopic kinetic equation of the reaction-diffusion type. The adiabatic approximation is used: We assume that the period of the ac force acting on the front in the system significantly exceeds the characteristic relaxation time of the system. By using the arguments based on the symmetry properties of the rate function in the governing equation of the ac driven front, we show that a close corelation (one-to-one correspondence) between the rate functions of the different symmetry, the symmetrical and asymmetrical ones, and the response functions performing the “input-output” conversion between the oscillatory forcing (input) function and the speed (output) function, which describes the temporal oscillations of the moment velocity of the ac driven BF, exists. Making use of the symmetry analysis we are able to show that the average characteristics of the ratchetlike transport of the ac driven BFs derivable by the symmetrical and asymmetrical rate functions radically differ. In particular, we find that depending on the symmetry of the rate function used, either symmetrical or asymmetrical one, the complete ensemble of the forward and backward running fronts propagating at the different initial velocities in the ac driven system remains either permanently at rest on average or it travels at some fixed nonzero velocity. We confirm our predictions being derived with the rate function of the general form by the direct calculations carried out by use of the cubic polynomial rate function and its piecewise linear emulations satisfying the different symmetry properties.