Noise-sustained propagation of unstable pulses due to exponential interaction between pulse fronts in bistable systems with flows

We study effects of noise on pulse propagation in two bistable systems with flows: a chain of unidirectionally coupled neurons and a reaction-diffusion-convection equation with cubic nonlinearity. Pulse propagation in the systems is described by a common kinematical equation, which has exponential interaction between adjacent pulse fronts. The propagation length of pulses is then dealt with as a first passage time problem on it. We show that additive spatiotemporal noise increases the propagation length of unstable pulses and sustains pulse propagation. The mean propagation length of a single pulse increases infinitely in the presence of noise of infinitesimal strength. The propagation length of a pulse is distributed in a power-law form of exponent -3/2 as noise strength increases. A resonancelike behavior is also shown by bounding pulse width, and the mean propagation length is maximal at intermediate noise strength. Further, the proportion of survival pulses at some fixed length takes a maximum value at optimal noise strength.