Subdiffusive fluctuations of “pulled” fronts with multiplicative noise

Abstract

We study the propagation of a “pulled” front with multiplicative noise that is created by a local perturbation of an unstable state. Unlike a front propagating into a metastable state, where a separation of time scales for sufficiently large t creates a diffusive wandering of the front position about its mean, we predict that for so-called pulled fronts, the fluctuations are subdiffusive with root mean square wandering $\Delta \mathrm{}\left(t\right)\mathrm{}\sim t^{1/4},$ not $t^{1/2}.$ The subdiffusive behavior is confirmed by numerical simulations: For $t<~600,$ these yield an effective exponent slightly larger than 1/4.