Mechanism of noise-induced wave-number selection in the stabilized Kuramoto-Sivashisky equation

We revisit the question of wave-number selection in pattern-forming systems by studying the one dimensional stabilized Kuramoto-Sivashinsky equation with additive Gaussian noise. It was found in previous work that the presence of noise leads the system to prefer one of many possible periodic steady states, establishing the critical role of noise in the selection process. However, the detailed mechanism by which the noise picked out the selected wave number was not understood. Here, we look at the ensemble averaged growth of each unstable Fourier mode from the spatially homogeneous state, with and without noise. We find drastic differences between the two cases. In particular, we find that noise opposes the growth of perturbations with wave numbers in a small band around the critical wave number and boosts the growth of perturbations with wave numbers much smaller than the critical wave number. This process determines the selected wave number. We further propose a partial explanation for this effect, which is confirmed by numerical simulations.