Dynamics of noise-induced wave-number selection in the stabilized Kuramoto-Sivashinsky equation

We revisit the question of wave-number selection in pattern-forming systems by studying the one-dimensional stabilized Kuramoto-Sivashinsky equation with additive noise. In earlier work, we found that a particular periodic state is more probable than all others at very long times, 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 address this issue by analyzing the noise-averaged time evolution of each unstable mode from the spatially homogeneous state, with and without noise. We find drastic differences between the nonlinear dynamics in the two cases. In particular, we find that noise opposes the growth of Eckhaus modes close to the critical wave number and boosts the growth of Eckhaus modes with wave numbers smaller than the critical wave number. We then hypothesize that the main factor responsible for this behavior is the excitation of long-wavelength (q→0) modes by the noise. This hypothesis is confirmed by extensive numerical simulations. We also examine the significance of the magnitude of the noise.