Dynamics of periodic pattern formation

The formation of a one-dimensional periodic pattern from an originally homogeneous infinite system is analyzed. We develop a mean-field-like theory for the structure function. The method gives predictions for the temporal evolution towards the final stationary state. It predicts a shift of the finally selected wave vector away from the maximum of the linear spectrum. Numerical simulation confirms this behavior for intermediate times but shows a ‘‘lock-in’’ of the pattern with subsequent conservation of ‘‘nodes.’’ Thus the final wave vector in general is neither the one predicted by our modified mean-field calculation nor one of those predicted by other selection criteria based on stationary solutions only. At long times a phase diffusion regime is observed where the node distances equilibrate. This results in a $t^{\mathrm{-}1/4}$ law for the width of the structure function which can be understood in terms of a linear diffusion equation for the phase by assuming a random distribution of the gradient of the phase at the lock-in time.