Noise and Pattern Selection in the One-Dimensional Swift-Hohenberg Model

The question of pattern selection in the presence of noise is addressed in the context of the one-dimensional Swift-Hohenberg equation, a model for the onset of convection. We show how noise destroys long-range order in the long-time patterns, so that characterization of the selected pattern in terms of the Fourier mode with the maximum spectral power is not always suitable. The number of zeros of the configurations turns out to be a better quantity. We consider also the decay process after an Eckhaus instability. It is shown that the selected pattern is close to the one of fastest growth during the linear regime, and not to the variationally preferred. This mechanism is robust to small noise.