Wave-number selection in pattern-forming systems

Abstract

Wavenumber selection in pattern forming systems remains a long standing puzzle in physics. Previous studies have shown that external noise is a possible mechanism for wavenumber selection. We conduct an extensive numerical study of the noisy stabilized Kuramoto-Sivashinsky equation. We use a fast spectral method of integration, which enables us to investigate long time behavior for large system sizes that could not be investigated by earlier work. We find that a state with a unique wavenumber has the highest probability of occurring at very long times. We also find that this state is independent of the strength of the noise and initial conditions, thus making a convincing case for the role of noise as a mechanism of state selection.

Global potential, topology, and pattern selection in a noisy stabilized Kuramoto–Sivashinsky equation

“…The prefactor P0(κ) is expected to vary slowly with κ, since the sum of eigenvalues and the number of negative eigenvalues do not vary with κ. The numerical results of Fourier Transform and Numerical Analysis confirm that the predicted most-probable states and the probability distribution agree precisely with direct stochastic simulations (29).…”

“…The role of noise in wavenumber selection in nonpotential systems is not well understood theoretically. Noise-induced wavenumber selection in directional solidification and in the noisy SKS equation has been studied numerically (21,27,(29)(30)(31) with results all confirming a shift of the most probable periodic state of wavenumber κs in the presence of external noise to κs < κc, the linearly most unstable state. In this work, we present a detailed account on how this shift can be explained naturally using a dynamically constructed global potential landscape which predicts a topological web of saddle points interconnected by unstable eigenmodes.…”

“…The pseudo-potential obtained in this way should have the same form as the potential landscape found by stochastic simulations. Assuming that the probability distribution is the usual Boltzmann distribution with as the temperature, the scale φ0 can be determined by fitting this to the probability distribution found from stochastic simulations (29). It turns out that φ0 ≈ 0.5 fits well for all α and larger for L = 512.…”

“…To obtain the periodic stationary solutions of the SKS equation in the noiseless limit, we use the semi-implicit algorithm of ref. 29. The Fourier transform N k (t) of the nonlinear term [ux (x , t)] 2 is obtained by combining the inverse and forward Fourier transforms as…”

“…In a narrow range of the control parameter, it exhibits a band of spatially periodic stationary states, each of which is characterized by a well-defined wavenumber, whose value depends on the control parameter. This makes it an ideal test system for studying phenomena like noise-induced state selection in the absence of a potential function (26)(27)(28)(29).…”