Effects of parametric disorder on a stationary bifurcation

Effects of a frozen random contribution to the control parameter are investigated in terms of the complex Ginzburg–Landau equation with real coefficients. The threshold of the bifurcation from the homogeneous basic state is reduced by a random contribution even with a vanishing spatial mean value, as shown by three different approaches, by a perturbation calculation, by a self-consistent iteration method and by a fully numerical solution of the linear part of the Ginzburg–Landau equation. For arbitrary random contributions the nonlinear stationary solutions are numerically determined and in the limit of small random amplitudes analytical expressions are derived in terms of two different perturbation expansions, which cover already several related trends beyond threshold. For instance, the spatial modulations of the solutions increase with the noise amplitude, but decrease with increasing distance from threshold.