Noise-sustained structures in coupled complex Ginzburg-Landau equations for a convectively unstable system

We investigate a pattern-forming system close to a Hopf bifurcation with broken translational symmetry. In one-dimensional geometries, its evolution is governed by two coupled complex Ginzburg-Landau equations which describe the amplitude of the counterpropagating traveling waves that develop beyond the instability. The convective and absolute instabilities of the possible steady states are analyzed. In the regime of strong cross coupling, where traveling waves are favored by the dynamics, the results of previous analysis are recovered. In the weak cross-coupling regime, where standing waves are favored by the dynamics, traveling waves nevertheless appear, in the absence of noise, between the uniform steady state and the standing-wave patterns. In this regime, standing waves are sustained by spatially distributed external noise for all values of the bifurcation parameter beyond the Hopf bifurcation. Hence, the noise is not only able to sustain spatiotemporal patterns, but also to modify pattern selection processes in regimes of convective instability. In this weak coupling regime we also give a quantitative statistical characterization of the transition between deterministic and noise-sustained standing waves when varying the bifurcation parameter. We show that this transition occurs at a noise-shifted point and it is identified by an apparent divergence of a correlation time and the saturation of a correlation length to a value given by the system size.