Spatial and temporal feedback control of traveling wave solutions of the two-dimensional complex Ginzburg–Landau equation

Previous work has shown that Benjamin–Feir unstable traveling waves of the complex Ginzburg–Landau equation (CGLE) in two spatial dimensions cannot be stabilized using a particular time-delayed feedback control mechanism known as ‘time-delay autosynchronization’. In this paper, we show that the addition of similar spatial feedback terms can be used to stabilize such waves. This type of feedback is a generalization of the time-delay method of Pyragas [K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (1992) 421–428] and has been previously used to stabilize waves in the one-dimensional CGLE by Montgomery and Silber [K. Montgomery, M. Silber, Feedback control of traveling wave solutions of the complex Ginzburg Landau equation, Nonlinearity 17 (6) (2004) 2225–2248]. We consider two cases in which the feedback contains either one or two spatial terms. We focus on how the spatial terms may be chosen to select the direction of travel of the plane waves. Numerical linear stability calculations demonstrate the results of our analysis.