A study of two-dimensional travelling waves of the complex Ginzburg-Landau equations indicates that only temporal feedback is insufficient to stabilize. Use of spatial feedback as well as temporal feedback can stabilize the travelling waves.

Abstract:

Previous work has shown that Benjamin-Feir unstable travelling waves of the complex Ginzburg-Landau equation (CGLE) in two spatial dimensions cannot be stabilised using a particular time-delayed feedback control mechanism known as `time-delay autosynchronisation'. In this paper, we show that the addition of similar spatial feedback terms can be used to stabilise such waves. This type of feedback is a generalization of the time-delay method of Pyragus (Phys. Letts. A 170, 1992) and has been previously used to tabilize waves in the one-dimensional CGLE by Montgomery and Silber (Nonlinearity 17, 2004). We consider two cases in which the feedback contains either one or two spatial terms. We give a numerical example to demonstrate our linear stability results.

Keywords:

Travelling waves; Ginzburg-Landau equation; Spatial feeback; Temporal feedback.

Lecture #13234

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