Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems

We study the coupled complex Ginzburg-Landau (CGL) equations for traveling wave systems, and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems. We focus on the regime in which sources and sinks separate patches of left and righttraveling waves, i.e., the case that these modes suppress each other. We present in detail the framework to analyze these coherent structures, and show that the theory predicts a number of general properties which can be tested directly in experiments. Our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations, there generally exists a symmetric stationary source solution, which sends out waves with a unique frequency and wave number. Sinks, on the other hand, occur in two-parameter families, and play an essentially passive role, being sandwiched between the sources. These simple but general results imply that sources are important in organizing the dynamics of the coupled CGL equations. Simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2D complex Ginzburg Landau equations; sources can send out stable waves, convectively unstable waves, or absolutely unstable waves. We show that there exists an additional dynamical regime where both singleand bimodal states are unstable; the ensuing chaotic states have no counterpart in single amplitude equations. A third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate ε. This is related to the fact that the standard coupled CGL equations are not uniform in ε. In particular, when the group velocity term dominates over the linear growth term, no stationary source can exist; however, sources displaying nontrivial dynamics can often survive here. Our results for the existence, multiplicity, wavelength selection, Preprint submitted to Elsevier Preprint 9 February 2008 dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments. We therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment. In addition, they bring up a large number of new research issues and open problems, which are listed explicitly in the concluding section. PACS: 47.54.+r; 03.40.Kf; 47.20.Bp; 47.20.Ky