Defects in Waves

Abstract

Among instabilities that lead to the formation of spatial periodic structures, Rayleigh- Bénard convection is one of the most famous and useful phenomena for studying systems driven far from equilibrium by a set of external parameters [1]. Another example of pattern formation is convection in liquid crystals [2], where many rolls can easily be observed. Such behaviors have been widely studied both experimentally and theoretically [3]. In those experiments, the periodic structure gives rise to spatial order, as the appearance of rhythms, in more confined systems, leads to temporal order. In the same way, spontaneous breaking of both space and time translational symmetries leads to spatio-temporal order, and corresponds to wave motion [4–12]. In real-life experiments, ordered states are the exception rather than the rule, and one typically sees many patches of locally ordered structures, linked together by narrow transition regions. Grain boundaries in convective experiments [13] or dislocations occurring in nonequilibrium cellular structures [14–17] can be regarded as defects of the spatial order. By analogy, one can expect the existence of spatio-temporal order breaking singularities. These are, for instance, defect lines connecting spatially extended regions of counter propagating waves, or dislocations of travelling and standing waves. The aim of this paper is the phenomenological study of defects in waves, using both topological arguments and numerical simulations, in the framework of amplitude equations.