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Stationary periodic patterns are widespread in natural sciences, ranging fromnano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemicaland biological media and to macro-scale vegetation and cloud patterns. Theirformation is usually due to a primary symmetry breaking of a uniform state tostripes, often followed by secondary instabilities to form zigzag andlabyrinthine patterns. These secondary instabilities are well studied underidealized conditions of an infinite domain, however, on finite domains, thesituation is more subtle since the unstable modes depend also on boundaryconditions. Using two prototypical models, the Swift-Hohenberg equation and theforced complex Ginzburg-Landau equation, we consider bounded domains with noflux boundary conditions transversal to the stripes, and reveal a distinctmixed-mode instability that lies in between the classical zigzag and theEckhaus lines. This explains the stability of stripes in the mildly zigzagunstable regime, and, after crossing the mixed-mode line, the evolution ofzigzag stripes in the bulk of the domain and the formation of defects near theboundaries. The results are of particular importance for problems with largetime scale separation, such as bulk-heterojunction deformations in organicphotovoltaic and vegetation in semi-arid regions, where early temporaltransients may play an important role.

Stripes on finite domains: Why the zigzag instability is only a partial story