Abstract

Three-wave interactions form the basis of our understanding of many pattern-forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, it is possible for two waves of the shorter wavelength to interact with one wave of the longer, as well as for two waves of the longer wavelength to interact with one wave of the shorter. Consideration of both types of three-wave interactions can generically explain the presence of complex patterns, such as quasipatterns, and spatiotemporal chaos. Two length scales arise naturally in some examples of polymer micelles and in the Faraday wave experiment, where a viscous fluid is subjected to vertical vibration. Our results enable some previously unexplained experimental observations of spatiotemporal chaos in the Faraday wave experiment to be interpreted in a new light; application to quasicrystals recently observed in self-assembled colloidal systems is more speculative.