Physical Review Letters 108 (2012) 074504. doi:10.1103/PhysRevLett.108.074504

Three-wave interactions and spatio-temporal chaos

A.M.Rucklidge(1), M. Silber(2) and A.C.Skeldon(3).

(1) Department of Applied Mathematics,
University of Leeds, Leeds, LS2 9JT, UK

(2) Department of Engineering Sciences and Applied Mathematics,
Northwestern University, Evanston, IL 60208, USA

(3) Department of Mathematics,
University of Surrey, Guildford, GU2 7XH, UK

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 and spatio-temporal chaos. Two length scales arise naturally in the Faraday wave experiment with multi-frequency forcing, and our results enable some previously unexplained experimental observations of spatio-temporal chaos to be interpreted in a new light. Our predictions are illustrated with numerical simulations of a model partial differential equation.

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