The presence of continuous symmetries, or coupling with a large-scale mode or mean flow, can strongly influence the dynamics of pattern-forming systems. After reviewing some aspects of pattern formation and spatiotemporal chaos in one-dimensional Kuramoto-Sivashinsky-type equations, I will focus on a 6th-order analogue, the Nikolaevskiy PDE, a model for short-wave pattern formation with Galilean invariance displaying spatiotemporal chaos with strong scale separation. I will discuss this PDE and its relation with the corresponding leading-order amplitude equations for the coupled long-wave and pattern modes. These equations, derived by Matthews and Cox, display unexpectedly rich, strongly system-size-dependent dynamics; I will describe their long-time behavior, which has a single stable Burgers-like viscous shock coexisting with a chaotic region, and the coarsening and collapse leading to this asymptotic state on sufficiently large domains.
Coarsening to Chaos-Stabilized Fronts in Pattern Formation with Galilean Invariance
Ralf Wittenberg, Simon Fraser University
Apr 28 2014 - 4:30pm