Harmonic Analysis for Risk Minimization on Coset Trees | Symmetry and spatiotemporal chaos with strong scale separation

*** Philip Poon
Title: Symmetry and spatiotemporal chaos with strong scale separation

I will discuss the effect of a continuous symmetry on pattern formation in one spatial dimension. In particular, I will present a study of the Nikolaevskiy equation, a sixth-order PDE, which is a paradigmatic model for pattern dynamics with continuous symmetries. This model exhibits spatiotemporal chaos with strong scale separation, due to the interaction of short-wave patterns with a long-wave mode. By a multiple-scale analysis, Matthews and Cox (2000) derived a consistent system of coupled amplitude equations for modulations of the underlying patterns; however, by extensive numerical investigations I have found anomalous scaling behaviours in the Nikolaevskiy PDE that cannot be captured by these leading-order Matthews-Cox equations, but that, surprisingly, can be recovered by adding next-order correction terms.
The Matthews-Cox equations can in their own right be considered as a pair of canonical equations for reflection- and Galilean-invariant systems. By extensive large-scale, long-time computations, I have discovered several unexpected properties of these equations. From small-amplitude arbitrary initial conditions, the long-wave mode coarsens to a metastable state with multiple “viscous Burgers shock”-like structures. Subsequently, a rapid transition occurs to a single-front state with no chaos within the front (“amplitude death”), which is stabilized by a coexisting spatiotemporally chaotic region and whose features are strongly system size-dependent.
This is joint work with Ralf Wittenberg (Simon Fraser University).