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

Deepti Pachauri | Philip Poon, Graduate student in CS and post-doc at WID

Date and Time: Apr 11, 2012 (12:30 PM)
Location: Orchard room (3280) at the Wisconsin Institute for Discovery Building


*** 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).