Pacific Institute for the Mathematical Sciences - PIMS

Coarsening to Chaos-Stabilized Fronts in One-Dimensional Partial Differential Equations

The presence of continuous symmetries, or coupling with a large-scale
mode or mean flow, considerably influences the behaviour of
pattern-forming systems. We focus on short-wave pattern formation in
Galilean-invariant systems, modelled by the (6th-order) Nikolaevskiy
PDE, in which the primary instability of a spatially homogeneous system
leads directly to a spatiotemporally chaotic state with strong scale
separation. We discuss this PDE and its relation with the leading-order
amplitude equations describing the coupling of the pattern and
long-wave modes in this context, derived by Matthews and Cox. Of
particular interest are the unexpectedly rich dynamics displayed by the
Matthews-Cox equations, and we describe the transition towards and
strongly system-size-dependent properties of their long-time asymptotic
state, which consists of a single stable Burgers-like viscous shock
coexisting with a chaotic state. For sufficiently large domains, an
initial rapid evolution yields multiple such viscous-shock structures,
which undergo slow coarsening until, after a long transient, there is a
relatively rapid collapse to the asymptotic single front state.

Location: LSK 301

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