Dynamics of the Nikolaevskiy and related equations

We study the Nikolaevskiy equation, a sixth-order PDE, as a paradigmatic model for pattern dynamics with translation, reflection and Galilean symmetries, with potential applications to complex spatiotemporal dynamics in various physical systems. This model exhibits spatiotemporal chaos with strong length-scale separation, due to the interaction of short-wave patterns with a long-wave mode. We characterize spatiotemporally chaotic states in the Nikolaevskiy model for large system size by computing the scaling of statistical measures of the solutions such as the amplitudes, power spectrum, correlation times and lengths. We perform numerical investigations of the coupled amplitude equations proposed by Matthews and Cox (2000), which were obtained via multiple-scale analysis by using an asymptotically consistent scaling different from the Ginzburg-Landau scaling typical in other pattern-forming systems. We find anomalous scaling behaviours for the long-wave mode in the Nikolaevskiy equation that cannot be captured by these leading-order, Matthews-Cox (MC) equations. However, we show that such behaviours can be recovered by adding next-order correction terms, in particular a Burgers-like term to the evolution equation for the long-wave mode. Detailed studies of the higher-order equations show, for instance, a strong spatial dependence of the correlation time. The MC 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, we discover 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, after which 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. Such a coarsening scenario leading to a state with spatially localized chaos appears to be novel, and is expected to attract further interest and investigation.