The presence of continuous symmetries can strongly influence the dynamics of pattern-forming systems. I will begin with an overview of pattern formation and spatiotemporal chaos in the Kuramoto-Sivashinsky (KS) equation for long-wave instability, a much-studied 4th-order scalar PDE in one space dimension. Of analytical interest is that while solutions of the KS equation have long been known to be bounded and indeed analytic, rigorous bounds on absorbing ball radii and attractor dimension which scale optimally with the system size have proved remarkably difficult to achieve; I will review the history and current status of such bounds.
My main focus will be the Nikolaevskiy PDE, a 6th-order analogue of the KS equation modelling short-wave pattern formation with Galilean invariance, in which coupling between long-wave and pattern modes leads to spatiotemporal chaos with strong scale separation. The corresponding leading-order amplitude equations, due to Matthews and Cox, display unexpectedly rich, strongly system-size-dependent dynamics. I will describe their long-time behaviour, 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.