Local control of globally competing patterns in coupled Swift–Hohenberg equations

We present analytical and numerical investigations of two anti-symmetrically coupled 1D Swift–Hohenberg equations (SHEs) with cubic nonlinearities. The SHE provides a generic formulation for pattern formation at a characteristic length scale. A linear stability analysis of the homogeneous state reveals a wave instability in addition to the usual Turing instability of uncoupled SHEs. We performed weakly nonlinear analysis in the vicinity of the codimension-two point of the Turing-wave instability, resulting in a set of coupled amplitude equations for the Turing pattern as well as left- and right-traveling waves. In particular, these complex Ginzburg–Landau-type equations predict two major things: there exists a parameter regime where multiple different patterns are stable with respect to each other and that the amplitudes of different patterns interact by local mutual suppression. In consequence, different patterns can coexist in distinct spatial regions, separated by localized interfaces. We identified specific mechanisms for controlling the position of these interfaces, which distinguish what kinds of patterns the interface connects and thus allow for global pattern selection. Extensive simulations of the original SHEs confirm our results. Originally set up to describe convection instabilities, the Swift–Hohenberg equation (SHE) generically captures the formation of stationary patterns in nonequilibrium systems. A scalar parameter in this equation controls the bifurcation of a spatially homogeneous steady state to a so-called Turing-pattern, a stable stationary pattern with a finite length scale. In this paper, we consider two antisymmetrically coupled SHEs in one dimension. For comparable intrinsic length scales, the coupling gives rise to wave solutions emerging from an additional linear instability. A linear stability analysis of the homogeneous state allowed us to identify the codimension-two point where both patterns become unstable. Moreover, we have performed a weakly nonlinear analysis in the vicinity of this point to derive so-called amplitude equations which describe the evolution of patterns on large temporal and spatial scales. The amplitude dynamics of left- and right-traveling waves as well as the Turing pattern is captured in three mutually coupled complex Ginzburg–Landau equations. Numerical simulations of these equations and the coupled SHEs coincide, but the former provide substantially more insight into the coexistence of patterns. Each pattern locally suppresses other patterns and tends to saturate to a constant amplitude solution. In a certain range of parameters, both wave and Turing patterns are stable. In this bistable regime, random initial conditions typically result in the coexistence of different patterns in different spatial domains which are separated by distinct interfaces: wave-Turing interfaces as well as sources and sinks which separate left- and right-traveling waves, respectively. Now, the properties of these interfaces are pivotal for local control of globally competing patterns: A small spatial dip in the scalar parameter mentioned above may pin these interfaces. Moving this dip shifts the interface position and results in the expansion of a pattern. As each type of interface has a different susceptibility to pinning, parameter dips of certain magnitudes pin interfaces selectively. Therefore, global control by local intervention can be achieved in the absence of spatial information