Secondary Turing-type instabilities due to strong spatial resonance

Abstract

We investigate the dynamics of pattern-forming systems in large domains near a codimension-two point corresponding to a ‘strong spatial resonance’ where competing instabilities with wavenumbers in the ratio 1 : 2 or 1 : 3 occur. We supplement the standard amplitude equations for such a mode interaction with Ginzburg–Landau-type modulational terms, appropriate to pattern formation in a large domain. In cases where the coefficients of these new diffusive terms differ substantially from each other, we show that spatially periodic solutions found near onset may be unstable to two long-wavelength modulational instabilities. Moreover, these instabilities generically occur near the codimension-two point only in the 1 : 2 and 1 : 3 cases, and not when weaker spatial resonances arise. The first instability is ‘amplitude-driven’ and is the analogue of the well-known Turing instability of reaction–diffusion systems. The second is a phase instability for which the subsequent nonlinear development is described, at leading order, by the Cahn–Hilliard equation. The normal forms for strong spatial resonances are also well known to permit uniformly travelling wave solutions. We also show that these travelling waves may be similarly unstable.