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Modal Interactions and Superlattice Patterns
Pattern formation in systems in many real world systems such as neural-field models, reaction-diffusion systems and fluid systems such as the Faraday wave system have separation of scales leading to nonlinear modal interactions. A general analysis of possible terms that can arise via modal interactions is subject to both the choice of a lattice grid and the ratio between the two length scales $q$. Motivated by the observance of different grid states and superlattice states in experiments of the Faraday wave system, here we consider a hexagonal lattice grid and identify families of amplitude equations for different values of the ratio in the range $0<q<1/2$. We find that the ratio of $q=1/\sqrt7$ gives rise to the maximum number of terms in the amplitude equations (up to third order terms) and that other families of amplitude equations can be recovered by setting the coefficients of certain modal interactions in this `general’ form to vanish. For the case with $q=1/\sqrt7$, we use homotopy methods to investigate the existence and stability of multiple co-existing superlattice patterns over a range of growth rates for both the length scales. By varying the relative strengths of the different quadratic interactions, we explore the effect of different types of quadratic modal interactions in making superlattice patterns the globally stable state, i.e. observations in experiments.