Priya Subramanian - Formation of complex spatial patterns in systems with two length scales

Pattern formation 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 $q$.

In the first half, we are motivated by the observance of different grid states and superlattice states in experiments of the Faraday wave system. This leads us to consider a hexagonal lattice grid and identify families of amplitude equations for different values of the ratio in the range $q \in (0,1/2)$. For a chosen case with $q=1/\sqrt{7}$, 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.

In the second half, we are motivated by the formation of complex self-organised quasicrystal patterns during crystallisation of soft matter. We can model these systems in terms of a conserved pattern forming system within a phase field crystal approach. For such a soft matter system, with the ratio of length scales in the range $q \in (1/2,1)$, we look to determine the conditions under which we can find both spatially extended and localised quasicrystals both in two and three dimensions.