Dynamic behaviour of zig-zag ladder-, double-, and broken-hexagonal superlattices in two-frequency parametric surface waves*

We study the nonlinear dynamics and patten formation associated with a planar interface vibrated vertically with two frequencies to form surface waves. We focus on the superlattice patterns, where a spatial mode as well as its spatial subharmonic coexist in the system. Superlattices have been studied extensively, both theoretically and experimentally [1], though focus was limited to time-independent bifurcations. Here, we ask whether the mode interactions underlying superlattices are time-dependent, and if so, how this influences the superlattice spatio-temporal evolution. Below the codimension-2 point, where the harmonic modes are unstable, we show that hexagons remain stable for a long time [1], unlike previous observations of alternating stripes and quasi-hexagons [2]. Upon increasing the forcing amplitude by 25% above its critical threshold, a series of metastable patterns are observed featuring broken- and double-hexagons resulting from mode competition between the primary harmonic mode and its spatial subharmonic. Finally, we obtain zig-zag ladder-like superlattice patterns, which is a hexagon symmetry-broken by its spatial subharmonic. To understand the dynamical behaviour of superlattices, we analyse the Fourier spectra, which shows that there are two competing routes of three-wave interactions; one involving primary mode interactions with its spatial subharmonic mode, and the other features two primary modes interacting to form the spatial subharmonic of their resultant.

[1] Arbell, H., & Fineberg, J. (2002). Pattern formation in two-frequency forced parametric waves. Physical Review E, 65(3), 036224.

[2] Périnet, N., Juric, D., & Tuckerman, L. S. (2012). Alternating hexagonal and striped patterns in Faraday surface waves. Physical Review Letters, 109(16), 164501.