Abstract
A recent Faraday wave experiment with two-frequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called `superlattice-two') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2√3 from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (√3 larger than the original scale) and apparently has 60° rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-two pattern presented in [Physica D 123 (1998) 99] and the subharmonic nature of the secondary instability, we show: (a) that a pattern with the same instantaneous symmetries as the superlattice-two pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in ]Physica D 123 (1998) 99]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60° rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude.
Original language | English (US) |
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Pages (from-to) | 367-387 |
Number of pages | 21 |
Journal | Physica D: Nonlinear Phenomena |
Volume | 146 |
Issue number | 1-4 |
DOIs | |
State | Published - Nov 15 2000 |
Funding
We would like to thank Jerry Gollub for inspiring this work, Jonathan Dawes, Marty Golubitsky, Paul Matthews and Michael Proctor for sharing their insights with us, and Jay Fineberg for showing us his unpublished experimental results. DPT is grateful to the Croucher Foundation for financial support. The research of AMR is supported by EPSRC. The research of RBH was supported by King’s College, Cambridge. The research of MS is supported by NSF grant DMS-9972059, NSF CAREER award DMS-9502266, and by NASA grant NAG3-2364.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics