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An Isaac Newton Institute Workshop

Developments in Experimental Pattern Formation

Grid States and Nonlinear Selection in Parametrically Excited Surface Waves

Author: Jay Fineberg (The Hebrew University of Jerusalem)


The nonlinear interactions of parametrically driven surface waves have been shown to yield a rich family of nonlinear states, when the system is driven by two commensurate frequencies. These patterns result from a variety of 3-wave resonant interactions. They include simple square or hexagonal patterns, different superlattices, and spatio-temporal chaotic states [1, 2, 3]. By perturbing the system with an additional (3rd) frequency, we can selectively favor either a desired temporal symmetry of the system or a selected nonlinear wave interaction. We will demonstrate the following: 1. In the region of phase space in which quadratic (three-wave) interactions are dominant, the only stable patterns correspond to “grid” states. Grid states are nonlinear states in which of two co-rotated sets of critical wavevectors are spanned by a sublattice whose basis states are linearly stable modes [4, 5]. A number of such states are observed. 2. By varying the phases of the driving frequencies, a variety of different superlattice states are selected. The selection is consistent with recent theoretical predictions of generalized phases which govern the pattern selection [6]. 3. Open loop control of the spatio-temporally disordered state can be achieved, with the controlled (ordered) state selected by the temporal parity of the perturbing frequency [7]. 1. H. Arbell and J. Fineberg, Phys. Rev. E65, 036224 (2002) 2. A. Kudrolli, B. Pier and J. P. Gollub, Physica D123, 9 (1998) 3. E. S. Edwards and S. Fauve, Phys. Rev. E47, R788 (1993). 4. M. Silber and M. R. E. Proctor, Phys. Rev. Lett. 81, 2450 (1998). 5. A. Rucklidge and W. J. Rucklidge, Physica D178, 62 (2003). 6. J. Porter, C. Topaz, and M. Silber, Phys. Rev. Lett. 93, 034502 (2004). 7. T. Epstein and J. Fineberg, Phys. Rev. Lett. 92, 244502 (2004).