Dynamic quasicrystalline patterns: Wave-mode–Turing-mode resonance with Turing-mode self-interaction


A quasicrystalline pattern is a fascinating but rather elusive form of complex order. Constructing such patterns may seem easy: in two dimensions they can be formed just by superposing four or more noncollinear modes @1#. Near a symmetry-breaking transition from a homogeneous to a patterned or crystalline state ~in both equilibrium and nonequilibrium systems! these modes appear formally as degenerate neutrally stable eigenmodes of linearized macroscopic equations, and may admit, in various contexts, different physical interpretations, e.g., density waves in equilibrium theory @2#. In practice, however, it has turned out rather difficult to find conditions that allow selection of a quasicrystalline pattern by nonlinear interactions. A possible source of quasicrystalline patterns is a superposition of two resonant triplets of Turing modes @3#. This pattern can be stabilized, however, only by quadratic interactions, which strongly inhibit multimode patterns, since interaction between the modes directed at small angles tends to be mutually damping. Quasicrystalline patterns of Turing type were, however, observed in experiments with parametric excitation of surface waves ~Faraday instability! @4,5#. Conditions suitable for formation of quasicrystalline Turing patterns were detected by the analysis of model equations @6# as well as of the amplitude equations of the Faraday instability @7#. Patterns formed by two resonant triplets were shown to be one of the possible states of Marangoni convection in a layer with a deformable interface @8#. A sure recipe for creating a quasicrystal is rotation of the optical field in a nonlinear cavity @9–14#. The number of modes is dictated then by the rotation angle. If it is commensurate with 2p , so that D52pn/N ~where n and N are integers that do not have common factors!, the basic planform is a rotationally invariant combination of N or N/2 plane waves ~respectively, for N odd or even!, which yields a quasicrystalline pattern at N55, 7, or more. Recently, Pismen and Rubinstein @14–16# suggested resonant interaction among Turing modes and wave modes near a degenerate bifurcation point as a powerful mechanism of formation of quasicrystalline patterns. Patterns involving resonant interactions are likely to exhibit complex dynamics

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@inproceedings{Musslimani2000DynamicQP, title={Dynamic quasicrystalline patterns: Wave-mode–Turing-mode resonance with Turing-mode self-interaction}, author={Ziad H. Musslimani and L. M. Pismen}, year={2000} }