Coexisting Pulses in a Model for Binary-Mixture Convection

We address the striking coexistence of localized waves (`pulses') of different lengths which was observed in recent experiments and full numerical simulations of binary-mixture convection. Using a set of extended Ginzburg-Landau equations, we show that this multiplicity finds a natural explanation in terms of the competition of two distinct, physical localization mechanisms; one arises from dispersion and the other from a concentration mode. This competition is absent in the standard Ginzburg-Landau equation. It may also be relevant in other waves coupled to a large-scale field.Comment: 5 pages revtex with 4 postscript figures (everything uuencoded

Pattern Selection in the Complex Ginzburg-Landau Equation with Multi-Resonant Forcing

We study the excitation of spatial patterns by resonant, multi-frequency forcing in systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. Using weakly nonlinear analysis we show that for small amplitudes only stripe or hexagon patterns are linearly stable, whereas square patterns and patterns involving more than three modes are unstable. In the case of hexagon patterns up- and down-hexagons can be simultaneously stable. The third-order, weakly nonlinear analysis predicts stable square patterns and super-hexagons for larger amplitudes. Direct simulations show, however, that in this regime the third-order weakly nonlinear analysis is insufficient, and these patterns are, in fact unstable

Phase Diffusion in Localized Spatio-Temporal Amplitude Chaos

We present numerical simulations of coupled Ginzburg-Landau equations describing parametrically excited waves which reveal persistent dynamics due to the occurrence of phase slips in sequential pairs, with the second phase slip quickly following and negating the first. Of particular interest are solutions where these double phase slips occur irregularly in space and time within a spatially localized region. An effective phase diffusion equation utilizing the long term phase conservation of the solution explains the localization of this new form of amplitude chaos.Comment: 4 pages incl. 5 figures uucompresse

Attractive Interaction Between Pulses in a Model for Binary-Mixture Convection

Recent experiments on convection in binary mixtures have shown that the interaction between localized waves (pulses) can be repulsive as well as {\it attractive} and depends strongly on the relative {\it orientation} of the pulses. It is demonstrated that the concentration mode, which is characteristic of the extended Ginzburg-Landau equations introduced recently, allows a natural understanding of that result. Within the standard complex Ginzburg-Landau equation this would not be possible.Comment: 7 pages revtex with 3 postscript figures (uuencoded

Parametric Forcing of Waves with Non-Monotonic Dispersion Relation: Domain Structures in Ferrofluids?

Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a non-monotonic dispersion relation. The effect of a parametric driving on such waves is studied within suitable coupled Ginzburg-Landau equations. Due to the non-monotonicity the neutral curve for the excitation of standing waves can have up to three minima. The stability of the waves with respect to long-wave perturbations is determined $via$ a phase-diffusion equation. It shows that the band of stable wave numbers can split up into two or three sub-bands. The resulting competition between the wave numbers corresponding to the respective sub-bands leads quite naturally to patterns consisting of multiple domains of standing waves which differ in their wave number. The coarsening dynamics of such domain structures is addressed.Comment: 23 pages, 6 postscript figures, composed using RevTeX. Submitted to PR

Modulation of Localized States in Electroconvection

We report on the effects of temporal modulation of the driving force on a particular class of localized states, known as worms, that have been observed in electroconvection in nematic liquid crystals. The worms consist of the superposition of traveling waves and have been observed to have unique, small widths, but to vary in length. The transition from the pure conduction state to worms occurs via a backward bifurcation. A possible explanation of the formation of the worms has been given in terms of coupled amplitude equations. Because the worms consist of the superposition of traveling waves, temporal modulation of the control parameter is a useful probe of the dynamics of the system. We observe that temporal modulation increases the average length of the worms and stabilizes worms below the transition point in the absence of modulation.Comment: 4 pages, 4 figure

A Non-Equilibrium Defect-Unbinding Transition: Defect Trajectories and Loop Statistics

In a Ginzburg-Landau model for parametrically driven waves a transition between a state of ordered and one of disordered spatio-temporal defect chaos is found. To characterize the two different chaotic states and to get insight into the break-down of the order, the trajectories of the defects are tracked in detail. Since the defects are always created and annihilated in pairs the trajectories form loops in space time. The probability distribution functions for the size of the loops and the number of defects involved in them undergo a transition from exponential decay in the ordered regime to a power-law decay in the disordered regime. These power laws are also found in a simple lattice model of randomly created defect pairs that diffuse and annihilate upon collision.Comment: 4 pages 5 figure

Temporal Modulation of the Control Parameter in Electroconvection in the Nematic Liquid Crystal I52

I report on the effects of a periodic modulation of the control parameter on electroconvection in the nematic liquid crystal I52. Without modulation, the primary bifurcation from the uniform state is a direct transition to a state of spatiotemporal chaos. This state is the result of the interaction of four, degenerate traveling modes: right and left zig and zag rolls. Periodic modulations of the driving voltage at approximately twice the traveling frequency are used. For a large enough modulation amplitude, standing waves that consist of only zig or zag rolls are stabilized. The standing waves exhibit regular behavior in space and time. Therefore, modulation of the control parameter represents a method of eliminating spatiotemporal chaos. As the modulation frequency is varied away from twice the traveling frequency, standing waves that are a superposition of zig and zag rolls, i.e. standing rectangles, are observed. These results are compared with existing predictions based on coupled complex Ginzburg-Landau equations

Temporal Modulation of Traveling Waves in the Flow Between Rotating Cylinders With Broken Azimuthal Symmetry

The effect of temporal modulation on traveling waves in the flows in two distinct systems of rotating cylinders, both with broken azimuthal symmetry, has been investigated. It is shown that by modulating the control parameter at twice the critical frequency one can excite phase-locked standing waves and standing-wave-like states which are not allowed when the system is rotationally symmetric. We also show how previous theoretical results can be extended to handle patterns such as these, that are periodic in two spatial direction.Comment: 17 pages in LaTeX, 22 figures available as postscript files from http://www.esam.nwu.edu/riecke/lit/lit.htm

Defect chaos and bursts: Hexagonal rotating convection and the complex Ginzburg-Landau equation

We employ numerical computations of the full Navier-Stokes equations to investigate non-Boussinesq convection in a rotating system using water as the working fluid. We identify two regimes. For weak non- Boussinesq effects the Hopf bifurcation from steady to oscillating (whirling) hexagons is supercritical and typical states exhibit defect chaos that is systematically described by the cubic complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the oscil- lations exhibit localized chaotic bursting, which is modeled by a quintic complex Ginzburg-Landau equation