arXiv subjects

Hermann Riecke

Publications and source records attributed to Hermann Riecke.

48 records · Page 3Linked to original sources

arXiv1996

Double Phase Slips and Spatio-Temporal Chaos in a Model for Parametrically Excited Standing Waves

We present results of numerical simulations of coupled Ginzburg-Landau equations that describe parametrically excited waves. In one dimension we focus on a new regime in which the Eckhaus sideband instability does not lead to an overall change in the wavelength via the occurrence of a single phase slip but instead leads to ``double phase slips''. They are characterized by the phase slips occurring in sequential pairs, with the second phase slip quickly following and negating the first. The resulting dynamics range from transient excursions from a fixed point resembling those seen in excitable media, to periodic solutions of varying complexity and chaotic solutions. In larger systems we find in addition localized spatio-temporal chaos, where the solution consists of a chaotic region with quiescent regions on each side. We explain the localization using an effective phase diffusion equation which can be viewed as arising from a homogenization of the chaotic state. In two dimensions the double phase slips are replaced by fluctuating bound defect pairs.

chao-dyn↗

arXiv1996

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.

patt-sol↗

arXiv1996

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.

chao-dyn↗

arXiv1996

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.

patt-sol↗

arXiv1995

Solitary Waves under the Influence of a Long-Wave Mode

The dynamics of solitons of the nonlinear Schrödinger equation under the influence of dissipative and dispersive perturbations is investigated. In particular a coupling to a long-wave mode is considered using extended Ginzburg-Landau equations. The study is motivated by the experimental observation of localized wave trains (`pulses') in binary-liquid convection. These pulses have been found to drift exceedingly slowly. The perturbation analysis reveals two distinct mechanisms which can lead to a `trapping' of the pulses by the long-wave concentration mode. The are given by the effect of the concentration mode on the local growth rate and on the frequency of the wave. The latter, dispersive mechanism has not been recognized previously, despite the fact that it dominates over the dissipative contribution within the perturbation theory. A second unexpected result is that the pulse can be accelerated by the concentration mode despite the reduced growth rate ahead of the pulse. The dependence of the pulse velocity on the Rayleigh number is discussed, and the hysteretic `trapping' transitions suggested by the perturbation theory are confirmed by numerical simulations, which also reveal oscillatory behavior of the pulse velocity in the vicinity of the transition. The derivation and reconstitution of the extended Ginzburg-Landau equations is discussed in detail.

patt-sol↗

arXiv1995

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.

patt-sol↗

arXiv1995

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.

patt-sol↗

arXiv1994

Domain Structures in Fourth-Order Phase and Ginzburg-Landau Equations

In pattern-forming systems, competition between patterns with different wave numbers can lead to domain structures, which consist of regions with differing wave numbers separated by domain walls. For domain structures well above threshold we employ the appropriate phase equation and obtain detailed qualitative agreement with recent experiments. Close to threshold a fourth-order Ginzburg-Landau equation is used which describes a steady bifurcation in systems with two competing critical wave numbers. The existence and stability regime of domain structures is found to be very intricate due to interactions with other modes. In contrast to the phase equation the Ginzburg-Landau equation allows a spatially oscillatory interaction of the domain walls. Thus, close to threshold domain structures need not undergo the coarsening dynamics found in the phase equation far above threshold, and can be stable even without phase conservation. We study their regime of stability as a function of their (quantized) length. Domain structures are related to zig-zags in two-dimensional systems. The latter are therefore expected to be stable only when quenched far enough beyond the zig-zag instability.

patt-sol↗

arXiv1994

Bound Pairs of Fronts in a Real Ginzburg-Landau Equation Coupled to a Mean Field

Motivated by the observation of localized traveling-wave states (`pulses') in convection in binary liquid mixtures, the interaction of fronts is investigated in a real Ginzburg-Landau equation which is coupled to a mean field. In that system the Ginzburg-Landau equation describes the traveling-wave amplitude and the mean field corrsponds to a concentration mode which arises due to the slowness of mass diffusion. For single fronts the mean field can lead to a hysteretic transition between slow and fast fronts. Its contribution to the interaction between fronts can be attractive as well as repulsive and depends strongly on their direction of propagation. Thus, the concentration mode leads to a new localization mechanism, which does not require any dispersion in contrast to that operating in the nonlinear Schrödinger equation. Based on this mechanism alone, pairs of fronts in binary-mixture convection are expected to form {\it stable} pulses if they travel {\it backward}, i.e. opposite to the phase velocity. For positive velocities the interaction becomes attractive and destabilizes the pulses. These results are in qualitative agreement with recent experiments. Since the new mechanism is very robust it is expected to be relevant in other systems as well in which a wave is coupled to a mean field.

patt-sol↗

arXiv1994

Front Structures in a Real Ginzburg-Landau Equation Coupled to a Mean Field

Localized traveling wave trains or pulses have been observed in various experiments in binary mixture convection. For strongly negative separation ratio, these pulse structures can be described as two interacting fronts of opposite orientation. An analytical study of the front solutions in a real Ginzburg-Landau equation coupled to a mean field is presented here as a first approach to the pulse solution. The additional mean field becomes important when the mass diffusion in the mixture is small as is the case in liquids. Within this framework it can lead to a hysteretic transition between slow and fast fronts when the Rayleigh number is changed.

patt-sol↗

arXiv1994

Domain Structures and Zig-Zag Patterns Modeled by a Fourth-Order Ginzburg-Landau Equation

Domain walls between spatially periodic patterns with different wave numbers, can arise in pattern-forming systems with a neutral curve that has a double minimum. Within the framework of the phase equation, the interaction of such walls is purely attractive. Thus, they annihilate each other and, if the total phase is not conserved, the final state will be periodic with a single wave number. Here we study the stability of arrays of domain walls (domain structures) in a fourth-order Ginzburg-Landau equation. We find a discrete set of domain structures which differ in the quantized lengths of the domains. They are stable even without phase conservation, as, for instance, in the presence of a subcritical ramp in the control parameter. We attribute this to the spatially oscillatory behavior of the wave number which should lead to an oscillatory interaction between the domain walls. These results are expected to shed also some light on the stability of two-dimensional zig-zag structures.

patt-sol↗

arXiv1993

Temporal Forcing of Small-Amplitude Waves in Anisotropic Systems

We investigate the effect of resonant temporal forcing on an anisotropic system that exhibits a Hopf bifurcation to obliquely traveling waves in the absence of this forcing. We find that the forcing can excite various phase-locked standing-wave structures: rolls, rectangles and cross rolls. At onset, at most one of the two - rolls or rectangles - is stable. The cross rolls can arise in a secondary bifurcation and can be stable. Experimentally, they would appear as a periodic switching between a structure in which the `zig'-component dominates and one with dominating `zag'-structure. Since there are two symmetry-related states of this kind one may expect disordered structures to arise due to the break-up of the pattern into domains. The results are consistent with recent experiments on electro-convection in nematic liquid crystals by de la Torre and Rehberg. We also apply the general analysis to a model of the behavior near a Lifshitz point, where the angle of obliqueness vanishes. This analysis indicates that phase-locked standing rectangles are always unstable in this parameter regime.

patt-sol↗