Subcritical instabilities in a convective fluid layer under a quasi-one-dimensional heating

Instabilities of conducting fluid layers in weak time-dependent magnetic fields

We present the experimental analysis of the instabilities generated on a large drop of liquid metal by a time-dependent magnetic field. The study is done exploring the range of tiny values of the control parameter (the ratio between the Lorentz forces and inertia) avoiding nonlinear effects. Two different instabilities break the symmetries generating spatial patterns that appear without a threshold for some specific frequencies (up to the experimental precision) and have been observed for parameter values two orders of magnitude lower than in previously published experiments [J. Fluid Mech. 239, 383 (1992)JFLSA70022-112010.1017/S0022112092004452]. One of the instabilities corresponds to a boundary condition oscillation that generates surface waves and breaks the azimuthal symmetry. The other corresponds to a parametric forcing through a modulation of the Lorentz force. The competition between these two mechanisms produces time-dependent patterns near codimension-2 points.

Non-unique results of collisions of quasi-one-dimensional dissipative solitons

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.

Class of compound dissipative solitons as a result of collisions in one and two spatial dimensions

We study the interaction of quasi-one-dimensional (quasi-1D) dissipative solitons (DSs). Starting with quasi-1D solutions of the cubic-quintic complex Ginzburg-Landau (CGL) equation in their temporally asymptotic state as the initial condition, we find, as a function of the approach velocity and the real part of the cubic interaction of the two counterpropagating envelopes: interpenetration, one compound state made of both envelopes or two compound states. For the latter class both envelopes show DSs superposed at two different locations. The stability of this class of compound states is traced back to the quasilinear growth rate associated with the coupled system. We show that this mechanism also works for 1D coupled cubic-quintic CGL equations. For quasi-1D states that are not in their asymptotic state before the collision, a breakup along the crest can be observed, leading to nonunique results after the collision of quasi-1D states.

The Kibble-Zurek mechanism in a subcritical bifurcation

We present a study of the freezing dynamics of topological defects in a subcritical system by testing the Kibble-Zurek (KZ) mechanism while crossing a tri-stable region in a one-dimensional quintic complex Ginzburg-Landau equation. The critical exponents of the KZ mechanism and the horizon (KZ-scaling regime) are predicted from the quasistatic study, and are in full accordance with the quenched study. The correlation length, in the KZ freezing regime, is corroborated from the number of topological defects and from the spatial correlation function of the order parameter. Furthermore, we characterize the dynamics to differentiate three out-of-equilibrium regimes: the adiabatic, the impulse and the free relaxation. We show that the impulse regime shares a common temporal domain with a fast exponential increase of the order parameter.

Quasi-one-dimensional solutions and their interaction with two-dimensional dissipative solitons

We describe the stable existence of quasi-one-dimensional solutions of the two-dimensional cubic-quintic complex Ginzburg-Landau equation for a large range of the bifurcation parameter. By quasi-one-dimensional (quasi-1D) in the present context, we mean solutions of fixed shape in one spatial dimension that are simultaneously fully extended and space filling in a second direction. This class of stable solutions arises for parameter values for which simultaneously other classes of solutions are at least locally stable: the zero solution, 2D fixed shape dissipative solitons, or 2D azimuthally symmetric or asymmetric exploding dissipative solitons. We show that quasi-1D solutions can form stable compound states with 2D stationary dissipative solitons or with azimuthally symmetric exploding dissipative solitons. In addition, we find stable breathing quasi-1D solutions near the transition to collapse. The analogy of several features of the work presented here to recent experimental results on convection by Miranda and Burguete [Phys. Rev. E 78, 046305 (2008); Phys. Rev. E 79, 046201 (2009)] is elucidated.

Moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation

We show and characterize numerically moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation. This class of stable moving breathing pulses has not been described before for this prototype envelope equation as it arises near the weakly hysteretic onset of traveling waves.

Experimentally observed route to spatiotemporal chaos in an extended one-dimensional array of convective oscillators

We report experimental evidence of the route to spatiotemporal chaos in a large one-dimensional array of hotspots in a thermoconvective system. As the driving force is increased, a stationary cellular pattern becomes unstable toward a mixed pattern of irregular clusters which consist of time-dependent localized patterns of variable spatiotemporal coherence. These irregular clusters coexist with the basic cellular pattern. The Fourier spectra corresponding to this synchronization transition reveal the weak coupling of a resonant triad. This pattern saturates with the formation of a unique domain of high spatiotemporal coherence. As we further increase the driving force, a supercritical bifurcation to a spatiotemporal beating regime takes place. The new pattern is characterized by the presence of two stationary clusters with a characteristic zig-zag geometry. The Fourier analysis reveals a stronger coupling than the previous mixed pattern and enables us to find out that this beating phenomenon is produced by the splitting of the fundamental spatiotemporal frequencies in a narrow band. Both secondary instabilities are phaselike synchronization transitions with global and absolute character. Far beyond this threshold, a new instability takes place when the system is not able to sustain the spatial frequency splitting, although the temporal beating remains inside these domains. These experimental results may support the understanding of other systems in nature undergoing similar clustering processes.