Coupled Complex Ginzburg-Landau Research Articles

Stability of plane wave solutions of complex Ginzburg-Landau equations

We consider the stability of plane wave solutions of both single and coupled complex Ginzburg-Landau equations and determine stability domains in the space of coefficients of the equations.

Dynamics of the globally coupled complex Ginzburg-Landau equation.

A discrete version of the complex Ginzburg-Landau equation is studied on a completely connected lattice of N sites. This can equivalently be described as a model of N identical globally coupled limit-cycle oscillators. The phase diagram is obtained by a combination of numerical and analytical techniques. A surprising variety of dynamical behaviors is found in the thermodynamic limit (N\ensuremath{\gg}1). Depending on the region of parameter space, one gets the following: (1) a simple homogeneous limit cycle; (2) a state with complete frequency locking but with no phase locking so that the global forcing term vanishes; (3) a breaking of the system into a few macroscopic clusters which can exhibit periodic or quasiperiodic dynamics; (4) surprisingly complex states where an individual oscillator behaves in a chaotic way but in a sufficiently coherent manner so that the average complex amplitude does not vanish in the thermodynamic limit. Moreover, in this last region, the dynamics of this natural order parameter is itself chaotic.

Counterpropagating quasilinear wave packets in binary-fluid convection.

I report observations of a new state of traveling-wave convection which consists of pairs of localized wave packets which propagate around an annular cell in opposite directions. During their periodic collisions, the wave packets experience a loss in amplitude due to nonlinear interaction which exactly compensates the linear gain they experience at other times. This leads to a regular, small-amplitude state with high phase and group velocities. This state offers the promise of being quantitatively explainable on first principles using the simplest model of coupled complex Ginzburg-Landau equations.

Localized Target Patterns in Oscillatory Media

Coupled complex Ginzburg-Landau equations representing interacting oscillatory fields were found to sustain a localized target pattern. The pattern emits phase waves from its center down to the boundary of localized structure. For a certain parameter range; we found analytically a one-dimensional solution which consists of a pair of localized waves, one of which is the so-called hole solution. Our computer simulation confirmed the existence of such a solution, but even for wider range of parameter values than it expected from the analytical calculations.

Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion

We consider the evolution of a gasless combustion front. We derive coupled complex Ginzburg-Landau type equations for the amplitudes of waves along the front as functions of slow temporal and spatial variables. The equations are written in characteristic variables and involve averaged terms which reflect the fact that in the slowest time scale, the effect on one wave, of a second wave traveling with the group velocity in the opposite direction on the intermediate time scale, enters only through its average. Solutions of the amplitude equations in the form of traveling, standing, and quasiperiodic waves are found, and regions of stability for these solutions are determined. In particular we find that the traveling and quasiperiodic (including standing) waves are not stable simultaneously. Finally, we observe that the stability analysis for coupled complex Ginzburg-Landau equations with averaged terms differs from that for coupled complex Ginzburg-Landau equations with the averages removed.

Amplitude equations for travelling wave convection

New asymptotically exact amplitude equations are derived for a dissipative system near a Hopf bifurcation. Unlike the usual coupled complex Ginzburg-Landau equations these are valid for O(1) group velocities.

Generation of counterpropagating nonlinear interacting traveling waves by localized noise

We investigate the interaction between external noise and spatial degrees of freedom for two coupled complex Ginzburg-Landau equations for counterpropagating traveling waves. Among the qualitatively new features caused by the nonlinear cross-coupling between the waves we find: transitions from convectively unstable to absolutely unstable behavior and vice versa, transition from subcritical to convectively or absolutely unstable supercritical behavior and a partial annihilation of colliding slugs, localized structures occurring between two stable laminar flow regimes. We also include a comparison with recent experimental results obtained near the convective onset in binary fluid mixtures.