Complex Ginzburg-Landau Equation

Brief Overview

The complex Ginzburg-Landau equation (CGLE) constitutes a generic model for extended systems at the onset of oscillations. Due to its generality it is capable of describing many phenomena like spatio-temporal chaos in reaction-diffusion systems, nonlinear waves, second-order phase transitions, superconductivity etc., thus constituting one of the most important nonlinear equations in physics.

 An underlying assumption is that information transport from one place of the system to another is governed by diffusional coupling. Since the diffusional coupling is proportional to the local curvature, it is of short-range nature.

 

Our Research

We investigate several modifications of the coupling function in the CGLE, to describe experimental setups like the electrodissolution of silicon or other electrochemical systems, where the coupling is indeed nonlocal or global. This gives rise to new dynamical states. A few examples are presented in the figures below. The goal is to connect theory and experiment and to gain more theoretical insight into pattern formation and dynamical instabilities in systems describable by the CGLE or its modifications.

Methods:

  • Simulations in one and two spatial dimensions
  • Bifurcation and stability analysis
  • Approximative methods
  • Dimensionality reduction methods

 

 

Transition to a chimera state in the CGLE with global coupling

Transition from two-phase clusters to a two-dimensional chimera state. (a) Two-phase clusters. Both phases are homogeneous. (b) Partial subclustering. (c) Chimera state: The coexistence of synchrony and incoherence. (d) Temporal evolution of the system in a one-dimensional cut at x = 0 in (c).

Self-organized alternating chimera state

Whereas synchrony and incoherence are initially confined to separate domains, the synchronized domains soon become infected with turbulence at the domain boundaries, followed by the synchronization of the initially turbulent domains.

Localized turbulence

The phase boundary between coherence and turbulence is not necessarily stationary, as for example in the localized turbulence shown above. States like these still raise many questions, and thus are topics of our current research.

 

References

V. García-Morales, K. Krischer. The complex Ginzburg–Landau equation: an introduction. Contemporary Physics 53(2), 79 (2012)

L. Schmidt, K. Schönleber, K. Krischer, V. García-Morales. Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling. Chaos 24, 013102 (2014)

S. W. Haugland, L. Schmidt, K. Krischer. Self-organized alternating chimera states in oscillatory media. Scientific Reports 5, 9883 (2015)