The complex Ginzburg–Landau equation: an introduction

Abstract

The complex Ginzburg–Landau equation (CGLE), probably the most celebrated nonlinear equation in physics, describes generically the dynamics of oscillating, spatially extended systems close to the onset of oscillations. Using symmetry arguments, this article gives an easy access to this equation and an introduction into the rich spatio-temporal behaviour it describes. Starting out from the familiar linear oscillator, we first show how the generic model for an individual nonlinear oscillator, the so-called Stuart–Landau equation, can be derived from symmetry arguments. Then, we extend our symmetry considerations to spatially extended systems, arriving at the CGLE. A comparison of diffusively coupled linear and nonlinear oscillators makes apparent the source of instability in the latter systems. A concise survey of the most typical patterns in 1D and 2D is given. Finally, more recent extensions of the CGLE are discussed that comprise external, time-periodic forcing as well as nonlocal and global spatial coupling.

Keywords:

• nonlinear physics
• spatially extended systems
• oscillations
• turbulence
• resonant forcing
• spatial coupling

Acknowledgements

The authors gratefully acknowledge Kathrin Kostorz and Lennart Schmidt for their comments on the manuscript. Financial support from the cluster of excellence Nanosystems Initiative Munich (NIM) is gratefully acknowledged. V.G.-M. acknowledges also financial support from the Technische Universität München – Institute for Advanced Study, funded by the German Excellence Initiative.