TY - JOUR

T1 - The complex Ginzburg-Landau equation

T2 - An introduction

AU - García-Morales, Vladimir

AU - Krischer, Katharina

N1 - Funding Information:
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.

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

KW - nonlinear physics

KW - oscillations

KW - resonant forcing

KW - spatial coupling

KW - spatially extended systems

KW - turbulence

UR - http://www.scopus.com/inward/record.url?scp=84857964225&partnerID=8YFLogxK

U2 - 10.1080/00107514.2011.642554

DO - 10.1080/00107514.2011.642554

M3 - Article

AN - SCOPUS:84857964225

SN - 0010-7514

VL - 53

SP - 79

EP - 95

JO - Contemporary Physics

JF - Contemporary Physics

IS - 2

ER -