The world of the complex Ginzburg-Landau equation

By Igor S. Aranson, L. Kramer. 2001

1,663 citations recorded by the source corpus.

The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The authors give an overview of various phenomena described by the complex Ginzburg-Landau equation in one, two, and three dimensions from the point of view of condensed-matter physicists. Their aim is to study the relevant solutions in order to gain insight into nonequilibrium phenomena in spatially extended systems.

Subfields: Atomic and Molecular Physics, and Optics, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/6a37bfe7928c6c66b37d3e89c3c432ab1ad06bd4

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In Atomic and Molecular Physics, and Optics

In Computer Networks and Communications

Citations in the corpus

(INVITED) Vortex solitons: Old results and new perspectives (2019, background): “[68] in a more general form, which applies as well to models based on complex GinzburgLandau equations (CGLEs), that include dissipation and gain, see Eq.”
Stochastic population dynamics in spatially extended predator–prey systems (2017, methodology): “This phenomenon was analyzed by combining lattice simulations, with a description in terms of stochastic partial differential equations and a complex Ginzburg–Landau equation [174] derived from Eqs.”
Evolutionary game theory: Theoretical concepts and applications to microbial communities (2010, background): “Here, we recognize the celebrated complex Ginzburg-Landau equation (CGLE), whose properties have been extensively studied in the past [114,115].”
Patterns and collective behavior in granular media: Theoretical concepts (2005, background): “Many pattern-forming systems are often described by generic amplitude equations such as Ginzburg-Landau or Swift-Hohenberg equations (Aranson and Kramer, 2002; Cross and Hohenberg, 1993).”
Front propagation into unstable states (2003, background): “The amplitude A is a complex quantity, and describes the slow space and time modulation of the finite wavelength mode that goes unstable [9,105,155,189,316,320,413,426,435].”
Dissipative solitons for mode-locked lasers (2012, methodology): “It turns out that the main master equation studied by Haus is similar to the cubic complex Ginzburg–Landau equation, a universal equation used to describe systems in the vicinity of bifurcation”
Phase reduction approach to synchronisation of nonlinear oscillators (2016, methodology): “This condition generally arises in the analysis of the uniformly synchronized state of a network of diffusively coupled Stuart-Landau oscillators, and is called the Benjamin-Feir condition [2, 81, 109, 110, 111].”
Higher Order Dynamic Mode Decomposition (2017, background): “The equation itself is a simple nonlinear equation that exhibits intrinsically complex dynamics [4], due to the modulational instability if αβ < −1 (Newell’s condition) and μ exceeds a threshold value.”
New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach (2018, methodology): “Among these methods; the improved fractional sub-equation method [8,9], Kudryashov method and its extended form [10–13], the unified method and its generalized scheme [14–19], the homotopy perturbation method [20,21], and the new extended trial equation method [22,23].”
The complex Ginzburg–Landau equation: an introduction (2012, background): “A number of reviews and important works have appeared during the last two decades, showing the generality of the CGLE in many physical situations at all scales [ 5 – 10 ].”
Active Turbulence (2021, background): “Near the instability threshold, modulations of these waves follow the so-called complex Ginzburg-Landau equation, originally proposed in the context of pattern formation [68, 77].”
Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model (2020, background): “Following studies of the complex GinzburgLandau equation [36], where the dynamics of the underlying oscillator provides clues to the onset of pattern formation, we study the underlying zero-dimensional system of two variables.”