The complex Ginzburg-Landau equation

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The complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation is one of the most-studied equations in applied mathematics. It describes qualitatively, and often quantitatively, a vast array of phenomena including nonlinear waves, second-order phase transitions, Rayleigh-Bénard convection and superconductivity. The equation describes the evolution of amplitudes of unstable modes for any process exhibiting a Hopf bifurcation, for which a continuous spectrum of unstable wavenumbers is taken into account. It can be viewed as a highly general normal form for a large class of bifurcations and nonlinear wave phenomena in spatially extended systems.

In this tutorial, a broad overview of the behaviour of the equation is given, with a focus on the one-dimensional case. The linear stability problem of plane wave solutions to the equation is expounded and the analysis is complemented with a graphical representation of the various observed behaviours. Two MATLAB programs are also provided, which simulate the 1D and 2D versions of the complex Ginzburg-Landau equation, allowing the reader to verify the results presented here and to conduct their own exploration of the equation.

Author Institutional Affiliation
David M. Winterbottom
School of Mathematical Sciences
University of Nottingham
Author Email
pmxdmw@nottingham.ac.uk
Tutorial LevelAdvanced Tutorial
DescriptionTutorial
Contest EntryYes

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