1 The equation

We consider here the conditions under which the complex Ginzburg-Landau equation arises. For simplicity we restrict attention to one spatial dimension; however, these results easily generalise to higher dimensions.

1.0.1 Finite-wavelength bifurcation

Let us consider a system with a control parameter that undergoes a finite-wavelength instability as is increased. Then, considering the growth of a Fourier mode in the linearised system, the growthrate behaves as shown.

For , all modes are decaying so the homogeneous state is stable. While, for , a critical wavenumber gains neutral stability and for , there is a narrow band of wavenumbers around the critical value where the growthrate is slightly positive. (We also define a critical wavefrequency .) Now since the width of the unstable wavenumber band is of order , there can be slow modulations over length scales of order .

Note that we have assumed that the instability is supercritical, meaning the nonlinearities saturate so that the realised patterns have small amplitude.

1.0.2 Stationary bifurcation

If (at the critical wavenumber) then the unstable modes are growing in time but stationary in space. Consequently, the resulting patterns take a stationary form. To describe the patterns of the system, we separate the dynamics into a fast component (varying over the original time and space scales) and an envelope that varies slowly in space and time. Thus, close to threshold, the dynamics can be described by

where is a complex amplitude. Now assuming the system is invariant under translations ('gauge invariance') and enjoys a reflectional symmetry, that is, the system is unchanged by the transformations:

then, to leading-order, and after rescaling, obeys the real Ginzburg-Landau equation (RGLE)

1.0.3 Oscillatory bifurcation

If then the unstable modes are of a travelling wave form. As before, we can separate the dynamics into two components:

In this case, after rescaling, is governed by the complex Ginzburg-Landau equation (CGLE)

where the parameters and measure linear and nonlinear dispersion (the dependence of the frequency of the waves on the wavenumber), respectively. Of course, the RGLE is simply a special case of the CGLE where .

At this stage, several comments are in order:

• Group velocity. In general, the group velocity of the wave envelopes will be non-zero and so, in the laboratory frame, the CGLE should include an advective term. However, the convention is to write the CGLE in the frame moving with the group velocity, thus eliminating the advection term.
• Coupled equations.The CGLE is not restricted to systems with a reflectional symmetry in ; however, if such a symmetry is present that both left- and right-travelling waves can exist. In this case, the appropriate amplitude description is a set of two coupled complex Ginzburg-Landau equations.
• Uniform oscillatory instability. In the case of , the CGLE is also applicable. Such instabilities are seen in oscillatory chemical reactions and also lasers (or passive nonlinear optical systems). Very rarely are such instabilities found in hydrodynamic systems due to the constraint of conservation of mass.