The Ginzburg–Landau equation, named after Vitaly Ginzburg and Lev Landau, describes the nonlinear evolution of small disturbances near a finite wavelength bifurcation from a stable to an unstable state of a system. At the onset of finite wavelength bifurcation, the system becomes unstable for a critical wavenumber k_c which is non-zero. In the neighbourhood of this bifurcation, the evolution of disturbances is characterised by the particular Fourier mode for k_c with slowly varying amplitude A (more precisely the real part of A). The Ginzburg–Landau equation is the governing equation for A. The unstable modes can either be non-oscillatory (stationary) or oscillatory. For non-oscillatory bifurcation, A satisfies the real Ginzburg–Landau equation which was first derived by Alan C. Newell and John A. Whitehead and by Lee Segel in 1969. For oscillatory bifurcation, A satisfies the complex Ginzburg–Landau equation which was first derived by Keith Stewartson and John Trevor Stuart in 1971. Here \alpha and \beta are real constants. When the problem is homogeneous, i.e., when A is independent of the spatial coordinates, the Ginzburg–Landau equation reduces to Stuart–Landau equation. The Swift–Hohenberg equation results in the Ginzburg–Landau equation. Substituting, where R=|A| is the amplitude and is the phase, one obtains the following equations

Some solutions of the real Ginzburg–Landau equation

Steady plane-wave type

If we substitute in the real equation without the time derivative term, we obtain This solution is known to become unstable due to Eckhaus instability for wavenumbers k^2>1/3.

Steady solution with absorbing boundary condition

Once again, let us look for steady solutions, but with an absorbing boundary condition A=0 at some location. In a semi-infinite, 1D domain, the solution is given by where a is an arbitrary real constant. Similar solutions can be constructed numerically in a finite domain.

Some solutions of the complex Ginzburg–Landau equation

Traveling wave

The traveling wave solution is given by The group velocity of the wave is given by The above solution becomes unstable due to Benjamin–Feir instability for wavenumbers

Hocking–Stewartson pulse

Hocking–Stewartson pulse refers to a quasi-steady, 1D solution of the complex Ginzburg–Landau equation, obtained by Leslie M. Hocking and Keith Stewartson in 1972. The solution is given by where the four real constants in the above solution satisfy

Coherent structure solutions

The coherent structure solutions are obtained by assuming where. This leads to where and