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
In the neighbourhood of this bifurcation, the evolution of disturbances is characterised by the particular Fourier mode for
with slowly varying amplitude
(more precisely the real part of
The unstable modes can either be non-oscillatory (stationary) or oscillatory.
[1][2] For non-oscillatory bifurcation,
satisfies the real Ginzburg–Landau equation which was first derived by Alan C. Newell and John A. Whitehead[3] and by Lee Segel[4] in 1969.
For oscillatory bifurcation,
satisfies the complex Ginzburg–Landau equation which was first derived by Keith Stewartson and John Trevor Stuart in 1971.
are real constants.
When the problem is homogeneous, i.e., when
is independent of the spatial coordinates, the Ginzburg–Landau equation reduces to Stuart–Landau equation.
is the phase, one obtains the following equations 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
Once again, let us look for steady solutions, but with an absorbing boundary condition
In a semi-infinite, 1D domain
is an arbitrary real constant.
Similar solutions can be constructed numerically in a finite domain.
The traveling wave solution is given by The group velocity of the wave is given by
d k = 2 ( α − β ) k .
The above solution becomes unstable due to Benjamin–Feir instability for wavenumbers
> ( 1 + α β )
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.
[6] The solution is given by where the four real constants in the above solution satisfy The coherent structure solutions are obtained by assuming
λ = 1 + i ω − ( 1 + i α )