Ginzburg–Landau equation

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 ${\displaystyle k_{c}}$ which is non-zero. In the neighbourhood of this bifurcation, the evolution of disturbances is characterised by the particular Fourier mode for ${\displaystyle k_{c}}$ with slowly varying amplitude ${\displaystyle A}$ (more precisely the real part of ${\displaystyle A}$). The Ginzburg–Landau equation is the governing equation for ${\displaystyle A}$. The unstable modes can either be non-oscillatory (stationary) or oscillatory.

For non-oscillatory bifurcation, ${\displaystyle A}$ satisfies the real Ginzburg–Landau equation

${\displaystyle {\frac {\partial A}{\partial t}}=\nabla ^{2}A+A-A|A|^{2}}$

which was first derived by Alan C. Newell and John A. Whitehead and by Lee Segel in 1969. For oscillatory bifurcation, ${\displaystyle A}$ satisfies the complex Ginzburg–Landau equation

${\displaystyle {\frac {\partial A}{\partial t}}=(1+ic_{1})\nabla ^{2}A+A-(1-ic_{3})A|A|^{2}}$

which was first derived by Keith Stewartson and John Trevor Stuart in 1971.

When the problem is homogeneous, i.e., when ${\displaystyle A}$ is independent of the spatial coordinates, the Ginzburg–Landau equation reduces to Stuart–Landau equation.