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 which is non-zero. 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 Ginzburg–Landau equation is the governing equation for . The unstable modes can either be non-oscillatory (stationary) or oscillatory.Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Reviews of modern physics, 65(3), 851.Cross, M., & Greenside, H. (2009). Pattern formation and dynamics in nonequilibrium systems. Cambridge University Press.
For non-oscillatory bifurcation, satisfies the real Ginzburg–Landau equation
which was first derived by Alan C. Newell and John A. WhiteheadNewell, A. C., & Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection. Journal of Fluid Mechanics, 38(2), 279-303. and by Lee SegelSegel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular convection. Journal of Fluid Mechanics, 38(1), 203-224. in 1969. For oscillatory bifurcation, satisfies the complex Ginzburg–Landau equation
which was first derived by Keith Stewartson and John Trevor Stuart in 1971.Stewartson, K., & Stuart, J. T. (1971). A non-linear instability theory for a wave system in plane Poiseuille flow. Journal of Fluid Mechanics, 48(3), 529-545. Here and 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. The Swift–Hohenberg equation results in the Ginzburg–Landau equation.
Substituting , where 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
Steady solution with absorbing boundary condition Once again, let us look for steady solutions, but with an absorbing boundary condition at some location. In a semi-infinite, 1D domain , the solution is given by
where 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.Hocking, L. M., & Stewartson, K. (1972). On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 326(1566), 289-313. 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
See also Davey–Stewartson equation Stuart–Landau equation Swift–Hohenberg equation Gross–Pitaevskii equation
References