Physics: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 [math]\displaystyle{ k_c }[/math] which is non-zero. In the neighbourhood of this bifurcation, the evolution of disturbances is characterised by the particular Fourier mode for [math]\displaystyle{ k_c }[/math] with slowly varying amplitude [math]\displaystyle{ A }[/math]. The Ginzburg–Landau equation is the governing equation for [math]\displaystyle{ A }[/math]. The unstable modes can either be non-oscillatory (stationary) or oscillatory.[1][2]
For non-oscillatory bifurcation, [math]\displaystyle{ A }[/math] satisfies the real Ginzburg–Landau equation
- [math]\displaystyle{ \frac{\partial A}{\partial t} = \nabla^2 A + A - A|A|^2 }[/math]
which was first derived by Alan C. Newell and John A. Whitehead[3] and by Lee Segel[4] in 1969. For oscillatory bifurcation, [math]\displaystyle{ A }[/math] satisfies the complex Ginzburg–Landau equation
- [math]\displaystyle{ \frac{\partial A}{\partial t} = (1+ic_1)\nabla^2 A + A - (1-ic_3) A|A|^2 }[/math]
which was first derived by Keith Stewartson and John Trevor Stuart in 1971.[5]
See also
References
- ↑ 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.
- ↑ Newell, A. C., & Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection. Journal of Fluid Mechanics, 38(2), 279-303.
- ↑ Segel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular convection. Journal of Fluid Mechanics, 38(1), 203-224.
- ↑ 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.
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