Ginzburg–Landau equation

Template:For 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

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

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

$\frac{\partial A}{\partial t}=(1+i\alpha ){\mathrm{\nabla }}^{2}A+A-(1+i\beta )A|A{|}^2$

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 $A(𝐱,t)=R{e}^{i\mathrm{\Theta }}$, where $R=|A|$ is the amplitude and $\mathrm{\Theta }=\mathrm{a}\mathrm{r}\mathrm{g}(A)$ is the phase, one obtains the following equations

$\begin{array}{cc}\frac{\partial R}{\partial t}& =[{\mathrm{\nabla }}^{2}R-R(\mathrm{\nabla }\mathrm{\Theta }{)}^2]-\alpha (2\mathrm{\nabla }\mathrm{\Theta }\cdot \mathrm{\nabla }R+R{\mathrm{\nabla }}^2\mathrm{\Theta })+(1-R^2)R,\\ R\frac{\partial \mathrm{\Theta }}{\partial t}& =\alpha [{\mathrm{\nabla }}^{2}R-R(\mathrm{\nabla }\mathrm{\Theta }{)}^2]+(2\mathrm{\nabla }\mathrm{\Theta }\cdot \mathrm{\nabla }R+R{\mathrm{\nabla }}^2\mathrm{\Theta })-\beta R^3.\end{array}$

Some solutions of the real Ginzburg–Landau equation

Steady plane-wave type

If we substitute $A=f(k)e^{i𝐤\cdot 𝐱}$ in the real equation without the time derivative term, we obtain

$A(𝐱)=\sqrt{1-k^2}{e}^{i𝐤\cdot 𝐱},|k|<1.$

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 $0\le x<\mathrm{\infty }$, the solution is given by

$A(x)=e^{ia}\tanh \frac{x}{\sqrt{2}},$

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

$A(𝐱,t)=\sqrt{1-k^2}{e}^{i𝐤\cdot 𝐱-\omega t},\omega =\beta +(\alpha -\beta )k^2,|k|<1.$

The group velocity of the wave is given by $d\omega /dk=2(\alpha -\beta )k.$ The above solution becomes unstable due to Benjamin–Feir instability for wavenumbers $k^2>(1+\alpha \beta )/(2{\beta }^2+\alpha \beta +3).$

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

$A(x,t)=\lambda L{e}^{i\nu t}(\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\lambda x{)}^{1+iM}$

where the four real constants in the above solution satisfy

${\lambda }^2(M^2+2\alpha -1)=1,{\lambda }^2(\alpha -\alpha M^2+2M)=\nu ,$

$2-M^2-3\alpha M=-L^2,2\alpha +3M-\alpha M^2=-\beta L^2.$

Coherent structure solutions

The coherent structure solutions are obtained by assuming $A=e^{i𝐤\cdot 𝐱-\omega t}B(\mathit{\xi },t)$ where $\mathit{\xi }=𝐱+𝐮t$. This leads to

$\frac{\partial B}{\partial t}+𝐯\cdot \mathrm{\nabla }B=(1+i\alpha ){\mathrm{\nabla }}^{2}B+\lambda B-(1+i\beta )B|B{|}^2$

where $𝐯=𝐮+(1+i\alpha )𝐤$ and $\lambda =1+i\omega -(1+i\alpha )k^2.$

See also

• Davey–Stewartson equation
• Stuart–Landau equation
• Swift–Hohenberg equation
• Gross–Pitaevskii equation