Ginzburg–Landau equation

For the superconductivity theory, see Ginzburg–Landau theory.

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

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

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

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

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 $A(\mathbf {x} ,t)=Re^{i\Theta }$, where $R=|A|$ is the amplitude and $\Theta =\mathrm {arg} (A)$ is the phase, one obtains the following equations

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

Some solutions of the real Ginzburg–Landau equation

Steady plane-wave type

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

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

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 $0\leq x<\infty$, 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

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

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

$A(x,t)=\lambda Le^{i\nu t}(\mathrm {sech} \lambda x)^{1+iM}$

where the four real constants in the above solution satisfy

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

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

Coherent structure solutions

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

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

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

See also

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