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.^[1][2]

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^[3] and by Lee Segel^[4] in 1969. For oscillatory bifurcation, ${\displaystyle A}$ satisfies the complex Ginzburg–Landau equation

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

which was first derived by Keith Stewartson and John Trevor Stuart in 1971.^[5] Here ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are real constants.

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. The Swift–Hohenberg equation results in the Ginzburg–Landau equation.

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

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

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

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

This solution is known to become unstable due to Eckhaus instability for wavenumbers ${\displaystyle k^{2}>1/3.}$

Once again, let us look for steady solutions, but with an absorbing boundary condition ${\displaystyle A=0}$ at some location. In a semi-infinite, 1D domain ${\displaystyle 0\leq x<\infty }$, the solution is given by

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

where ${\displaystyle a}$ is an arbitrary real constant. Similar solutions can be constructed numerically in a finite domain.

The traveling wave solution is given by

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

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

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.^[6] The solution is given by

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

where the four real constants in the above solution satisfy

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

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

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

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

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

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