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Ginzburg–Landau equation

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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 {\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 k c {\displaystyle k_{c}} with slowly varying amplitude A {\displaystyle A} (more precisely the real part of A {\displaystyle A} ). The Ginzburg–Landau equation is the governing equation for A {\displaystyle A} . The unstable modes can either be non-oscillatory (stationary) or oscillatory.

For non-oscillatory bifurcation, A {\displaystyle A} satisfies the real Ginzburg–Landau equation

∂ A ∂ t = ∇ 2 A + A − A | A | 2 {\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 and by Lee Segel in 1969. For oscillatory bifurcation, A {\displaystyle A} satisfies the complex Ginzburg–Landau equation

∂ A ∂ t = ( 1 + i α ) ∇ 2 A + A − ( 1 + i β ) A | A | 2 {\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. Here α {\displaystyle \alpha } and β {\displaystyle \beta } are real constants.

When the problem is homogeneous, i.e., when A {\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 A ( x , t ) = R e i Θ {\displaystyle A(\mathbf {x} ,t)=Re^{i\Theta }} , where R = | A | {\displaystyle R=|A|} is the amplitude and Θ = a r g ( A ) {\displaystyle \Theta =\mathrm {arg} (A)} is the phase, one obtains the following equations

∂ R ∂ t = [ ∇ 2 R − R ( ∇ Θ ) 2 ] − α ( 2 ∇ Θ ⋅ ∇ R + R ∇ 2 Θ ) + ( 1 − R 2 ) R , R ∂ Θ ∂ t = α [ ∇ 2 R − R ( ∇ Θ ) 2 ] + ( 2 ∇ Θ ⋅ ∇ R + R ∇ 2 Θ ) − β R 3 . {\displaystyle {\begin{aligned}{\frac {\partial R}{\partial t}}&=-\alpha (2\nabla \Theta \cdot \nabla R+R\nabla ^{2}\Theta )+(1-R^{2})R,\\R{\frac {\partial \Theta }{\partial t}}&=\alpha +(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 k ⋅ x {\displaystyle A=f(k)e^{i\mathbf {k} \cdot \mathbf {x} }} in the real equation without the time derivative term, we obtain

A ( x ) = 1 − k 2 e i k ⋅ x , | k | < 1. {\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 k 2 > 1 / 3. {\displaystyle 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 {\displaystyle A=0} at some location. In a semi-infinite, 1D domain 0 ≤ x < ∞ {\displaystyle 0\leq x<\infty } , the solution is given by

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

where a {\displaystyle 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 ) = 1 − k 2 e i k ⋅ x − ω t , ω = β + ( α − β ) k 2 , | k | < 1. {\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 d ω / d k = 2 ( α − β ) k . {\displaystyle d\omega /dk=2(\alpha -\beta )k.} The above solution becomes unstable due to Benjamin–Feir instability for wavenumbers k 2 > ( 1 + α β ) / ( 2 β 2 + α β + 3 ) . {\displaystyle 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 ) = λ L e i ν t ( s e c h λ x ) 1 + i M {\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

λ 2 ( M 2 + 2 α − 1 ) = 1 , λ 2 ( α − α M 2 + 2 M ) = ν , {\displaystyle \lambda ^{2}(M^{2}+2\alpha -1)=1,\quad \lambda ^{2}(\alpha -\alpha M^{2}+2M)=\nu ,}
2 − M 2 − 3 α M = − L 2 , 2 α + 3 M − α M 2 = − β L 2 . {\displaystyle 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 k ⋅ x − ω t B ( ξ , t ) {\displaystyle A=e^{i\mathbf {k} \cdot \mathbf {x} -\omega t}B({\boldsymbol {\xi }},t)} where ξ = x + u t {\displaystyle {\boldsymbol {\xi }}=\mathbf {x} +\mathbf {u} t} . This leads to

∂ B ∂ t + v ⋅ ∇ B = ( 1 + i α ) ∇ 2 B + λ B − ( 1 + i β ) B | B | 2 {\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 v = u + ( 1 + i α ) k {\displaystyle \mathbf {v} =\mathbf {u} +(1+i\alpha )\mathbf {k} } and λ = 1 + i ω − ( 1 + i α ) k 2 . {\displaystyle \lambda =1+i\omega -(1+i\alpha )k^{2}.}

See also

References

  1. Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Reviews of modern physics, 65(3), 851.
  2. Cross, M., & Greenside, H. (2009). Pattern formation and dynamics in nonequilibrium systems. Cambridge University Press.
  3. Newell, A. C., & Whitehead, J. A. (1969). Finite bandwidth, finite amplitude convection. Journal of Fluid Mechanics, 38(2), 279-303.
  4. Segel, L. A. (1969). Distant side-walls cause slow amplitude modulation of cellular convection. Journal of Fluid Mechanics, 38(1), 203-224.
  5. 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.
  6. 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.
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