Complex Ginzburg–Landau equation

The complex Ginzburg–Landau equation is a nonlinear complex-amplitude evolution equation with linear and cubic terms and generally complex coefficients. The driven-condensate form ${\Psi }_t=(\alpha -\beta |\Psi {|}^2)\Psi +i({\nabla }^2-g|\Psi {|}^2+s)\Psi$ has gain/loss coefficients $\alpha ,\beta$ and conservative interaction/frequency coefficients $g,s$. A uniform nonzero state has number density $\alpha /\beta$ when $\alpha ,\beta >0$.

A complex Ginzburg–Landau equation describes a complex number disturbance amplitude through linear amplification, advection, complex diffusion equation terms, and often cubic amplitude saturation. A constant-coefficient example is ${\psi }_t+U{\psi }_x=\mu \psi +(1+i{c}_d){\psi }_{xx}-|\psi {|}^2\psi$. Here real $U,\mu ,c_d$ describe transport, linear growth rate and dispersive diffusion equation terms. Omitting the cubic term gives the linear complex Ginzburg-Landau equation.

This nondispersive cubic complex Ginzburg–Landau equation balances linear growth against nonlinear damping. Its real-wavenumber Ginzburg-Landau plane wave has amplitude $\sqrt{\mu -k^2}$ when $\mu >k^2$. Relaxation within that single-mode amplitude family does not by itself establish stability against general perturbations.

With real $k,U$ and $\mu >k^2$, substitution into the nonlinear Ginzburg-Landau equation cancels the material derivative and gives $Q^2=\mu -k^2$. A complex $k$ makes the modulus spatially varying, so it generally does not admit this constant-amplitude nonlinear Ginzburg-Landau equation ansatz.

For $a>0$, the positive amplitude ordinary differential equation $R^{\prime }=aR-R^3$ has solution $R(t)=\sqrt{a}[1+(a/R_0^2-1)e^{-2at}{]}^{-1/2}$. This follows from the logistic differential equation for $R^2$. Every $R_0>0$ approaches $\sqrt{a}$ as a monotone function, and the positive distance to this limit decreases; an oscillating complex wave carrying this amplitude has no ordinary monotone ordering.

A normal mode $e^{i(kx-\omega t)}$ has dispersion relation $\omega =Uk+(c_d-i)k^2+i\mu$. The absolute wavenumber is the saddle where $d\omega /dk=0$. The Green function of the linear complex Ginzburg-Landau equation distinguishes convective hydrodynamic instability from absolute hydrodynamic instability.

For $t>0$ this Green function is the response to a point impulse on the infinite line, using the continuous square root of $1+i{c}_d$. The Fourier transform multiplies the initial transform by $e^{\mu t-ikUt-(1+i{c}_d)k^{2}t}$; its Gaussian inverse gives the displayed expression. At fixed $x$, its exponential rate is $\mu -U^2/[4(1+c_d^2)]$, whereas along $x=Ut$ it is $\mu$. Thus a temporally growing flow with $0<\mu <U^2/[4(1+c_d^2)]$ is convectively rather than absolutely unstable.

For $[{\nabla }^2+\xi (1-|\psi {|}^2)]\psi =0$ and a regular unit-charge core $\psi =f(r)e^{i[\theta +\chi (r)]}$, write current velocity $u_r=2{\chi }^{\prime }$ when the time-dependent kinetic operator is $-{\nabla }^2$. Separating the imaginary part givesWith $f=ar+O(r^3)$ and no singular core flux, integration gives $u_r=-\mathrm{Im}\xi r/2+O(r^3)$. If $\xi =g{n}_{\infty }-i\alpha$, the slope is $\alpha /2$. The phase-gradient convention $k={\chi }^{\prime }$ has half this slope, $\alpha /4$. These are local regularity results, not a global vortex-existence assertion.