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An absolute wavenumber is a complex wavenumber at the selected zero-group velocity saddle of a dispersion relation. For the linear complex Ginzburg-Landau equation, $k_0=-U/[2(c_d-i)]$; the positive real part of the diffusion coefficient makes the Gaussian saddle accessible from the real Fourier contour.

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.

For the whole-line linear complex Ginzburg-Landau equation with real positive diffusion $\gamma$ and $\mu (x)={\mu }_0-\nu {ϵ}^2{x}^2$, removing drift and scaling $\xi =\sqrt{ϵ}(\nu /\gamma {)}^{1/4}x$ gives the Hermite differential equation. Decay at both infinities quantizes the spectral parameter to $2j+1$. The strongest mode is $j=0$, giving global instability iff ${\mu }_0>U^2/(4\gamma )+ϵ\sqrt{\gamma \nu }$. The confinement correction vanishes in the slow-profile limit, recovering the homogeneous absolute wave-packet instability threshold.

Assume $\gamma >0$, the positive diffusivity requirement for the forward heat kernel. A normal mode $e^{ikx-i\omega t}$ of the linear complex Ginzburg-Landau equation givesFor real $k$, the maximum temporal growth rate is $\mu$. The Briggs-Bers criterion starts the inverse time Laplace transform above the temporal growth spectrum and follows the two spatial roots as its frequency contour is lowered. A contributing spatial pinch point occurs when roots continued from opposite sides of the spatial contour coalesce and prevent its deformation. An arbitrary algebraic saddle is not sufficient without this branch selection.

Here $D=D_k=0$ givesTo check the pinch, start at $\omega =i\Omega$ with $\Omega$ large. The roots are $k=-iU/(2\gamma )\pm i\sqrt{U^2+4\gamma (\Omega -\mu )}/(2\gamma )$ and initially lie in opposite half-planes. They meet at $\Omega =\mu -U^2/(4\gamma )$, giving the causal saddle. Positive imaginary absolute frequency means fixed-position exponential growth. ThereforeFor $\mu \le 0$ there is no positive temporal growth. At $\mu =U^2/(4\gamma )$ the fixed-position exponential rate is zero; its algebraic prefactor, derived below, decays. When $U=0$ the convective interval is empty.

The source specifies only real coefficients. For $\gamma <0$, temporal growth is unbounded at high real wavenumber, so the usual causal forward-diffusion construction is ill posed and the above criterion does not apply. For $\gamma =0$ the equation reduces to transport with amplification; its point impulse remains a moving delta, rather than a spreading heat kernel. These cases cannot be included by division by $\gamma$.

Use the real diffusive interpretation $\gamma >0$, $U,{\mu }_0\in \mathbb{R}$, with $\nu >0$. Without a positive real diffusion coefficient the printed ordering inequality and real oscillator coordinate would need different hypotheses. Remove the drift by $\eta =e^{Ux/(2\gamma )}v$. Direct differentiation reduces the linear complex Ginzburg-Landau equation toSet $v=b(\xi )e^{-i({\omega }_0+ϵ{\omega }_1)t}$ and $\xi =\sqrt{ϵ}fx$. The resulting equation isCanceling its order-one coefficient and normalizing the quadratic term givesThe supplied decay quantization has $\lambda =2j+1$, $j=0,1,2,\dots$. Equivalently the Hermite functions $b_j=e^{-{\xi }^2/2}{H}_j(\xi )$ solve this harmonic-oscillator problem, where $H_j$ is the physicists' Hermite polynomial. Thus ${\omega }_1=-i(2j+1)\sqrt{\nu \gamma }$ and the exact temporal growth rates areThe global modes of a quadratically confined Ginzburg-Landau equation areFor every fixed positive $ϵ$, the negative quadratic spatial exponent dominates the linear drift exponent and the polynomial at both infinities; all these normal modes are spatially bounded. The largest growth rate is $s_0$, so the least-node normal mode is temporally non-growing, and all normal modes are non-growing, whenEquality is a neutral ground-state normal mode; strict inequality gives decay. This proves the printed sufficient condition for a bounded stable global normal mode and identifies its stronger meaning: stability of the leading global normal mode. Existence of merely some stable higher-$j$ normal mode is not equivalent to the inequality, since sufficiently large $j$ gives $s_j<0$ even above the leading-mode threshold. In the quadratic-coefficient model the reduction and eigenvalues are exact, not just a first-order approximation to a more general profile.

For a normal mode, ${\partial }_t\mapsto -i\omega$, ${\partial }_x\mapsto ik$, and ${\partial }_x^2\mapsto -k^2$. Substitution into the linear complex Ginzburg-Landau equation givesTherefore the dispersion relation isFor real $k$, its temporal growth rate is ${\omega }_i=\mu -k^2$, with unique maximum $\mu$ at $k=0$. Its real angular frequency is $Uk+c_d{k}^2$, displaying the separate advection and wave dispersion terms.

In the linear complex Ginzburg-Landau equation, take $U,c_d,\mu$ real. $U$ gives advection, carrying the envelope downstream at speed $U$. $c_d$ gives wave dispersion: the imaginary part of the second-derivative coefficient changes wave phase without directly dissipating a real-wavenumber Fourier mode. $\mu$ gives local linear production or damping: a spatially uniform perturbation grows for $\mu >0$ and decays for $\mu <0$.

The real part of the diffusion coefficient is the scaled value one. It smooths the envelope and damps a real-wavenumber Fourier mode at rate $k^2$. Thus the linear complex Ginzburg-Landau equation models competing advection, smoothing through a diffusion equation, wave dispersion and local growth, rather than nonlinear saturation.

Take the physical coefficients $U,\mu ,c_d$ real. Substitution of $e^{i(kx-\omega t)}$ into the linear complex Ginzburg-Landau equation gives $-i\omega +ikU-\mu +(1+i{c}_d)k^2=0$, henceLet $A=c_d-i$. Completing the square yields $\omega =A(k+U/(2A){)}^2+i\mu -U^2/(4A)$. Therefore the absolute wavenumber and absolute frequency areTheir explicitly separated parts areThus $d\omega /dk=U+2(c_d-i)k$ vanishes at $k_0$. The complex saddle describes the localized Green function response; it is different from selecting a real wavenumber for a Fourier mode for temporal amplification.

When $R\ll Q$, the ratio of cubic damping to linear amplification is $R^2/Q^2\ll 1$. The cubic amplitude saturation equation therefore giveswhile the small-amplitude regime lasts. With $c_d=0$, the matching real-wavenumber normal mode of the linear complex Ginzburg-Landau equation has $\omega =Uk+i(\mu -k^2)$ and hence amplitude proportional to $e^{(\mu -k^2)t}$. ThusThis is the temporal growth rate of the same Fourier component. It is not the fixed-position absolute frequency rate $\mu -U^2/4$ of a localized impulse. Nonlinear damping becomes important when $R$ is comparable with $Q$ and then arrests exponential amplification.

For the linear complex Ginzburg-Landau equation with real $U,c_d,\mu$ and positive real diffusion coefficient scaled to one, the maximum temporal growth rate is $\mu$ and the absolute growth rate is $\mu -U^2/[4(1+c_d^2)]$. For $U>0$, $\mu <0$ is stable, $0<\mu <{\mu }_a$ has convective hydrodynamic instability, and $\mu >{\mu }_a$ has absolute hydrodynamic instability. Both equality boundaries are marginal in exponential rate.