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A localized perturbation has positive asymptotic exponential growth at a fixed observation point in the chosen reference frame. The relevant spatial pinch point of the Briggs-Bers criterion has positive imaginary frequency. Merely finding a growing temporal mode or a formal complex saddle is not sufficient; the saddle must contribute to the localized impulse response.

For $\omega (k)=i\alpha k^2-\beta k^3-i\gamma$, real-wavenumber growth is $\mathrm{Re}\alpha k^2-\mathrm{Im}\beta k^3-\mathrm{Re}\gamma$. A nonzero cubic coefficient is unbounded above on one side; after it vanishes, a positive quadratic coefficient is unbounded above on both sides. The displayed conditions are therefore necessary and sufficient for a finite temporal upper bound, which is then $-\mathrm{Re}\gamma$. The bound permits the initial causal temporal contour required by the Briggs-Bers criterion.

A localized disturbance grows along some moving rays but decays at its original fixed location. This wave-packet meaning differs from buoyancy-driven stellar convection and is therefore given a distinct canonical title. Changing the observation frame can change the absolute/convective classification. The Briggs-Bers criterion uses the laboratory-frame impulse response to distinguish this case from absolute wave-packet instability.

With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\mathrm{Im}\omega$ allows the inverse temporal Laplace transform contour to start above all temporal singularities. An unbounded growth rate at high real wavenumber prevents the standard causal Green function construction used in the Briggs-Bers criterion. For $\mathrm{Im}\omega =\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta >0$.

The real-wavenumber dispersion relation has the explicit branchIts frequencies are real. A real-axis Fourier transform propagator multiplies every component by $e^{-i\omega (k)t}$, of modulus one, so a square-integrable localized disturbance has constant $L^2$ norm. For sufficiently smooth localized data with an integrable Fourier transform, its pointwise amplitude also has a time-independent upper bound. In particular, there is no exponential growth along a fixed or moving ray.

The Briggs-Bers criterion makes the same conclusion while enforcing causality. Start the temporal inversion line at $\mathrm{Im}\omega =\sigma >0$ and the spatial inversion contour on the real $k$ axis. The spatial roots can be followed explicitly:Choose the principal logarithm along this upper temporal half-plane. Since $\mathrm{Im}(1-2\omega )<0$, the $j=0$ root lies above the real $k$ axis; the $j=-1$ root lies below it. Successive roots remain separated by $i\pi /a$. As $\sigma$ is lowered, $k_0$ approaches the real axis when $\omega <1/2$ is real. The spatial contour is then indented beneath that root, as required by continuation from positive $\sigma$. No lower root meets it: their separation stays nonzero.

Algebraically,for every finite complex $k$, so there is no finite spatial pinch point. The logarithmic singularity at $\omega =1/2$ sends roots to $\mathrm{Re}k=+\infty$; it is on the real-frequency boundary, not a positive-growth pinch in the upper temporal half-plane. The continuum of neutral poles is the real curve $\omega <1/2$ in the $\omega$ plane. It is reached as a causal boundary value, rather than producing an upper-half-plane singularity.

The figure follows neighboring spatial roots while lowering horizontal temporal contours. It shows why merely finding roots in both spatial half-planes does not establish an instability: those roots never collide across the inversion contour.

These modes have neither absolute wave-packet instability nor convective wave-packet instability; they are temporally neutral. Their positive real group velocity transports wave packets downstream. Nonreal spatial roots describe spatial continuation and evanescence, not temporal amplification. The exponential growth criteria concern the causal impulse response, not an unrestricted choice of a complex root.

Select the causal Green function, zero for $t<0$; otherwise the impulse equation permits arbitrary homogeneous additions. Use the spatial Fourier transform convention $e^{-ikx}$ and the temporal Laplace transform boundary convention $e^{i\omega t}$. The transformed impulse equation isIf $\mu >0$, a literal real-frequency Fourier transform of the growing response need not exist. Its causal inverse instead starts on $\mathrm{Im}\omega =c>\max (\mu ,0)$ and is continued from there. For $t>0$ close the frequency contour below; its clockwise orientation and the pole residue $1/(-i)=i$ give $e^{-i\omega (k)t}$. For $t<0$ the upper closure contains no pole. The remaining inverse spatial transform is a Gaussian integral:The unit-mass Gaussian heat kernel tends to $\delta (x)$ at $t\downarrow 0$, giving the required impulse normalization.

At the advected center $x=Ut$, its amplitude is $e^{\mu t}/\sqrt{4\pi \gamma t}$. At a fixed spatial point,Thus $\mu >0$ permits growth of the moving wave packet, but it is convective hydrodynamic instability if the fixed-position rate is negative. A positive fixed-position rate is absolute hydrodynamic instability. At its equality threshold the $t^{-1/2}$ factor decays, explaining the marginal exponential classification. This directly verifies the physically selected Briggs-Bers criterion saddle rather than assuming its accessibility.

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 $\gamma$.

The Briggs-Bers technique starts with a temporal Laplace transform and spatial Fourier transform of a localized disturbance. The temporal inversion contour must initially lie above every temporal singularity. Lower it while deforming the spatial contour away from the roots of the dispersion relation. A collision of spatial branches originating on opposite sides of that contour can create a spatial pinch point. A contributing pinch with $\mathrm{Im}{\omega }_{\ast }>0$ gives absolute wave-packet instability: exponential growth at a fixed position. Growth that is transported away while a fixed position decays is convective wave-packet instability. Solving $D=D_k=0$ finds candidate double roots, but does not by itself establish a pinch.

Substitution of the normal mode givesFor real wavenumber $k$, the temporal growth rate isA nonzero real cubic coefficient in this expression makes it unbounded above on one end of the real axis. Once that coefficient vanishes, a positive quadratic coefficient also makes it unbounded above. Hence the bounded temporal growth condition for cubic dispersion isThe maximum is then $-\mathrm{Re}\gamma$, attained at $k=0$ and, when $\mathrm{Re}\alpha =0$, at all real $k$. A finite upper bound is necessary for choosing the initial temporal inversion contour; arbitrarily fast high-wavenumber growth prevents the usual Briggs-Bers criterion from defining that causal inversion for generic localized data.

For $\beta \ne 0$, the stationary candidates satisfysoEach candidate must pass the spatial-branch pinch test before its imaginary frequency is used. In particular, under the finite-growth conditions the real-wavenumber evolution has multiplier modulus at most $e^{-t\mathrm{Re}\gamma }$. For integrable transformed initial data this bounds its fixed-position amplitude by a constant times that exponential. Thus a further necessary condition isThe false complex saddle in dissipative cubic dispersion illustrates why the second formal candidate cannot override this bound. For $\mathrm{Re}\alpha <0$, the neighborhood $k=0$ gives a fixed-position impulse response proportional to $e^{-\gamma t}/\sqrt{t}$: the cubic term is smaller on its $k=O(t^{-1/2})$ scale. It therefore supplies the usual growing pinch when $\mathrm{Re}\gamma <0$. The purely dispersive cases have stationary-phase algebraic prefactors instead. If $\beta =0$, only the quadratic stationary point remains; if both derivative coefficients vanish, the equation is local multiplication and its growth is directly $e^{-\gamma t}$.

Substitution of a normal mode gives the dispersion relationFor real $k$, $\sigma$ is the temporal growth rate. Since $\beta \ne 0$, its supremum is finite exactly whenFor $\beta <0$, arbitrarily short wavelengths grow arbitrarily fast. For $\beta >0$, the finite maximum temporal growth rate isThe Briggs-Bers criterion starts the inverse temporal Laplace transform above all temporal singularities and then deforms its contour downward while following the spatial roots. A finite growth bound supplies such an initial contour and a causal, high-frequency-controlled Green function. Unbounded temporal growth prevents that standard construction.

For absolute wave-packet instability, a candidate spatial pinch point must satisfy $D=0$, $D_k=0$, and $\mathrm{Im}\omega >0$. HeregivesThus candidate growing saddles require $\gamma <0$ at $k_0$, or $\gamma <{\alpha }^2/(4\beta )$ at $k_{\pm }$. Together with $\beta >0$, existence of at least one such candidate requires $\gamma <{\alpha }^2/(4\beta )$.

A growing double root is not sufficient: the roots must pinch the spatial inversion contour from opposite sides. Collisions of branches originating in the same spatial half-plane do not obstruct the relevant deformation. This distinction is part of the Briggs-Bers criterion; it is stated, for example, in the primary study doi.org/10.1017/jfm.2016.195.

An explicit false spatial saddle in quartic dispersion is $\alpha =-2$, $\beta =1$, $\gamma =1/2$. Its candidates $k=\pm i$ have $\omega =i/2$, but all real modes have $\sigma =-k^4-2k^2-1/2<0$. Each imaginary collision joins two branches in the same half-plane; neither is a relevant pinch. For $\omega =i\Omega$, the spatial roots obey $k^2=-1\pm \sqrt{1/2-\Omega }$, making those same-half-plane collisions transparent as $\Omega \downarrow 1/2$.

For this particular real, even dispersion relation one can also establish the actual threshold directly. At the origin its impulse Green function isLaplace method selects the real maximum, and gives a positive prefactor times $t^{-1/2}{e}^{{\sigma }_{\max }t}$ because $\alpha \ne 0$. Therefore the actual absolute wave-packet instability condition isThese are sufficient for this model as well as necessary. They follow after identifying relevant real saddles; the earlier algebraic double-root test alone lacks the pinch information. Equality is marginal, not exponential absolute growth.

For fixed complex frequency $\omega$, a spatial root is a complex wavenumber $k$ satisfying the dispersion relation. Analytic continuation of these branches from a causal temporal contour is essential in the Briggs-Bers criterion; their collision alone does not decide whether a spatial pinch point occurs.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.