= Solution
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 $\operatorname{Im}\omega_*>0$ 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> gives
$$
D(k,\omega)=-i\omega-\alpha k^2-i\beta k^3+\gamma=0,\qquad \boxed{\omega(k)=i\alpha k^2-\beta k^3-i\gamma.}
$$
For real <wavenumber> $k$, the temporal <growth rate> is
$$
\operatorname{Im}\omega(k)=(\operatorname{Re}\alpha)k^2-(\operatorname{Im}\beta)k^3-\operatorname{Re}\gamma.
$$
A 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> is
$$
\boxed{\operatorname{Im}\beta=0,\qquad \operatorname{Re}\alpha\leq0.}
$$
The maximum is then $-\operatorname{Re}\gamma$, attained at $k=0$ and, when $\operatorname{Re}\alpha=0$, 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 $\beta\ne0$, the stationary candidates satisfy
$$
\omega'(k)=k(2i\alpha-3\beta k)=0,
$$
so
$$
k_*=0,\quad \omega_*=-i\gamma;\qquad k_*^{(1)}=\frac{2i\alpha}{3\beta},\quad \omega_*^{(1)}=-i\left(\gamma+\frac{4\alpha^3}{27\beta^2}\right).
$$
Each 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 $e^{-t\operatorname{Re}\gamma}$. For integrable transformed initial data this bounds its fixed-position amplitude by a constant times that exponential. Thus a further necessary condition is
$$
\boxed{\operatorname{Re}\gamma<0.}
$$
The <false complex saddle in dissipative cubic dispersion> illustrates why the second formal candidate cannot override this bound. For $\operatorname{Re}\alpha<0$, 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 $\operatorname{Re}\gamma<0$. 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 $e^{-\gamma t}$.
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