Bounded temporal growth condition for cubic dispersion ($\mathrm{Im}\beta =0,\mathrm{Re}\alpha \le 0$)

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.