= Bounded temporal growth condition for cubic dispersion
{title2=$\operatorname{Im}\beta=0,\quad\operatorname{Re}\alpha\leq0$}
For $\omega(k)=i\alpha k^2-\beta k^3-i\gamma$, real-wavenumber growth is $\operatorname{Re}\alpha\,k^2-\operatorname{Im}\beta\,k^3-\operatorname{Re}\gamma$. 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 $-\operatorname{Re}\gamma$. The bound permits the initial causal temporal contour required by the <Briggs-Bers criterion>.
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