= Finite maximum temporal growth rate
{title2=$\sup_{k\in\mathbb R}\operatorname{Im}\omega(k)<\infty$}
With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\operatorname{Im}\omega$ 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 $\operatorname{Im}\omega=\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta>0$.
Back to article page
