Stability diagram of the linear complex Ginzburg-Landau equation (${\mu }_a=U^2/[4(1+c_d^2)]$)

For the linear complex Ginzburg-Landau equation with real $U,c_d,\mu$ and positive real diffusion coefficient scaled to one, the maximum temporal growth rate is $\mu$ and the absolute growth rate is $\mu -U^2/[4(1+c_d^2)]$. For $U>0$, $\mu <0$ is stable, $0<\mu <{\mu }_a$ has convective hydrodynamic instability, and $\mu >{\mu }_a$ has absolute hydrodynamic instability. Both equality boundaries are marginal in exponential rate.