Airy global modes of a linearly confined Ginzburg-Landau equation (${\omega }_n=i[{\mu }_0-U^2/(4\gamma )+({ϵ}^2\gamma {\lambda }^2{)}^{1/3}{z}_n]$)

On $x>0$ with a homogeneous wall condition, positive diffusion $\gamma$ and growth $\mu (x)={\mu }_0-ϵ\lambda x$, remove constant drift by $A=e^{Ux/(2\gamma )-i\omega t}f$. The scale $\xi =(ϵ\lambda /\gamma {)}^{1/3}x$ reduces the eigenvalue problem to the Airy ordinary differential equation. Decay selects $f=\mathrm{Ai}(\xi +z_n)$, where $\mathrm{Ai}(z_n)=0$ enforces the wall. The least negative zero $z_1$ sets the global instability threshold ${\mu }_0>U^2/(4\gamma )+|z_1|({ϵ}^2\gamma {\lambda }^2{)}^{1/3}$. A locally absolutely unstable pocket can therefore exist without an unstable global mode.