= Airy global modes of a linearly confined Ginzburg-Landau equation
{c}
{title2=$\omega_n=i[\mu_0-U^2/(4\gamma)+(\epsilon^2\gamma\lambda^2)^{1/3}z_n]$}
On $x>0$ with a homogeneous wall condition, positive <diffusion> $\gamma$ and growth $\mu(x)=\mu_0-\epsilon\lambda x$, remove constant drift by $A=e^{Ux/(2\gamma)-i\omega t}f$. The scale $\xi=(\epsilon\lambda/\gamma)^{1/3}x$ reduces the <eigenvalue problem> to the <Airy ordinary differential equation>. Decay selects $f=\operatorname{Ai}(\xi+z_n)$, where $\operatorname{Ai}(z_n)=0$ enforces the wall. The least negative zero $z_1$ sets the <global instability> threshold $\mu_0>U^2/(4\gamma)+|z_1|(\epsilon^2\gamma\lambda^2)^{1/3}$. A locally absolutely unstable pocket can therefore exist without an unstable <global mode>.
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