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For a solution with a continuous gradient at a quadrant corner, the prescribed derivative directions are $(-\sin {\beta }_1,\cos {\beta }_1)$ and $(\cos {\beta }_2,-\sin {\beta }_2)$. When ${\beta }_1+{\beta }_2=\pi /2$, these vectors are negatives of one another. Their corner data must therefore be opposite. Imposing $h_1(0)=h_2(0)$ as well forces both to vanish. Smooth decay of separate boundary functions alone does not ensure this local compatibility condition for an overdetermined linear system.

// Target: analysis.bigb

The system prescribes both first derivatives of the same vector $\psi$, so it is overdetermined: the two mixed derivatives must agree. Differentiating givesThus every solution satisfiesFor the system to possess arbitrary nontrivial local solution data, the coefficient matrix must vanish:This is the zero-curvature condition, or the compatibility condition for an overdetermined linear system.

Differentiating the first auxiliary equation in $t$, the second in $x$, and equating ${\psi }_{xt}$ and ${\psi }_{tx}$ leaves preciselyso the Airy equation is their compatibility condition.

Multiplication of ${\psi }_x-ik\psi =q$ by $e^{-ikx}$ givesHenceFor $\mathrm{Im}k\ge 0$, the kernel $e^{ik(x-y)}$ is bounded when $y\le x$; dominated differentiation makes ${\psi }_{+}$ analytic for $\mathrm{Im}k>0$. Rapid decrease gives

For real $k$, multiply the time equation by $e^{-ikx}$ and let $x\to +\infty$. The rapidly decreasing terms on its right vanish, leavingThereforeand the Fourier inversion theorem gives