= Finite-interval Airy global relation
{title2=$\widehat q_0-e^{-ik^3t}\widehat q=\mathcal F-e^{-ikL}\mathcal G$}
For the <Airy equation> $q_t+q_{xxx}=0$ on $0<x<L$, define $F_j=\int_0^t e^{-ik^3s}\partial_x^jq(0,s)ds$ and $G_j$ analogously at $L$. Set $\mathcal F=k^2F_0-ikF_1-F_2$ and $\mathcal G=k^2G_0-ikG_1-G_2$. Integrating the <local relation> with flux $-q_{xx}-ikq_x+k^2q$ proves the displayed <global relation>. The <dispersion relation> is invariant under multiplication of $k$ by a cube root of unity; the resulting three equations determine the three missing boundary transforms for data $q(0,t),q(L,t),q_x(L,t)$.
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