= Dirichlet spectral formula for Airy flow with negative transport
{c}
{title2=$q_t+q_{xxx}-q_x=0$}
Let $F(k)=\int_0^\infty e^{-ikx}q_0(x)dx$ and $G_0(k,t)=\int_0^t e^{w(k)s}g_0(s)ds$. Using the roots of <cubic dispersion symmetry with negative transport>, put $\mathcal T F=[(k-\nu_2)F(\nu_1)+(\nu_1-k)F(\nu_2)]/(\nu_1-\nu_2)$. The <Fokas method> formula is an initial real-line integral minus an integral over $\partial D_+$ of $e^{ikx-wt}[\mathcal TF+(3k^2+1)G_0]$, both divided by $2\pi$. The boundary is oriented with $D_+$ on the left. Its unknown final-state transform has zero integral by analyticity and upper-half-plane decay.
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