= Dirichlet spectral representation for the linear dispersive Stokes equation
Let $A_j$ be the weights of the <cubic Stokes dispersion symmetry> and set $F(k)=\sum_jA_j(k)\widehat q_0(\nu_j(k))$. Orient $\partial D_+$ with $D_+$ on its left. Evaluating the <global relation> at its two lower-half-plane roots and interpolating the linear expression $ikG_1+G_2$ removes both unknown boundary traces. The resulting <Fokas method> formula is
$$
q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega t}\widehat q_0(k)\,dk+\frac1{2\pi}\int_{\partial D_+}e^{ikx-\omega t}\bigl[(1-3k^2)G_0(k,t)-F(k)\bigr]dk.
$$
The discarded transform of the solution is <holomorphic> in $D_+$ and has zero <integral> after closing its <contour> for $x>0$. Initial compatibility and sufficient smoothness and spatial decay are required. At the boundary, use a fixed time horizon later than $t$ before <Fourier inversion>, to avoid an endpoint half-value.
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