= Half-line linear dispersive Stokes global relation
{title2=$\widehat q_0+(1-k^2)G_0+ikG_1+G_2=e^{\omega t}\widehat q$}
For the <linear dispersive Stokes equation> $q_t+q_x+q_{xxx}=0$, let $\omega(k)=i(k-k^3)$. Its <local relation> is obtained with $X=q_{xx}+ikq_x+(1-k^2)q$. Integrating $\partial_t(e^{-ikx+\omega t}q)+\partial_x(e^{-ikx+\omega t}X)=0$ over the half-line gives the displayed <global relation>, with negative-exponential <Half-range Fourier transforms> and $G_j(k,t)=\int_0^t e^{\omega(k)s}\partial_x^jq(0,s)\,ds$. The spatial transforms are <holomorphic> for $\operatorname{Im}k<0$, whereas the finite-time boundary transforms are <entire functions>.
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