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Define the negative-exponential Half-range Fourier transformswhich are holomorphic for $\mathrm{Im}k<0$. Write $g_j(s)={\partial }_x^{j}q(0,s)$ and define the finite-time spectral boundary transformsIntegrate the local relation over $0<x<\infty$, $0<s<t$. The spatial boundary term at infinity vanishes by decay, and the term at $x=0$ is subtracted. Consequently the half-line linear dispersive Stokes global relation isFourier inversion on the real axis gives an initial-data integral plus an integral of $B$. To put the boundary term on a complex contour, setFor $k=u+iv$, $\mathrm{Re}\omega =v(3u^2-v^2-1)$, so $D_{+}$ is the connected upper domain $v>0$, $v^2>3u^2-1$. Orient $\partial D_{+}$ with this domain on the left: from upper-left infinity down to $-1/\sqrt{3}$, along the real segment to $1/\sqrt{3}$, and then to upper-right infinity.

The finite-time spectral boundary transforms are entire functions. In the upper regions outside $D_{+}$, where $\mathrm{Re}\omega \ge 0$, their time-integrated factors satisfy $|e^{-\omega t}{e}^{\omega s}|\le 1$ for $0\le s\le t$. Together with $e^{ikx}$, this permits contour deformation of the two remaining real-axis pieces to the curved boundary of $D_{+}$. The resulting representation isThe contour orientation accounts for the plus sign in this convention. This formula still contains the two unknown derivative traces through $G_1,G_2$.