= Solution
Define the negative-exponential <Half-range Fourier transforms>
$$
\widehat q_0(k)=\int_0^\infty e^{-ikx}q_0(x)\,dx,
\qquad\widehat q(k,t)=\int_0^\infty e^{-ikx}q(x,t)\,dx,
$$
which are <holomorphic> for $\operatorname{Im}k<0$. Write $g_j(s)=\partial_x^jq(0,s)$ and define the <finite-time spectral boundary transforms>
$$
G_j(k,t)=\int_0^t e^{\omega(k)s}g_j(s)\,ds\qquad(j=0,1,2).
$$
Integrate 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> is
$$
\boxed{e^{\omega(k)t}\widehat q(k,t)=\widehat q_0(k)+B(k,t),\qquad
B=(1-k^2)G_0+ikG_1+G_2,\quad\operatorname{Im}k\leq0.}
$$
<Fourier inversion> on the real axis gives an initial-data <integral> plus an <integral> of $B$. To put the boundary term on a complex <contour>, set
$$
D_+=\{k:\operatorname{Im}k>0,\ \operatorname{Re}\omega(k)<0\}.
$$
For $k=u+iv$, $\operatorname{Re}\omega=v(3u^2-v^2-1)$, 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 $-1/\sqrt3$, along the real segment to $1/\sqrt3$, and then to upper-right infinity.
The <finite-time spectral boundary transforms> are <entire functions>. In the upper regions outside $D_+$, where $\operatorname{Re}\omega\geq0$, their time-integrated factors satisfy $|e^{-\omega t}e^{\omega s}|\leq1$ for $0\leq s\leq t$. Together with $e^{ikx}$, this permits <contour deformation> of the two remaining real-axis pieces to the curved boundary of $D_+$. The resulting representation is
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega(k)t}\widehat q_0(k)\,dk
+\frac1{2\pi}\int_{\partial D_+}e^{ikx-\omega(k)t}\bigl[(1-k^2)G_0+ikG_1+G_2\bigr]dk.}
$$
The <contour> orientation accounts for the plus sign in this convention. This formula still contains the two unknown derivative traces through $G_1,G_2$.
Back to article page
