= Solution
Apply <Fourier inversion> to the <global relation> on the real axis, extending $u$ by zero to negative $x$. At an interior point $x>0$ this gives
$$
u(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega t}\widehat u_0(k)\,dk
-\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega t}[G_1+(ik+\alpha)G_0](k,t)\,dk.
$$
The <finite-time spectral boundary transforms> are <entire functions> of $k$. Define the upper decay domain
$$
D^+=\{k=a+ib:b>0,\ \operatorname{Re}\omega(k)<0\}
=\{a+ib:b>\alpha,\ a^2<b(b-\alpha)\}.
$$
Orient $\partial D^+$ from its left infinite end through $i\alpha$ to its right infinite end, so that $D^+$ lies to the left. Its finite point is $i\alpha$, not zero. On the contour, $\operatorname{Re}\omega=0$; the factor $e^{ikx}$ decays because $x>0$.
For the boundary term, write $e^{-\omega t}G_j=\int_0^t e^{-\omega(t-s)}g_j(s)\,ds$. Between the real axis and $\partial D^+$, $\operatorname{Re}\omega\geq0$, so this factor has no exponential growth. <Contour deformation> and <Jordan lemma>, with the usual cutoffs for oscillatory integrals, therefore give
$$
\boxed{u(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-\omega t}\widehat u_0(k)\,dk
-\frac1{2\pi}\int_{\partial D^+}e^{ikx-\omega t}[G_1+(ik+\alpha)G_0](k,t)\,dk.}
$$
This is the requested complex-plane representation with the unknown <finite-time spectral boundary transform> still present. The decay domain must agree with the sign of the drift in the <dispersion relation>.
Back to article page
