= Solution
Use the <Half-range Fourier transform> and the <finite-time spectral boundary transforms>
$$
\widehat q(k,t)=\int_0^\infty e^{-ikx}q(x,t)\,dx,\quad Q_0(k)=\widehat q(k,0),\quad
G_j(k,t)=\int_0^t e^{w(k)s}g_j(s)\,ds,
$$
where $g_j=\partial_x^jq(0,\cdot)$ for $j=0,1,2$. The spatial transforms are <holomorphic> in $\operatorname{Im}k<0$, while these finite-time transforms are <entire functions>. For now $g_2$ is an unknown boundary trace. Integrating the <local relation> over the half-strip, using spatial decay at infinity, gives the <global relation>
$$
e^{w(k)t}\widehat q(k,t)=Q_0(k)+F(k,t),\qquad
F=(1+k^2)G_0-ikG_1-G_2.
$$
The boundary sign follows from integrating the flux derivative: its contribution is minus the flux at zero.
<Fourier inversion> first gives the initial real-line integral plus a real-line integral of $F$. To put the boundary contribution on its useful complex contour, define
$$
D_+=\{k=u+iv:v>0,\ \operatorname{Re}w(k)<0\}
=\{u+iv:0<v<\sqrt{1+3u^2}\}.
$$
Orient its full boundary with $D_+$ on the left. It consists of the real axis from left to right and the upper curve $C:k=u+i\sqrt{1+3u^2}$ from right to left. The real-axis piece must not be discarded for this dispersion. The <contour integration> representation is
$$
\boxed{q(x,t)=\frac1{2\pi}\int_{\mathbb R}e^{ikx-w(k)t}Q_0(k)\,dk
+\frac1{2\pi}\int_{\partial D_+}e^{ikx-w(k)t}F(k,t)\,dk.}
$$
To justify the replacement, the integral of $e^{ikx-wt}F$ along $C$ is zero by closing above $C$. In that region $\operatorname{Re}w\ge0$, and each boundary term is an integral of $e^{-w(t-s)}$ over $0\le s\le t$ times a polynomial in $k$. This has no growing temporal exponential there; $e^{ikx}$ suppresses the upper closing arc for $x>0$. The <Cauchy integral theorem> then applies. The figure shows the full contour and, separately, the sectors used in the nonlinear problem.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-61-spectral-domains.png]
{title=Full cubic-dispersion contour and bounded-column sectors for reverse-dispersion mKdV}
One may replace $F(k,t)$ by $F(k,T)$ for any finite $T\ge t$. Their difference is a future-time integral over $t<s<T$; its factor $e^{w(s-t)}$ decays inside $D_+$, so its integral around $\partial D_+$ vanishes. All contour formulas are understood as limits of truncated contours, with standard small displacements if a boundary transform needs an Abel limit. Compatibility of the initial and boundary data controls the corner behavior. This is the <two-trace half-line cubic dispersion representation>.
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