= Two-trace half-line cubic dispersion representation
{title2=$w(k)=i(k^3+k)$}
For $q_t+q_x-q_{xxx}=0$ on $x>0$, its <local relation> has flux $(1+k^2)q-ikq_x-q_{xx}$. With $Q_0(k)=\int_0^\infty e^{-ikx}q(x,0)dx$ and $G_j(k,T)=\int_0^T e^{w(k)s}\partial_x^jq(0,s)ds$, its <global relation> is $e^{wT}\widehat q=Q_0+(1+k^2)G_0-ikG_1-G_2$. In $D_+=\{\operatorname{Im}k>0,\operatorname{Re}w(k)<0\}$, choose the other cubic root $\nu(k)$ in the lower half-plane. Then the known spectral density is $-Q_0(\nu)+(k^2-\nu^2)G_0-i(k-\nu)G_1$. The remaining final-state transform has zero integral around $\partial D_+$ by the <Cauchy integral theorem> and exponential decay. The full boundary includes the real axis and the hyperbola $\operatorname{Im}k=\sqrt{1+3(\operatorname{Re}k)^2}$, oriented with the domain on the left.
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