Two-trace half-line cubic dispersion representation ($w(k)=i(k^3+k)$)

For $q_t+q_x-q_{xxx}=0$ on $x>0$, its local relation has flux $(1+k^2)q-ik{q}_x-q_{xx}$. With $Q_0(k)={\int }_0^{\infty }{e}^{-ikx}q(x,0)dx$ and $G_j(k,T)={\int }_0^T{e}^{w(k)s}{\partial }_x^{j}q(0,s)ds$, its global relation is $e^{wT}\hat{q}=Q_0+(1+k^2)G_0-ik{G}_1-G_2$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}w(k)<0\}$, 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 $\mathrm{Im}k=\sqrt{1+3(\mathrm{Re}k{)}^2}$, oriented with the domain on the left.