= Solution
Put $w(k)=-i(k^3+k)$ and define the spectral flux
$$
\boxed{X(x,t,k)=q_{xx}+ikq_x-(k^2+1)q.}
$$
Writing $E=e^{-ikx+w(k)t}$, division of the proposed <divergence form> by $E$ gives $q_t+wq+X_x-ikX$. Direct differentiation yields
$$
X_x-ikX=q_{xxx}-q_x+i(k^3+k)q.
$$
The last term cancels $wq$, so the <divergence form> is exactly the normalized <partial differential equation>. This calculation fixes both the sign of the <dispersion relation> and the sign of the boundary flux used below.
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