= Solution
Let $E=e^{-ikx+w(k)t}$. Expanding the proposed <local relation> gives $E[q_t+wq+X_x-ikX]$. Choose the derivative terms in $X$ to cancel the mixed spectral terms:
$$
\boxed{w(k)=i(k+k^3),\qquad X=(1+k^2)q-ikq_x-q_{xx}.}
$$
Indeed,
$$
X_x-ikX=q_x-q_{xxx}-i(k+k^3)q,
$$
so the resulting divergence is exactly $E(q_t+q_x-q_{xxx})$. This establishes the identity for every complex <spectral parameter> $k$, with the given differentiability. The cubic <dispersion relation> has the reverse sign from the usual $q_t+q_{xxx}=0$ convention; that sign affects both the spectral sectors and the number of required <boundary conditions>.
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