 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:Indeed,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.