 Solution

Put $w(k)=-i(k^3+k)$ and define the spectral fluxWriting $E=e^{-ikx+w(k)t}$, division of the proposed divergence form by $E$ gives $q_t+wq+X_x-ikX$. Direct differentiation yieldsThe 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.