Solution
= Solution
Set $\omega(k)=k^2-i\beta k$ and $E=e^{-ikx+\omega t}$. Differentiation gives $(Eu)_t=E(u_t+\omega u)$, while
$$
\bigl(E[(ik+\beta)u+u_x]\bigr)_x
=E\left[u_{xx}+\beta u_x+(k^2-i\beta k)u\right].
$$
The two sides are equal precisely by the stated <advection-diffusion equation>. This supplies its divergence form in the $(x,t)$ plane.