PDE from dispersion relation?

Suppose I have knowledge of a system's dispersion relation $f(\omega,k)$. Is it possible to recover the underlying PDE describing the system? Can I simply use the replacement $k=-i\nabla$, $\omega=i\frac{d}{dt}$ to go back?

I came across one source which claimed that the original PDE could only be recovered to a certain extent. An example given was the Dirac and Klein Gordon equations which both satisfy the same dispersion relation. But I didn't quite follow.

Example: I have a polynomial dispersion relation of the form $\omega^2+c^2k^2=1$. Can I automatically say that the underlying PDE is $\frac{d^2u(x,t)}{dt^2}+c^2\nabla^2u(x,t)+u(x,t)=0$? Or is there a subtlety I'm missing?

Thanks