= Solution
For a <normal mode>, $\partial_t\mapsto-i\omega$, $\partial_x\mapsto ik$, and $\partial_x^2\mapsto-k^2$. Substitution into the <linear complex Ginzburg-Landau equation> gives
$$
-i\omega+ikU-\mu+(1+ic_d)k^2=0.
$$
Therefore \b[the dispersion relation is]
$$
\boxed{D(k,\omega)=-i\omega+ikU-\mu+(1+ic_d)k^2=0,\qquad
\omega=Uk+(c_d-i)k^2+i\mu.}
$$
For real $k$, its <temporal growth rate> is $\omega_i=\mu-k^2$, with unique maximum $\mu$ at $k=0$. Its real <angular frequency> is $Uk+c_dk^2$, displaying the separate <advection> and <wave dispersion> terms.
Back to article page
