Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 331 / 3 / a

Take the physical coefficients $U,\mu ,c_d$ real. Substitution of $e^{i(kx-\omega t)}$ into the linear complex Ginzburg-Landau equation gives $-i\omega +ikU-\mu +(1+i{c}_d)k^2=0$, henceLet $A=c_d-i$. Completing the square yields $\omega =A(k+U/(2A){)}^2+i\mu -U^2/(4A)$. Therefore the absolute wavenumber and absolute frequency areTheir explicitly separated parts areThus $d\omega /dk=U+2(c_d-i)k$ vanishes at $k_0$. The complex saddle describes the localized Green function response; it is different from selecting a real wavenumber for a Fourier mode for temporal amplification.

At the absolute wavenumber, the group velocity $d\omega /dk$ is zero. The corresponding accessible saddle therefore governs the disturbance seen at a fixed position. Its exponential growth rate isSaddle accessibility can be checked directly here. With $D=1+i{c}_d$, whose real part is positive, Fourier inversion gives the Green function of the linear complex Ginzburg-Landau equation,At fixed $x$ its exponential rate is $\mathrm{Im}{\omega }_0$, while along the packet centre $x=Ut$ it is $\mu$. For real wavenumbers of Fourier modes, the temporal growth rate is $\mathrm{Im}\omega (k)=\mu -k^2$, so the flow is temporally unstable exactly when $\mu >0$.

Consequently the classifications for a localized disturbance in this laboratory frame areFor convective hydrodynamic instability, a travelling wave packet amplifies but the response at each fixed position decays. For absolute hydrodynamic instability, that fixed-position response amplifies. The equality is the marginal absolute threshold: its exponential rate is zero and the impulse response has a $t^{-1/2}$ prefactor. For $\mu <0$ there is no temporal growth; $\mu =0$ is temporally marginal. When $U=0$ the convective window is empty. These conclusions use the infinite-line impulse problem; a general dispersion relation requires its own spatial-branch or saddle selection.