Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 331 / 3 / b

In the linear complex Ginzburg-Landau equation, take $U,c_d,\mu$ real. $U$ gives advection, carrying the envelope downstream at speed $U$. $c_d$ gives wave dispersion: the imaginary part of the second-derivative coefficient changes wave phase without directly dissipating a real-wavenumber Fourier mode. $\mu$ gives local linear production or damping: a spatially uniform perturbation grows for $\mu >0$ and decays for $\mu <0$.

The real part of the diffusion coefficient is the scaled value one. It smooths the envelope and damps a real-wavenumber Fourier mode at rate $k^2$. Thus the linear complex Ginzburg-Landau equation models competing advection, smoothing through a diffusion equation, wave dispersion and local growth, rather than nonlinear saturation.

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 givesTherefore the dispersion relation isFor real $k$, its temporal growth rate is ${\omega }_i=\mu -k^2$, with unique maximum $\mu$ at $k=0$. Its real angular frequency is $Uk+c_d{k}^2$, displaying the separate advection and wave dispersion terms.

The dispersion relation gives the zero-group velocity saddleThus the absolute growth rate is $\sigma (0)=\mu -U^2/[4(1+c_d^2)]$. More generally the saddle growth rate along a ray isso a frame travelling at $V=U$ sees the maximal rate $\mu$.

For $U>0$, the three open regions areThe line $\mu =0$ is the temporal marginal boundary; the parabola $\mu =U^2/[4(1+c_d^2)]$ is the convective-to-absolute boundary. The Green function of the linear complex Ginzburg-Landau equation has the prefactor $t^{-1/2}$, so zero exponential rate on the latter boundary still allows algebraic decay of the impulse response. This is the stability diagram of the linear complex Ginzburg-Landau equation.