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

For this nonlinear ansatz, take $k$ real as well as $U,\mu$ real. Then $|e^{ik(x-Ut)}|=1$, and the advective time derivative vanishes. Substitution into the nonlinear Ginzburg-Landau equation givesThe stipulated positive-amplitude Ginzburg-Landau plane wave therefore hasThe strict inequality $\mu >k^2$ guarantees that this is real and positive. The zero solution of the nonlinear Ginzburg-Landau equation also exists but is not the required positive wave. Although complex $k$ is useful in the linear impulse analysis, it cannot generally be carried into this constant-amplitude nonlinear ansatz: its spatially varying modulus would make the cubic term carry a different spatial factor.

Put $a=\mu -k^2=Q^2>0$. The shared travelling phase again cancels advection in the nonlinear Ginzburg-Landau equation; the remaining amplitude ordinary differential equation isFor $R>0$, $Y=R^2$ obeys the logistic differential equation $Y^{\prime }=2Y(a-Y)$. Equivalently, $Z=1/Y$ solves $Z^{\prime }=-2aZ+2$, whence $Z(t)=a^{-1}+(R_0^{-2}-a^{-1})e^{-2at}$. The nonconstant cubic amplitude saturation solution isIt takes the specified $R_0$ at $t=0$. The denominator of the cubic amplitude saturation solution remains positive for all future time: if $R_0<Q$ its bracket exceeds one, and if $R_0>Q$ it lies between $Q^2/R_0^2$ and one. Thus the positive solution is global forward in time. $R_0=Q$ would give the excluded constant solution; $R_0=0$ would remain at zero rather than relax to the positive wave.

The cubic amplitude saturation equation $R^{\prime }=R(Q^2-R^2)$ and the Picard-Lindelöf theorem imply that positive solutions cannot cross the equilibrium $Q$. If $0<R_0<Q$, then $R$ stays below $Q$ and increases. If $R_0>Q$, then it stays above $Q$ and decreases. The explicit cubic amplitude saturation solution tends to $Q$ in both cases, soThe same relation makes the distance a decreasing monotone function in any spatial norm for which the plane wave has finite norm, for example on a periodic domain. At large time, $R-Q\sim -\frac{Q}{2}(Q^2/R_0^2-1)e^{-2Q^{2}t}$.

The printed monotonicity wording must be interpreted as monotone amplitude relaxation, or monotone decay of the distance between the waves. Complex numbers have no ordinary monotone ordering, and the real part at a fixed point can oscillate. For instance, $k=U=1$, $\mu =2$ and $R_0=1/2$ give $\mathrm{Re}{\psi }_R(0,t)=R(t)\cos t$, which is not monotone despite the monotone amplitude. This result concerns the common-phase single-mode family, not stability against arbitrary spatial perturbations.

When $R\ll Q$, the ratio of cubic damping to linear amplification is $R^2/Q^2\ll 1$. The cubic amplitude saturation equation therefore giveswhile the small-amplitude regime lasts. With $c_d=0$, the matching real-wavenumber normal mode of the linear complex Ginzburg-Landau equation has $\omega =Uk+i(\mu -k^2)$ and hence amplitude proportional to $e^{(\mu -k^2)t}$. ThusThis is the temporal growth rate of the same Fourier component. It is not the fixed-position absolute frequency rate $\mu -U^2/4$ of a localized impulse. Nonlinear damping becomes important when $R$ is comparable with $Q$ and then arrests exponential amplification.