Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 74 / 3

The real-wavenumber dispersion relation has the explicit branchIts frequencies are real. A real-axis Fourier transform propagator multiplies every component by $e^{-i\omega (k)t}$, of modulus one, so a square-integrable localized disturbance has constant $L^2$ norm. For sufficiently smooth localized data with an integrable Fourier transform, its pointwise amplitude also has a time-independent upper bound. In particular, there is no exponential growth along a fixed or moving ray.

The Briggs-Bers criterion makes the same conclusion while enforcing causality. Start the temporal inversion line at $\mathrm{Im}\omega =\sigma >0$ and the spatial inversion contour on the real $k$ axis. The spatial roots can be followed explicitly:Choose the principal logarithm along this upper temporal half-plane. Since $\mathrm{Im}(1-2\omega )<0$, the $j=0$ root lies above the real $k$ axis; the $j=-1$ root lies below it. Successive roots remain separated by $i\pi /a$. As $\sigma$ is lowered, $k_0$ approaches the real axis when $\omega <1/2$ is real. The spatial contour is then indented beneath that root, as required by continuation from positive $\sigma$. No lower root meets it: their separation stays nonzero.

Algebraically,for every finite complex $k$, so there is no finite spatial pinch point. The logarithmic singularity at $\omega =1/2$ sends roots to $\mathrm{Re}k=+\infty$; it is on the real-frequency boundary, not a positive-growth pinch in the upper temporal half-plane. The continuum of neutral poles is the real curve $\omega <1/2$ in the $\omega$ plane. It is reached as a causal boundary value, rather than producing an upper-half-plane singularity.

The figure follows neighboring spatial roots while lowering horizontal temporal contours. It shows why merely finding roots in both spatial half-planes does not establish an instability: those roots never collide across the inversion contour.

These modes have neither absolute wave-packet instability nor convective wave-packet instability; they are temporally neutral. Their positive real group velocity transports wave packets downstream. Nonreal spatial roots describe spatial continuation and evanescence, not temporal amplification. The exponential growth criteria concern the causal impulse response, not an unrestricted choice of a complex root.

Expanding the exponential for $|ka|\ll 1$ gives $e^{-2ka}=1-2ka+2k^2{a}^2+O((ka{)}^3)$. Substitution of the plane wave $A=e^{i(kx-\omega t)}$ in the model giveswhich is exactly the quadratic approximation to the dispersion relation.

For the slowly varying width, put $X=ϵx$ and use the WKB approximationThe derivatives areAt leading order $\Omega =ak-a^2{k}^2$. On the downstream small-wavenumber branch writeso $q$ is independent of $X$. The next order gives the slowly varying triangular-jet envelope transport equationIntegration therefore yieldsIndeed, $q/(1-2q)=\Omega +O({\Omega }^2)$. The requested power is the leading low-frequency envelope, not the exact finite-frequency exponent of the quadratic model. The corresponding leading field is this wave envelope times $\exp (i{\int }_0^{x}q/a(ϵs)ds-i\Omega t)$. Positive slowly varying width and separation from the zero of the local group velocity, $1-2q=0$, are needed for this WKB approximation. Extremely large $|\log [a(X)/a(0)]|$ can also invalidate replacing the exact exponent by $\Omega$.