= Solution
The real-wavenumber <dispersion relation> has the explicit branch
$$
\omega(k)=\tfrac12(1-e^{-2ak}),\qquad \omega'(k)=ae^{-2ak}>0\quad(k\in\mathbb R).
$$
Its <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 $\operatorname{Im}\omega=\sigma>0$ and the spatial inversion contour on the real $k$ axis. The spatial roots can be followed explicitly:
$$
k_j(\omega)=-\frac{1}{2a}\operatorname{Log}(1-2\omega)+\frac{i\pi j}{a},\qquad j\in\mathbb Z.
$$
Choose the principal logarithm along this upper temporal half-plane. Since $\operatorname{Im}(1-2\omega)<0$, 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,
$$
D_k=-2ae^{-2ak}\ne0
$$
for every finite complex $k$, so there is no finite <spatial pinch point>. The logarithmic singularity at $\omega=1/2$ sends roots to $\operatorname{Re}k=+\infty$; 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-74-causal-roots.png]
{title=Causal temporal contours and nonpinching spatial roots of the triangular-jet dispersion relation}
\b[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.
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