= Solution
Above the threshold, $c^2k_j^2>1/(4\tau^2)$, the square root in part (d) is real. Both fields oscillate, but the first has envelope $\boxed{e^{-t/(2\tau)}}$ and the second $\boxed{e^{+t/(2\tau)}}$. Thus the first is damped and the second amplified.
Below the threshold, set $\sigma_j=\sqrt{1/(4\tau^2)-c^2k_j^2}$. The temporal factors are
$$
\boxed{\phi_1:\ e^{[-1/(2\tau)\pm\sigma_1]t},\qquad\phi_2:\ e^{[+1/(2\tau)\pm\sigma_2]t}.}
$$
They are nonoscillatory. For nonzero $k_j$, $0<\sigma_j<1/(2\tau)$, so both first-field branches decay and both second-field branches grow. At $k_j=0$, one branch is constant; the other is $e^{-t/\tau}$ for $\phi_1$ or $e^{t/\tau}$ for $\phi_2$.
At equality, the repeated temporal root gives $\boxed{(A+Bt)e^{-t/(2\tau)}}$ for the first field and $\boxed{(C+Dt)e^{t/(2\tau)}}$ for the second. The full two-field system is generically unstable on both sides because of the amplified sector. This is compatible with its autonomous Lagrangian: the total conserved <energy> is an <indefinite quadratic form>, rather than a positive energy forcing every component to remain bounded.
Back to article page
