= Damped and antidamped fields from an antisymmetric velocity coupling
{title2=$\omega_{1,2}=\mp i/(2\tau)\pm\sqrt{c^2k^2-1/(4\tau^2)}$}
Adding $(\phi_1\phi_{2,t}-\phi_2\phi_{1,t})/(2\tau)$ to the cross-coupled scalar <Lagrangian density> produces independent equations $\phi_{1,tt}-c^2\phi_{1,xx}+\phi_{1,t}/\tau=0$ and $\phi_{2,tt}-c^2\phi_{2,xx}-\phi_{2,t}/\tau=0$. For the convention $e^{i(kx-\omega t)}$, their frequencies are respectively $-i/(2\tau)\pm\sqrt{c^2k^2-1/(4\tau^2)}$ and $+i/(2\tau)\pm\sqrt{c^2k^2-1/(4\tau^2)}$. Above the threshold they oscillate with decaying or growing envelopes; below it they have nonoscillatory real temporal exponents. At equality the repeated temporal root supplies $(A+Bt)e^{\mp t/(2\tau)}$. At $k=0$ each equation also has a constant solution. The full system is generically unstable because of the amplified second field, even though it arises from an autonomous variational system with conserved, indefinite total energy.
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