  We study a semilinear hyperbolic system of PDEs which arises as a continuum
approximation of the discrete nonlinear dimer array model introduced by Hadad,
Vitelli and Alu (HVA) in \cite{HVA17}. We classify the system's traveling
waves, and study their stability properties. We focus on traveling pulse
solutions (``solitons'') on a nontrivial background and moving domain wall
solutions (kinks); both arise as heteroclinic connections between spatially
uniform equilibrium of a reduced dynamical system. We present analytical
results on: nonlinear stability and spectral stability of supersonic pulses,
and spectral stability of moving domain walls. Our stability results are in
terms of weighted $H^1$ norms of the perturbation, which capture the phenomenon
of {\it convective stabilization}; as time advances, the traveling wave
``outruns'' the \underline{growing} disturbance excited by an initial
perturbation; the non-trivial spatially uniform equilibria are linearly
exponentially unstable. We use our analytical results to interpret phenomena
observed in numerical simulations.
