  The propagation of extremely short pulses of electromagnetic field
(electromagnetic spikes) is considered in the framework of a model where the
material medium is represented by anharmonic oscillators with cubic
nonlinearities (Duffing model) and waves can propagate only in the right
direction. The system of reduced Maxwell-Duffing equations admits two families
of exact analytical solutions in the form of solitary waves. These are bright
spikes propagating on a zero background, and bright and dark spikes,
propagating on a nonzero background. Direct simulations demonstrate that these
pulses are very robust against perturbations. We find that a high frequency
modulated electromagnetic pulse evolves into a breather-like one. Conversely a
low frequency pulse transforms into a quasi-harmonic wave.
