We derive a nonlinear Schroedinger equation with a radical term, in the form
of the square root of (1-|V|^2), as an asymptotic model of the optical medium
built as a periodic set of thin layers of two-level atoms, resonantly
interacting with the electromagnetic field and inducing the Bragg reflection. A
family of bright solitons is found, which splits into stable and unstable
parts, exactly obeying the Vakhitov-Kolokolov criterion. The soliton with the
largest amplitude, which is |V| = 1, is found in an explicit analytical form.
It is a "quasi-peakon", with a discontinuity of the third derivative at the
center. Families of exact cnoidal waves, built as periodic chains of
quasi-peakons, are found too. The ultimate solution belonging to the family of
dark solitons, with the background level |V| = 1, is a dark compacton, also
obtained in an explicit analytical form. Those bright solitons which are
unstable destroy themselves (if perturbed) attaining the critical amplitude,
|V| = 1. The dynamics of the wave field around this critical point is studied
analytically, revealing a switch of the system into an unstable phase.
Collisions between bright solitons are investigated too. The collisions between
fast solitons are quasi-elastic, while slowly moving ones merge into breathers,
which may persist or perish (in the latter case, also by attaining |V| = 1).Comment: Physical Review A, in pres