Nonpolynomial Schrödinger equation for resonantly absorbing gratings

We derive a nonlinear Schrödinger equation with a radical term, ∼1-|V|2, as an asymptotic model of the resonantly absorbing Bragg reflector (RABR), i.e., 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, (|V|)_max=1, is a "quasipeakon," i.e., a solution with a discontinuity of the third derivative at the center. Families of exact cnoidal waves, built as periodic chains of quasipeakons, are found too. The ultimate solution belonging to the family of dark solitons, with the background level V=1, is a dark compacton. Those bright solitons that 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, in terms of the RABR model. Collisions between bright solitons are investigated too. The collisions between fast solitons are quasielastic, while slowly moving ones merge into breathers, which may persist or perish (in the latter case, also by attaining |V|=1).