A nonpolynomial Schroedinger equation for resonantly absorbing gratings

TL;DR

This paper derives a novel nonlinear Schrödinger equation with a radical term to model resonantly absorbing gratings, revealing new soliton solutions, their stability properties, and collision dynamics in the optical medium.

Contribution

It introduces a nonpolynomial Schrödinger equation with a radical term as an asymptotic model for resonant optical gratings, and finds explicit soliton solutions including bright, dark, and quasi-peakons.

Findings

Found a family of bright solitons obeying the Vakhitov-Kolokolov criterion.

Derived explicit forms of the largest amplitude quasi-peakon and dark compacton.

Analyzed soliton collisions, showing quasi-elastic interactions and breather formation.

Abstract

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).