Propagation of few-cycle pulses in a nonlinear medium and an integrable generalization of the sine-Gordon equation

Abstract

The generalized sine-Gordon equation is obtained under the theoretical investigation of interaction of few-cycle pulses in a nonlinear medium modeled by a set of four-level atoms. This equation is derived without the use of the slowly varying envelope approximation and is shown to be integrable in the frameworks of the inverse scattering transformation method. Its solutions describing the propagation of the solitons and breathers and their interaction are investigated. In the case of different signs of the parameters of the equation considered, it is revealed, in particular, that the collision of solitons with opposite polarities can lead to an appearance of the short-living pulse having extraordinarily large amplitude, whose dynamics is similar to that of rogue waves. Also, the solitons of ``rectangular'' form and the breathers with rectangular oscillations exist in the case of the same signs of the parameters.