A model equation for ultrashort optical pulses around the zero dispersion frequency

The nonlinear Schrodinger equation (NSE) based on the Taylor approximation of the material dispersion can become invalid for ultrashort and few-cycle optical pulses. Instead, we use a rational fit to the dispersion function around the zero dispersion frequency where the transition between anomalous and normal dispersion regimes occurs. This approach allows us to derive a simple non-envelope model for pulses propagating in time within a transparency window of a nonlinear dispersive medium with an instantaneous cubic nonlinearity. For this model we investigate integrals of motion and demonstrate that a uniformly moving non-envelope soliton does not exist. The only possible localized solution is the solitary breather with some intrinsic dynamics in the comoving frame. Classical envelope solitons oscillating in the comoving frame appear for a longer pulse for which the model is equivalent to the standard NSE. For an ultrashort pulse the model provides a natural bridge between the known non-envelope equations for the purely normal and anomalous dispersion regimes.