Super and subluminal propagation in nonlinear Schrödinger equation model with self-steepening and self-frequency shift

We investigate exact traveling wave solutions of higher order nonlinear Schrödinger equation (NLSE) in the absence of third-order dispersion, which exhibit nontrivial self-phase modulation. It is shown that the corresponding dynamical equation, governing the evolution of intensity in the femtosecond regime, is that of NLSE with a source. The exact localized solutions to this system can have both super and subluminal propagation belonging to two distinct classes. A number of these solitons exhibit chirality, thereby showing preferential propagation behavior determined by group velocity dispersion. Both localized bright and dark solitons are found in complementary velocity and experimental parameter domains, which can exist for anomalous and normal dispersion regimes. It is found that dark solitons in this system propagate with nonzero velocity, unlike their counterpart in nanosecond regime. Interestingly, subluminal propagation is observed for solitons having a nontrivial Padé type intensity profile.