W -shaped and bright optical solitons in negative indexed materials

We investigate a generalized nonlinear Schrödinger equation with higher-order effects such as pseudo-quintic nonlinearity and self-steepening effect. The model applies to the description of ultrashort pulse propagation in nonlinear materials exhibiting a negative index of refraction. Three new types of nonlinearly chirped W-shaped soliton solutions are derived for the first time by using the traveling-wave method. The obtained structures have new functional forms that are distinct from the usual W-shaped soliton solution reported within the context of optical fibers. An important characteristic of these envelope solitons is the nonlinear chirp that is directly proportional to the intensity of the pulse. It is shown that these chirped W-shaped structures are formed as a result of the exact balance among the group velocity dispersion, the self-steepening effect, and the pseudo-quintic nonlinearity. Exact chirped bright soliton solutions of the model were also obtained under appropriate conditions. Our results may raise the possibility of some experiments and potential applications related to left-handed materials in the presence of self-steepening nonlinearity.