ABSTRACT
In this chapter, we study the propagation of a wave envelop with fractional temporal evolution by considering the transverse surface in the non-linear dynamic system as optical metamaterials. Using variable separation, we get a system of two equations: the first one is of Bessel type, and the second is a non-linear Schr\“{o}dinger equation. Solving these equations, when the fractional time-derivative is taken in the modified Riemann--Liouville sense, provides three kinds of chirped solitons. The obtained soliton solutions are new and have physical meaning. Through their graphical representations, we can deduce that it is possible to get a pulse that propagates in a dynamic system as optical metamaterials.
