Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons

Abstract

In this work, an efficient novel technique, namely, the q-homotopy analysis transform method (q-HATM) is applied to obtain analytical solutions for a system of time-fractional coupled nonlinear Schrödinger (TF-CNLS) equations with the time-fractional derivative taken in the Caputo sense. This system of equations incorporate nonlocality behaviors which cannot be modeled under the framework of classical calculus. With numerous important applications in nonlinear optics, it describes interactions between waves of different frequencies or the same frequency but belonging to different polarizations. We first establish existence and uniqueness of solutions for the considered time-fractional problem via a fixed point argument. To demonstrate the effectiveness and efficiency of the q−HATM, two cases each of two time-fractional problems are considered. One important feature of the q−HATM is that it provides reliable algorithms which can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series. Numerical simulation are provided to capture the behavior of the state variables for distinct values of the fractional order parameter. The results demonstrate that the general response expression obtained by the q−HATM contains the fractional order parameter which can be varied to obtain other responses. Particularly, as this parameter approaches unity, the responses obtained for the considered fractional equations approaches that of the corresponding classical equations.