Fractional Solitons in Optical Twin-Core Couplers with Kerr Law Nonlinearity and Local M-Derivative Using Modified Extended Mapping Method

This study focuses on optical twin-core couplers, which facilitate light transmission between two closely aligned optical fibers. These couplers operate based on the principle of coupling, allowing signals in one core to interact with those in the other. The Kerr effect, which describes how a material’s refractive index changes in response to the intensity of light, induces the nonlinear behavior essential for generating solitons—self-sustaining wave packets that preserve their shape and speed. In our research, we employ fractional derivatives to investigate how fractional-order variations influence wave propagation and soliton dynamics. By utilizing the modified extended mapping method (MEMM), we derive solitary wave solutions for the equations governing the behavior of optical twin-core couplers under Kerr nonlinearity. This methodology produces novel fractional traveling wave solutions, including dark, bright, singular, and combined bright–dark solitons, as well as hyperbolic, Jacobi elliptic function (JEF), periodic, and singular periodic solutions. To enhance understanding, we present physical interpretations through contour plots and include both 2D and 3D graphical representations of the results.