The nonlinear Schrödinger equation and its generalizations are fundamental in soliton theory and nonlinear wave dynamics of optical systems. This paper deals with the integrable Fokas-Lenells equation, which describes the nonlinear propagation of ultrashort optical pulses in optical fibers. We obtain precise optical soliton solutions to the Fokas–Lenells equation with the help of an extended Riccati equation mapping technique and an undetermined coefficients method, with a systematic scheme for obtaining analytical solutions under some parameter conditions. To ensure accuracy and reliability in the analytical solutions, the differential transform method is utilized to find numerical solutions for comparison purposes. Physical aspects and behavior of the soliton solutions are also represented by visualizing 2-dimensional, 3-dimensional, and density plots of the impact of various parameters on the profiles of waves. The research illustrates a number of new families of traveling wave solutions, such as dark, bright, and combinations of dark and bright solitons, presenting useful insights on nonlinear wave propagation, interaction among solitons, and possibilities of their implementation in optical communications and photonic technologies. The originality of this paper is in the combination of an extended Riccati equation mapping method and the method of undetermined coefficients to obtain novel exact soliton solutions of the Fokas–Lenells equation, tested by numerical comparison and thorough visualization. This research not only enhances the theoretical basis of the Fokas–Lenells equation but also provides new avenues for future work in nonlinear optics and mathematical physics.
Rehman et al. (Tue,) studied this question.