Analytical investigation of soliton dynamics in the Fokas–Lenells equation with generalized self-phase modulation and nonlinear dispersion

Abstract This work investigates the impact of a generalized quadratic-cubic form of self-phase modulation together with nonlinear chromatic dispersion on perturbed quiescent optical solitons governed by the Fokas–Lenells equation. The model integrates both generalized and linear temporal evolution to depict the whole range of nonlinear interactions affecting pulse dynamics. Soliton production, frequency shifting, spectrum broadening, and pulse compression are some of the phenomena that emerge from the interaction of self-phase modulation and nonlinear dispersion, which significantly alter the pulse phase, spectral properties, and temporal profile. These processes are essential to many photonic applications, including the creation of supercontinuum and advanced medical imaging methods like optical coherence tomography. Analytical solutions to this generalized model are obtained using the modified extended tanh method, the extended simple equation method, and the exp $$(-\varphi (\eta ))$$ -expansion technique. These approaches yield a rich and diverse family of soliton solutions, including dark, periodic, singular, dark-singular, and singular-periodic structures. Among the methods used, the extended simple equation approach provides a straightforward and effective process, while the modified extended tanh method generates a richer spectrum of solutions in a compact and non-redundant form. The physical behaviour of selected soliton profiles is illustrated through real and imaginary contour plots, as well as both two-dimensional and three-dimensional visualizations. The two-dimensional plots are used to illustrate the comparisons between different values of n. Overall, this work demonstrates a novel analytical framework for regulating soliton behavior in nonlinear optical media, which has theoretical and practical implications for applications including fiber-optic communications, ultrafast lasers, and optical signal processing.