Analytical study of optical soliton solutions in nonlinear Schrödinger model with self-steepening, self-frequency shift, and Kerr nonlinearity

This paper investigates the nonlinear Schrödinger equation incorporating Kerr nonlinearity, self-steepening, self-frequency shift, and spatiotemporal effects using the new extended direct mapping method. The proposed model and its analytical solutions offer significant potential for applications in internet traffic regulation and high-speed data transmission systems. A class of novel optical soliton solutions has been systematically derived, encompassing bright solitons, dark solitons, wave solitons, and W-shaped solitons. These solutions are not only theoretically established but also visually demonstrated to capture their intricate dynamical behaviors and structural features. To this end, a comprehensive set of graphical representations, including contour plots along with detailed two-dimensional and three-dimensional simulations, is employed to vividly capture the evolution and spatiotemporal characteristics of each soliton type. These visualizations effectively highlight the distinctive structural patterns and dynamic behaviors exhibited by the solutions. Moreover, the influence of the conformable derivative parameter and the temporal parameter on the obtained soliton profiles is thoroughly examined through graphical analysis. The results demonstrate how variations in these parameters alter the soliton shape, amplitude, and propagation dynamics, providing deeper insight into their physical significance.