In this study, we investigate the pure-cubic optical Schrӧdinger equation with Kerr law nonlinearity, which plays a crucial role in modeling pulse propagation in optical fibers and nonlinear wave phenomena. Despite numerous analytical methods proposed in the literature, there remains a need for innovative approaches capable of generating diverse families of soliton solutions efficiently. To address this gap, we apply for the first time a novel soliton method (SM) combined with the traveling wave transformation to systematically derive optical soliton solutions. By employing this method, we successfully obtain twenty distinct families of exact soliton solutions encompassing exponential, trigonometric, hyperbolic, and polynomial function combinations. The dynamical behaviors of these solutions are illustrated through comprehensive 2D and 3D graphical representations, revealing wave solitons, dark solitons, and other nonlinear wave structures. Furthermore, the constraint conditions relating the physical parameters are established to ensure the existence of these solutions. The obtained results demonstrate that the proposed method is not only effective and straightforward but also significantly enriches the solution structure of the model. These findings have direct applications in optical communications, fiber optics technology, nonlinear optics, and the design of photonic devices where soliton propagation is essential. The novel methodology presented in this work contributes a powerful analytical tool to the scientific community for investigating other nonlinear evolution equations in mathematical physics and opens new avenues for understanding complex wave phenomena in nonlinear optical systems.
Mahmood et al. (Thu,) studied this question.