On certain novel numerical and analytical solutions for the pure-cubic Schrödinger equation in optical fibers with Kerr nonlinearity

This study examines the pure-cubic nonlinear Schrödinger equation associated with Kerr-type optical responses and formulates an integrated analytical–computational strategy for exploring its nonlinear wave behavior. The approach is based on transforming the governing model into an ordinary differential equation through an appropriate traveling-wave reduction; constructing exact waveform expressions using a refined ansatz procedure; and validating these solutions through a high-precision split-step numerical solver supported. The analysis yields several families of localized and periodic wave patterns, including bright solitons, dark solitons, kink-type solitons, periodic waves, singular solitons, and oscillatory profiles generated under varying parameter choices. The novelty of the work lies in providing a single, coherent framework capable of producing multiple solitary-wave configurations within the pure-cubic Kerr setting, together with a structured verification pipeline that has not been previously applied in this consolidated manner. The close agreement between analytical formulas and numerical propagation results confirms the robustness of the proposed methodology and establishes a set of reference solutions for future investigations involving extended or perturbed nonlinear optical models.