Exact Solutions for Optical Solitons in the Dimensionless Time-Dependent Paraxial Equation via Two Analytical Approaches

Abstract

In this study, we investigate the dimensionless time-dependent paraxial equation by two powerful analytical methods: the generalized exponential rational function method (GERFM) and the \(\left( {m + \frac{1}{{G'}}} \right)\)-expansion method. The GERFM method gives us six distinct solution sets which include bright, dark, singular, mixed, and kink-type soliton solutions expressed in terms of hyperbolic functions such as \({\text{csch}}(\xi )\), \({\text{sech}}(\xi )\), \({\text{tanh}}(\xi )\), and \({\text{coth}}(\xi )\). The \(\left( {m + \frac{1}{{G'}}} \right)\)-expansion method provides additional solution forms involving exponential rational functions. We show these solutions through 2D and 3D graphical representations, demonstrating their temporal stability and localization properties over time evolution. The bright solitons maintain their symmetric peak structure, dark solitons exhibit stable trough formations, and mixed solitons combine multiple wave components coherently during propagation. The stability analysis confirms that all obtained soliton solutions preserve their shape and amplitude characteristics under time evolution, indicating their physical robustness. This model is useful for applications in nonlinear fiber optics, optical communication systems, supercontinuum generation, plasma wave propagation, and nonlinear waveguide design. The results provide valuable insights for understanding complex wave phenomena in dispersive nonlinear media and offer practical applications for future photonic technologies and optical signal processing systems.