Exploring the new classes of optical soliton solutions with diverse structure for the (2+1)–dimensional paraxial equation in fiber optics via two analytical methods

This research explores the analytical soliton solutions of the dimensionless time-dependent paraxial equation (DTPE). We applied the improved Sardar sub-equation and improved Riccati equation analytical techniques to develop numerous soliton solutions for the DTPE, providing insights into the behavior of optical solitons in nonlinear evolution equations. The two techniques are applied for the first time to generate vigorous solutions to the DTPE. Numerous types of exponential, trigonometric, hyperbolic, dark, periodic, anti-kink, peakon, kink, and bright soliton solutions are obtained for DTPE. Although constructing an effective scheme to solve the DTPE has been our primary aim. Moreover, the obtained analytical solutions offer a useful foundation for comprehending the complex dynamics of optical solitons in nonlinear evolution equations, enabling enhancements in different applications that include signal processing, optical communication, beam shaping, Gaussian beams, lenses and mirrors, optical fibers, and predicting beam behaviors.