In this paper, we explore applications of the third-order nonlinear Schrödinger’s equation (NLSE) using the rational exp( \(-\varphi \) ( \(\zeta \) ))-expansion method (REEM) and the modified exponential function method (MEFM), offering insights into wave propagation and soliton behavior in optical communications, nonlinear optics, plasma physics, quantum mechanics, and engineering. These two methods share the premise that it is first essential to use a new wave definition to transform the equation with partial derivatives into a form of the equation with ordinary derivatives. By using these methods, we are able to produce novel soliton solutions that are represented in terms of trigonometric and hyperbolic functions. The results can be utilized to comprehend and clarify the physical properties of waves propagating through a dispersive medium. Graphical shapes are generated using relevant parameter values to visually present the obtained results, including 3D, 2D, and contour plots, providing an insightful visualization of the discovered solutions. All computations in this research were exclusively conducted utilizing the symbolic software Mathematica. The discovered solutions lead to a wide range of exact solutions, including dark, bright, mixed dark-bright, periodic, singular, peakons, singular periodic soliton solutions. These solutions have diverse applications, from enhancing optical signal transmission in telecommunications to modeling complex wave behaviors in plasma physics and quantum mechanics. These fundamental, consistent, and effective methods can be used to solve a wide range of different models in physics, field of engineering, and other useful disciplines.
