Optical Solitons of the Nonlinear Schrödinger Equation Without Self-Phase Modulation: Two Different Techniques

In this paper, the nonlinear Schrödinger equation (NLSE) without self-phase modulation (SPM) is investigated. The new Kudryashov method and the auxiliary equation method are employed to derive various classes of exact analytical solutions. The obtained solutions include bright and dark soliton structures, periodic waves, rational singular solutions, and plane-wave profiles, depending on the parameter choices. The bright-soliton solution obtained through the Kudryashov method recovers, under a reduced-dispersion limit, a previously reported solution in Ref. [6]. The hyperbolic bright/dark solution families obtained by the auxiliary equation method partially reproduce known results while extending the corresponding solution structures, whereas, to the best of our knowledge, the periodic and rational singular solution families have not been previously reported for the considered higher-order NLSE. Representative graphical illustrations are provided to demonstrate the qualitative behavior of the obtained wave structures, and the physical characteristics and admissibility of the localized solutions are discussed.