Presentation of Soliton Solutions and Modulation Instability of a Nonlinear Schrödinger Equation

In this study, the nonlinear Schrödinger equation (NLSE) is analytically investigated by employing the Extended Rational Sine–Cosine method, the Extended Rational Sinh–Cosh method, and the Modified F-Expansion Method (MFEM). Through the application of these techniques, a broad class of exact travelling-wave solutions with rational–trigonometric and rational–hyperbolic structures is systematically constructed. The obtained solutions include bright solitons, dark solitons, periodic waves, singular solutions, and hybrid composite wave profiles. All solutions are verified by direct substitution into the governing equation, thereby ensuring their mathematical consistency. The spatiotemporal behavior of the solutions is illustrated in detail through two- and three-dimensional graphical representations. In addition, a modulation instability (MI) analysis is carried out for the continuous-wave background solution of the equation, and the corresponding MI gain spectrum is explicitly derived. The results reveal the influence of dispersion and nonlinear parameters on the modulational behavior of the system and demonstrate that the continuous-wave solution remains stable within the considered parameter regime. Overall, the findings extend the solution space of the NLSE and provide deeper physical insight into nonlinear wave dynamics in optical media.