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We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.

Optical solitons is one of the fastest growing areas of research in the field of nonlinear fiber optics. In the current work, the modified extended direct algebraic method is implemented to obtain optical solitons for the nonlinear Schrödinger’s equation with group velocity dispersion and second order spatiotemporal dispersion that describe the propagation of optical solitons in nonlinear media. Bright soliton solutions, dark soliton solutions and singular soliton solutions are obtained. Also, periodic solutions, exponential solutions, Jacobi elliptic solutions and Weierstrass elliptic solutions are extracted. Under specific parameter values, 2D, 3D and contour graphs are introduced to visualize the model and demonstrate their accurate physical behaviors.