The modulation instability gain spectrum assessment and generalized propagating wave structures of nonlinear cubic-quartic Schrödinger equation

This study investigates a cubic-quartic nonlinear Schrödinger model with third- and fourth-order dispersions without the disturbance in parabolic law media and group velocity dispersion (GVD). The analytical method is used to produce traveling wave solutions because the inverse scattering transform is unable to solve the Cauchy problem for this equation. The utilized analytical approach is novel and providing generalized families of solutions as compared to other, thus, prior to this study, there was not existing any work in which suck kinds of solutions were exist. The propagating solitons are driven using the unified auxiliary equation approach. As a result, the aforementioned technique is learned in the following areas: singular soliton, shock solution, mixed-singular, mixed trigonometric, mixed shock single solution, plane solutions, exponential, trigonometry, mixed-hyperbolic solution, periodic and mixed periodic solutions, and trigonometry. The stability of the topic under consideration is shown by the modulational instability (MI) study. The appropriate parametric values are shown alongside the graphical representation.