This research addresses the pure-cubic nonlinear Schrödinger equation augmented by thirdorder dispersion, characterized by quadratic–cubic nonlinearities and explicitly excluding the group-velocity dispersion term. Such an equation is well suited for representing the behavior of ultrashort optical pulses in nonlinear fibers, in which higher-order effects strongly influence the dynamics. By applying the new Kudryashov method and the Jacobi elliptic function method, we systematically derive a wide family of soliton solutions, including bright solitons, W-shaped solitons, and kink solitons, each exhibiting distinct amplitude profiles and propagation characteristics. The role of essential system parameters, including quadratic and cubic nonlinear coefficients along with third-order dispersion strength, is systematically explored, revealing the physical conditions under which these nonlinear waveforms emerge. The results highlight the rich diversity of soliton structures possible in this system and underscore the potential applications in the control of high-speed optical pulses. In the absence of the group-velocity dispersion term, this version of the equation has not been previously explored; here, we present the first modulation instability analysis, incorporating the effects of quadratic and cubic nonlinearities, with results expected to offer valuable input to the field.
Durmus et al. (Fri,) studied this question.