Stability analysis and extraction of newly cubic–quintic soliton wave structure for the generalized derivative resonant nonlinear Schrödinger equation through an analytical technique

This study investigates the analytical solutions of the generalized derivative resonant nonlinear Schrödinger equation, a fundamental model governing wave propagation in nonlinear dispersive media with applications in nonlinear optics, quantum mechanics, and plasma physics. We employ a modified extended direct algebraic method to systematically derive exact solutions, extending traditional approaches to incorporate cubic–quintic nonlinearity, which accounts for the interplay between self-focusing and defocusing effects.

By transforming the generalized derivative resonant nonlinear Schrödinger equation into an integrable algebraic form, we obtain a diverse set of analytical solutions, including bright and dark solitons, singular solitons, periodic waves, rational solutions, exponential solutions, Jacobi and Weierstrass elliptic functions. These solutions provide deep insights into the formation and dynamics of localized wave structures in nonlinear systems. We then conduct a rigorous stability analysis of the obtained solitons through linear perturbation theory, identifying critical parameter regimes where stable propagation occurs. The stability thresholds are shown to depend fundamentally on the balance between cubic self-focusing and quintic defocusing effects. To validate our results, we present numerical simulations and graphical representations of selected solutions, illustrating their stability and propagation characteristics. Our findings contribute to the theoretical understanding of nonlinear wave phenomena and offer potential applications in optical communications, quantum information, and ultrafast laser physics. This work not only advances the analytical treatment of the generalized derivative resonant nonlinear Schrödinger equation but also provides a robust framework for exploring other high-order nonlinear evolution equations in mathematical physics.