Linear stability and dispersive soliton propagation in nonlinear media subject to parabolic phase modulation

This work investigates dispersive optical solitons governed by a perturbed cubic-quartic nonlinear Schrödinger equation with parabolic self-phase modulation, a model of direct relevance to high-capacity fiber-optic systems where simultaneous higher-order dispersion and nonlinear perturbations shape pulse dynamics. The model is physically motivated by fibers with intensity-dependent refractive index profiles, where the interplay between fourth-order chromatic dispersion and parabolic (cubic-quintic) nonlinearity generates wave structures that the standard Kerr approximation cannot capture. To extract exact traveling-wave solutions, we employ the improved modified extended tanh-function method (IMETFM), which is selected for its ability to handle multi-parameter auxiliary equations and yield a wider diversity of solution families than classical expansion methods such as the tanh-function or [Formula: see text]-expansion approaches, without requiring the integrability of the underlying system. Our analysis produces five families of exact solutions: bright solitons, dark solitons, exponential-type solutions, singular periodic waves, and solutions expressed in terms of Weierstrass elliptic functions. For each family, explicit existence conditions and free-parameter restrictions are stated. The parametric constraints governing solution validity are derived and physically interpreted in terms of the dispersion, nonlinearity, and perturbation coefficients. Graphical representations of the spatial and temporal profiles illustrate the distinct propagation features of each solution type. A linear stability analysis, conducted via perturbation theory, yields an explicit eigenvalue dispersion relation and identifies a critical wavenumber threshold at which modulational instability sets in. The stability criteria provide actionable guidelines for maintaining soliton integrity under weak disturbances in practical optical environments. The results have direct implications for optical fiber communications, ultrafast signal processing, and dispersion-engineered photonic waveguides. The novelty lies in the simultaneous treatment of the parabolic law nonlinearity, fourth-order dispersion, and perturbative effects within a unified algebraic framework, yielding solution families including Weierstrass elliptic solutions that have not previously been reported for this model. Future work will address numerical validation, extension to stochastic and variable-coefficient models, and higher-dimensional soliton dynamics.