Optical solitons and modulation instability of the fourth-order perturbed Schrödinger–Hirota equation: parabolic law in the presence of nonlinear chromatic dispersion

In this study, the optical soliton solutions of the perturbed fourth-order Schrödinger–Hirota equation with a parabolic law in the presence of nonlinear chromatic dispersion are investigated. To obtain analytic solutions, the modified new Kudryashov method, which provides effective solutions for fiber-optic models, and the modified extended tanh function method, which yields a wide variety of solutions through its multiple auxiliary equations, are employed. The combined use of these approaches enables the systematic derivation of a rich family of exact solutions despite the complex structure of the governing model. The analysis reveals the existence of various types of optical solitons, including bright, dark, singular, and kink-shaped structures. Furthermore, modulation instability analysis is performed to explore the stability characteristics of continuous-wave solutions, and the corresponding gain spectrum is examined in detail. The influence of the model parameters on both the soliton profiles and the gain spectrum is thoroughly illustrated through an extensive set of 2D and 3D graphical representations. With no restrictions on the scope of investigation, the findings provide a comprehensive understanding of the dynamical behavior associated with the studied equation, highlighting its potential relevance to nonlinear wave propagation and optical fiber communication systems.