In this paper, we study the Schrödinger–Hirota equation in birefringent fibers incorporating cubic–quartic dispersion. The model studied comes with a cubic–quintic nonlinear structure. The governing model introduced in this study is novel and original, and the obtained solitons have not been reported before. The Schrödinger–Hirota equation is an important modified structure within the higher-order NLSE that garnered significant attention in the fields of mathematics and physics, especially in the context of the transmission of solitons in water waves and optical fibers. To investigate optical solitons for this model, three widely applied in various research studies are implemented for our governing model. These methods are the Unified Riccati expansion method, the addendum to Kudryashov’s method, and the addendum to the modified Sub-ODE method. A variety of solutions have emerged with this governing model, including bright, dark, singular solitons and the hybrid of these solitons. Also, Jacobi and Weierstrass doubly periodic-type solutions are recovered, which, with selected parameters, these solutions give solitons. Furthermore, for more illustration of the obtained solutions, a detailed discussion with a graphical presentation is presented.
