Traveling Waves, Bifurcation Analysis and Benjamin–Feir Instability in Schrödinger–Hirota Model with Power-Law of Self-Phase Modulation

We critically reexamine traveling-wave solutions and Benjamin–Feir instability for a Schrödinger–Hirota equation with power-law nonlinearity. A traveling-wave reduction, Hamiltonian phase-plane analysis, direct substitution, and symbolic residual checks are used to establish the compatibility of candidate solutions with the original complex equation. The previously reported unrestricted extended-Kudryashov solution family is shown to be invalid; only a restricted singular branch survives on a half-line and cannot represent a globally regular optical soliton. A regular bright solitary pulse is verified, and sign-changing dark waves are characterized through an implicit real quadrature rather than ambiguous fractional powers. The corrected Jacobian determinants yield consistent classifications of equilibria and their homoclinic or heteroclinic trajectories. For continuous-wave backgrounds, the perturbation dispersion relation determines the instability band and the corrected maximum growth rate. Explicit parameter sets and reproducibility checks distinguish valid traveling waves from singular profiles and clarify that instability of the background does not establish instability of localized solitons.