Bifurcations and Exact Traveling Wave Solutions of the Schrödinger–Hirota Equation with Triple-Power Law Nonlinearity

This work investigates the traveling wave solution of the dispersive Schrödinger–Hirota (SH) equation with triple-power law nonlinearity. An appropriate transformation reduces the equation to a singular integrable traveling-wave system. Utilizing the dynamical systems method, we obtain bifurcations of the phase portraits of the singular integrable traveling-wave system under different parameter conditions. Furthermore, a bifurcation analysis is carried out by considering the equation’s symmetric equilibrium structure. Corresponding to the various phase portrait trajectories, exact explicit parametric representations of solutions are derived, including periodic, singular periodic, solitary, periodic peakon, and kink wave solutions. Some of these newly obtained solutions and their associated trajectories are illustrated graphically. Additionally, this work generalizes and extends earlier findings in the literature. These results contribute to a deeper understanding of the dynamics of the nonlinear SH equation model with triple-power law nonlinearity.