Chaos, Lyapunov exponent, and sensitivity demonstration of the coupled nonlinear integrable model with soliton solutions

The stochastic Schrödinger–Hirota equation is a well-established model for describing nonlinear pulse propagation in magneto-optic waveguides and fiber optics. While prior research has primarily concentrated on isolated soliton solutions, broader chaotic dynamics affecting signal stability remain less explored. In this work, we applied the Generalized Arnous method, planar dynamical system theory, and numerical simulations with the Runge–Kutta method to construct and analyze the solutions of the stochastic Schrödinger–Hirota equation. The result yielded tan-type bright, cot-type dark, and periodic solutions. Chaotic behavior was characterized via bifurcation diagrams, positive Lyapunov exponents, Poincaré sections, and time-series analyses. Sensitivity to initial conditions confirmed the presence of deterministic chaos. These results demonstrate that the stochastic Schrödinger–Hirota equation displays a rich spectrum of dynamical behavior, with the stability strongly influenced by parameter variations and perturbations. These insights offer a theoretical foundation for advancing the stability of optical soliton transmission and reducing the adverse effects of chaos in nonlinear communication systems.