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

Khizar Farooq; Aljethi Reem Abdullah; Ejaz Hussain; Muhammad Amin S. Murad; Khizar Farooq; Aljethi Reem Abdullah; Ejaz Hussain; Muhammad Amin S. Murad

doi:10.3934/math.20251019

AIMS Mathematics

2025, Volume 10, Issue 10: 22929-22957. doi: 10.3934/math.20251019

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

1.
Centre for High Energy Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan
2.
Department of mathematics and statistics, college of science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan
4.
Department of Cybersecurity, College of Engineering Technology, Alnoor University, Mosul 41012, Iraq

Received: 10 August 2025 Revised: 13 September 2025 Accepted: 24 September 2025 Published: 10 October 2025
MSC : 26A48, 26A51, 33B10, 37K40, 39B62

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.
- the generalized stochastic Schödinger-Hirota equation,
- generalized Arnous method,
- chaotic phenomena,
- bifurcation analysis,
- multistability,
- Lyapunov exponents,
- sensitivity analysis
Citation: Khizar Farooq, Aljethi Reem Abdullah, Ejaz Hussain, Muhammad Amin S. Murad. Chaos, Lyapunov exponent, and sensitivity demonstration of the coupled nonlinear integrable model with soliton solutions[J]. AIMS Mathematics, 2025, 10(10): 22929-22957. doi: 10.3934/math.20251019

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Abstract

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.

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