Study of Soliton Solutions, Bifurcation, Quasi-Periodic, and Chaotic Behaviour in the Fractional Coupled Schrödinger Equation

Manal Alharbi, Adel Elmandouh () and Mamdouh Elbrolosy
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Manal Alharbi: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Adel Elmandouh: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Mamdouh Elbrolosy: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Mathematics, 2025, vol. 13, issue 19, 1-24

Abstract: This study presents a qualitative analysis of the fractional coupled nonlinear Schrödinger equation (FCNSE) to obtain its complete set of solutions. An appropriate wave transformation is applied to reduce the FCNSE to a fourth-order dynamical system. Due to its non-Hamiltonian nature, this system poses significant analytical challenges. To overcome this complexity, the dynamical behavior is examined within a specific phase–space subspace, where the system simplifies to a two-dimensional, single-degree-of-freedom Hamiltonian system. The qualitative theory of planar dynamical systems is then employed to characterize the corresponding phase portraits. Bifurcation analysis identifies the physical parameter conditions that give rise to super-periodic, periodic, and solitary wave solutions. These solutions are derived analytically and illustrated graphically to highlight the influence of the fractional derivative order on their spatial and temporal evolution. Furthermore, when an external generalized periodic force is introduced, the model exhibits quasi-periodic behavior followed by chaotic dynamics. Both configurations are depicted through 3D and 2D phase portraits in addition to the time-series graphs. The presence of chaos is quantitatively verified by calculating the Lyapunov exponents. Numerical simulations demonstrate that the system’s behavior is highly sensitive to variations in the frequency and amplitude of the external force.

Keywords: Schrödinger equation; conformable fractional; bifurcation theory; wave solutions; quasi-periodic behavior; chaotic behavior (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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