Dynamics of the generalized derivative resonant NLS equation: bifurcation, exact solutions, and the transition from periodic to chaotic waves

Adel Emandouh; Adel Emandouh

doi:10.3934/math.2026123

AIMS Mathematics

2026, Volume 11, Issue 1: 3096-3131. doi: 10.3934/math.2026123

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Dynamics of the generalized derivative resonant NLS equation: bifurcation, exact solutions, and the transition from periodic to chaotic waves

Adel Emandouh ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 22 November 2025 Revised: 11 January 2026 Accepted: 16 January 2026 Published: 30 January 2026
MSC : 34H10, 35B20, 35C05, 35C075

This work presents a inclusive inspection of the nonlinear dynamics of the generalized derivative resonant nonlinear Schrödinger equation (GD-RNLS), a fundamental model for wave phenomena in dispersive media. We use a tailored transformation to recast the GD-RNLS as a two-dimensional conservative dynamic system, which is equivalent to a Hamiltonian system with a single degree of freedom. Leveraging the Hamiltonian framework, we perform a detailed bifurcation analysis. The corresponding phase portraits are constructed, and we studied the degeneracy of the phase plane trajectories as the bifurcation parameter varies. Within this dynamic systems approach, we derive a family of novel exact solutions, categorizing them as periodic, super-periodic, kink (and anti- kink), and solitary wave solutions; selected solutions are illustrated graphically to clarify their properties. Furthermore, we examine a perturbed variant of the GD-RNLS, incorporating an external periodic forcing term modeled by a Jacobi elliptic function. The influence of this perturbation is explored numerically through two-dimensional (2D) and three- dimensional (3D) phase portraits, as well as the time series. Our simulations reveal a sequence of behavioral transitions: the system's initially periodic state gives way to quasi-periodic dynamics as the forcing frequency increases. Upon a further increase in frequency, chaotic behavior emerges. This transition to chaos is quantitatively confirmed by calculating the largest Lyapunov exponent and qualitatively visualized by the irregular structure of the Poincaré surface of section, both of which underscore the complex, stochastic nature of the system's dynamics under strong forcing.
- Shrödinger equation,
- bifurcation theory,
- wave solutions,
- quasi-periodic,
- chaotic pattern
Citation: Adel Emandouh. Dynamics of the generalized derivative resonant NLS equation: bifurcation, exact solutions, and the transition from periodic to chaotic waves[J]. AIMS Mathematics, 2026, 11(1): 3096-3131. doi: 10.3934/math.2026123

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Abstract

This work presents a inclusive inspection of the nonlinear dynamics of the generalized derivative resonant nonlinear Schrödinger equation (GD-RNLS), a fundamental model for wave phenomena in dispersive media. We use a tailored transformation to recast the GD-RNLS as a two-dimensional conservative dynamic system, which is equivalent to a Hamiltonian system with a single degree of freedom. Leveraging the Hamiltonian framework, we perform a detailed bifurcation analysis. The corresponding phase portraits are constructed, and we studied the degeneracy of the phase plane trajectories as the bifurcation parameter varies. Within this dynamic systems approach, we derive a family of novel exact solutions, categorizing them as periodic, super-periodic, kink (and anti- kink), and solitary wave solutions; selected solutions are illustrated graphically to clarify their properties. Furthermore, we examine a perturbed variant of the GD-RNLS, incorporating an external periodic forcing term modeled by a Jacobi elliptic function. The influence of this perturbation is explored numerically through two-dimensional (2D) and three- dimensional (3D) phase portraits, as well as the time series. Our simulations reveal a sequence of behavioral transitions: the system's initially periodic state gives way to quasi-periodic dynamics as the forcing frequency increases. Upon a further increase in frequency, chaotic behavior emerges. This transition to chaos is quantitatively confirmed by calculating the largest Lyapunov exponent and qualitatively visualized by the irregular structure of the Poincaré surface of section, both of which underscore the complex, stochastic nature of the system's dynamics under strong forcing.

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