Analytical dynamics of the Kuralay-II equation: Bifurcation analysis, chaotic behavior, and soliton solution

Alaaeddin Moussa; Lama Alhakim; Abdelnaby S. Saad; Yazid Mati; Boubekeur Gasmi; Alaaeddin Moussa; Lama Alhakim; Abdelnaby S. Saad; Yazid Mati; Boubekeur Gasmi

doi:10.3934/math.2026326

AIMS Mathematics

2026, Volume 11, Issue 3: 7910-7933. doi: 10.3934/math.2026326

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Analytical dynamics of the Kuralay-II equation: Bifurcation analysis, chaotic behavior, and soliton solution

1.
Department of Management Information Systems, College of Business & Economics, Qassim University, P.O. Box 6666, Buraidah, 51452, Saudi Arabia
2.
Department of Operations Management, College of Business & Economics, Qassim University, P.O. Box 6666, Buraidah, 51452, Saudi Arabia
3.
Higher School of Management and Digital Economy, Kolea, Tipaza, Algeria

Received: 04 February 2026 Revised: 12 March 2026 Accepted: 18 March 2026 Published: 25 March 2026
MSC : 34C23, 35C07, 35C08, 34Hxx

This paper presents a comprehensive investigation into the complex behavior of the Kuralay-II equation. We begin by analyzing the bifurcation structure and corresponding phase portraits of the equation to gain insight into the underlying dynamical transitions. Next, we examine the chaotic behavior of the governing system through a Lyapunov stability analysis, which provides valuable insights into the sensitivity and stability characteristics of the model. Additionally, we derive several explicit solutions to the equation using an exact analytical method and visualize some of these solutions to highlight their structural properties and dynamical implications. The findings provide a solid framework for future research aimed at understanding the complex behavior of nonlinear systems.
- Kuralay-II equation,
- bifurcation analysis,
- phase portraits,
- chaos,
- Lyapunov exponents,
- exact solutions,
- nonlinear dynamics,
- partial differential equations
Citation: Alaaeddin Moussa, Lama Alhakim, Abdelnaby S. Saad, Yazid Mati, Boubekeur Gasmi. Analytical dynamics of the Kuralay-II equation: Bifurcation analysis, chaotic behavior, and soliton solution[J]. AIMS Mathematics, 2026, 11(3): 7910-7933. doi: 10.3934/math.2026326

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Abstract

This paper presents a comprehensive investigation into the complex behavior of the Kuralay-II equation. We begin by analyzing the bifurcation structure and corresponding phase portraits of the equation to gain insight into the underlying dynamical transitions. Next, we examine the chaotic behavior of the governing system through a Lyapunov stability analysis, which provides valuable insights into the sensitivity and stability characteristics of the model. Additionally, we derive several explicit solutions to the equation using an exact analytical method and visualize some of these solutions to highlight their structural properties and dynamical implications. The findings provide a solid framework for future research aimed at understanding the complex behavior of nonlinear systems.

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