Bifurcation analysis, stability analysis, modulation instability and soliton structures of the nonlinear Klein–Gordon model in particle physics

Kalim U. Tariq; S. M. Raza Kazmi; Sajawal A. Baloch; Abdulaziz Khalid Alsharidi; Naif Almusallam; Luai Abdulla Aldoghan; Kalim U. Tariq; S. M. Raza Kazmi; Sajawal A. Baloch; Abdulaziz Khalid Alsharidi; Naif Almusallam; Luai Abdulla Aldoghan

doi:10.3934/math.2026353

AIMS Mathematics

2026, Volume 11, Issue 3: 8584-8617. doi: 10.3934/math.2026353

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Research article Special Issues

Bifurcation analysis, stability analysis, modulation instability and soliton structures of the nonlinear Klein–Gordon model in particle physics

1.
Department of Mathematics, Mirpur University of Science and Technology, Mirpur-10250 (AJK), Pakistan
2.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3.
National Institute for Theoretical and Computational Sciences (NITheCS), Stellenbosch University, South Africa
4.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
5.
Department of Management Information Systems, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia
6.
Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 23 January 2026 Revised: 18 March 2026 Accepted: 27 March 2026 Published: 30 March 2026
MSC : 35Rxx, 35Qxx, 35Q51

In this study, the nonlinear Klein-Gordon model is investigated analytically as an extension of the classical Klein-Gordon equation involving nonlinear effects. This relativistic wave equation plays an important role in describing scalar field dynamics in particle physics, condensed matter physics, and cosmology. Although several studies have reported soliton solutions for nonlinear wave equations, comprehensive analyses combining exact solutions with stability, modulation instability, and chaotic dynamics remain limited for this model. To address this gap, we employ analytical techniques to obtain different types of solutions, including solitary, periodic, and V-shaped wave structures. Graphical illustrations demonstrate the rich dynamical behavior of the system for various parameter regimes. In addition, a stability analysis is performed to determine the conditions under which the system preserves its dynamical behavior. The modulation instability and bifurcation structures are also examined through phase portraits, revealing transitions between regular and complex dynamics. Furthermore, periodic perturbations are introduced to explore chaotic behavior in the system. The results reveal previously unreported solitary and periodic solutions whose stability and chaotic phases offer insight into the underlying nonlinear mechanisms, with potential applications in wave propagation and field dynamics.
- nonlinear dynamics,
- periodic wave solitons,
- travelling wave solitons,
- stability analysis,
- bifurcation analysis,
- modulation instability,
- nonlinear Klein-Gordon equation,
- solitons
Citation: Kalim U. Tariq, S. M. Raza Kazmi, Sajawal A. Baloch, Abdulaziz Khalid Alsharidi, Naif Almusallam, Luai Abdulla Aldoghan. Bifurcation analysis, stability analysis, modulation instability and soliton structures of the nonlinear Klein–Gordon model in particle physics[J]. AIMS Mathematics, 2026, 11(3): 8584-8617. doi: 10.3934/math.2026353

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Abstract

In this study, the nonlinear Klein-Gordon model is investigated analytically as an extension of the classical Klein-Gordon equation involving nonlinear effects. This relativistic wave equation plays an important role in describing scalar field dynamics in particle physics, condensed matter physics, and cosmology. Although several studies have reported soliton solutions for nonlinear wave equations, comprehensive analyses combining exact solutions with stability, modulation instability, and chaotic dynamics remain limited for this model. To address this gap, we employ analytical techniques to obtain different types of solutions, including solitary, periodic, and V-shaped wave structures. Graphical illustrations demonstrate the rich dynamical behavior of the system for various parameter regimes. In addition, a stability analysis is performed to determine the conditions under which the system preserves its dynamical behavior. The modulation instability and bifurcation structures are also examined through phase portraits, revealing transitions between regular and complex dynamics. Furthermore, periodic perturbations are introduced to explore chaotic behavior in the system. The results reveal previously unreported solitary and periodic solutions whose stability and chaotic phases offer insight into the underlying nonlinear mechanisms, with potential applications in wave propagation and field dynamics.

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