Soliton solutions of the nonlinear dynamics in the Boussinesq equation with bifurcation analysis and chaos

Muhammad Shakeel; Amna Mumtaz; Abdul Manan; Marouan Kouki; Nehad Ali Shah; Muhammad Shakeel; Amna Mumtaz; Abdul Manan; Marouan Kouki; Nehad Ali Shah

doi:10.3934/math.2025484

AIMS Mathematics

2025, Volume 10, Issue 5: 10626-10649. doi: 10.3934/math.2025484

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Soliton solutions of the nonlinear dynamics in the Boussinesq equation with bifurcation analysis and chaos

1.
Department of Mathematics, Faculty of Basic Sciences, University of Wah, Wah Cantt 47040, Pakistan; muhammad.shakeel@uow.edu.pk; amna.mumtazsyed@gmail.com
2.
Department of Mathematics, Faculty of Sciences, Superior University, Main Campus Lahore, Pakistan; abdul.manan@superior.edu.pk
3.
Department of Information System, Faculty of Computing and Information Technology, Northern Border University, Rafha, Saudi Arabia
4.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Republic of Korea
^a These authors contributed equally to this work and are co-first authors

Received: 13 February 2025 Revised: 08 April 2025 Accepted: 18 April 2025 Published: 09 May 2025
MSC : 39A33, 35A20, 34G20

In this study, we examined the nonlinear dynamics of the Boussinesq equation, a foundational equation in ocean engineering to model and investigate the behavior of waves in shallow water. The novel (G'/G²)-expansion method was employed to obtain different soliton solutions, including periodic, bright, W-type, and bell-shaped soliton solutions. These solutions are illustrated through 2D, 3D, and contour plots. We discovered different dynamical behavior, including periodic, quasi-periodic, and weak chaos, depending on the choice of initial conditions and parameters. The important outcomes included the detection of multistable attractors and the presence of weak chaotic behavior supported by Lyapunov exponents. These understandings have important effects in practical uses such as energy harvesting and wave control in ocean systems, where handling and understanding system transitions and stability is crucial. These findings also give a framework for further examination of stability and control in nonlinear wave systems.
- Boussinesq equation,
- novel (G'/G²)-expansion method,
- phase portrait,
- chaotic analysis,
- Lyapunov exponents
Citation: Muhammad Shakeel, Amna Mumtaz, Abdul Manan, Marouan Kouki, Nehad Ali Shah. Soliton solutions of the nonlinear dynamics in the Boussinesq equation with bifurcation analysis and chaos[J]. AIMS Mathematics, 2025, 10(5): 10626-10649. doi: 10.3934/math.2025484

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Abstract

In this study, we examined the nonlinear dynamics of the Boussinesq equation, a foundational equation in ocean engineering to model and investigate the behavior of waves in shallow water. The novel (G'/G²)-expansion method was employed to obtain different soliton solutions, including periodic, bright, W-type, and bell-shaped soliton solutions. These solutions are illustrated through 2D, 3D, and contour plots. We discovered different dynamical behavior, including periodic, quasi-periodic, and weak chaos, depending on the choice of initial conditions and parameters. The important outcomes included the detection of multistable attractors and the presence of weak chaotic behavior supported by Lyapunov exponents. These understandings have important effects in practical uses such as energy harvesting and wave control in ocean systems, where handling and understanding system transitions and stability is crucial. These findings also give a framework for further examination of stability and control in nonlinear wave systems.

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