Exact and numerical solutions of the generalized breaking soliton system: Insights into non-linear wave dynamics

Amer Ahmed; Abdulghani R. Alharbi; Ishak Hashim; Amer Ahmed; Abdulghani R. Alharbi; Ishak Hashim

doi:10.3934/math.2025235

AIMS Mathematics

2025, Volume 10, Issue 3: 5124-5142. doi: 10.3934/math.2025235

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Exact and numerical solutions of the generalized breaking soliton system: Insights into non-linear wave dynamics

1.
Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
2.
Department of Mathematics, College of Science, Taibah University, 42353, Medina, Saudi Arabia
3.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, P.O. Box 346, United Arab Emirates

Received: 02 January 2025 Revised: 08 February 2025 Accepted: 27 February 2025 Published: 06 March 2025
MSC : 35A24, 35B35, 35Q51, 35Q92, 65N06, 65N40, 65N45, 65N50

In this paper, we examined the (2+1)-dimensional generalized breaking soliton system (GBSS), an adaptable framework that accurately describes the three-dimensional, wave-dominated interactions occurring in many non-linear media, i.e., fluids, plasmas, and optical fibers. We used an improved F-expansion technique to generate new families of exact solitonic and periodic wave solutions, significantly enlarging the well-studied solution space and providing insight into the complicated interaction between multi-pulse solitons. Validation of these results and an assessment of their stability were carried out by developing a numerical scheme based on finite difference and undertaking a detailed error and stability analysis, demonstrating unconditional stability across a range of parameter values. The results provide new insights into the interplay of dispersion, non-linearity, and cross-wave coupling in governing soliton formation and energy transport in multidimensional systems. In addition to its theoretical importance, this work can provide valuable practical information on engineering applications such as soliton-based communications and wave control applications in fluid systems. This study offers a new methodology to investigate more complex non-linear wave phenomena by integrating the power of symbolic computation with that of robust numerical verification, opening new opportunities for further developments in soliton-driven technologies.
- non-linear wave dynamics,
- exact solutions,
- numerical solutions,
- modified F-expansion method,
- finite difference method,
- soliton performance,
- energy transportation,
- wave interactions
Citation: Amer Ahmed, Abdulghani R. Alharbi, Ishak Hashim. Exact and numerical solutions of the generalized breaking soliton system: Insights into non-linear wave dynamics[J]. AIMS Mathematics, 2025, 10(3): 5124-5142. doi: 10.3934/math.2025235

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Abstract

In this paper, we examined the (2+1)-dimensional generalized breaking soliton system (GBSS), an adaptable framework that accurately describes the three-dimensional, wave-dominated interactions occurring in many non-linear media, i.e., fluids, plasmas, and optical fibers. We used an improved F-expansion technique to generate new families of exact solitonic and periodic wave solutions, significantly enlarging the well-studied solution space and providing insight into the complicated interaction between multi-pulse solitons. Validation of these results and an assessment of their stability were carried out by developing a numerical scheme based on finite difference and undertaking a detailed error and stability analysis, demonstrating unconditional stability across a range of parameter values. The results provide new insights into the interplay of dispersion, non-linearity, and cross-wave coupling in governing soliton formation and energy transport in multidimensional systems. In addition to its theoretical importance, this work can provide valuable practical information on engineering applications such as soliton-based communications and wave control applications in fluid systems. This study offers a new methodology to investigate more complex non-linear wave phenomena by integrating the power of symbolic computation with that of robust numerical verification, opening new opportunities for further developments in soliton-driven technologies.

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