Exact and numerical approaches for solitary and periodic waves in a (2+1)-dimensional breaking soliton system with adaptive moving mesh

Amer Ahmed; A. R. Alharbi; Haza S. Alayachi; Ishak Hashim; Amer Ahmed; A. R. Alharbi; Haza S. Alayachi; Ishak Hashim

doi:10.3934/math.2025380

AIMS Mathematics

2025, Volume 10, Issue 4: 8252-8276. doi: 10.3934/math.2025380

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Exact and numerical approaches for solitary and periodic waves in a (2+1)-dimensional breaking soliton system with adaptive moving mesh

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: 20 January 2025 Revised: 27 March 2025 Accepted: 31 March 2025 Published: 10 April 2025
MSC : 35A25, 35B35, 35Q51, 35Q92, 65M06, 65M12, 65M50

In this study, we examined a (2+1)-dimensional generalized breaking soliton system (GBSS) using both analytical and numerical methods. By applying a generalized direct algebraic method, we derived exact solutions that displayed a variety of solitary and periodic wave patterns. These solutions illuminated the interplay between nonlinearity and dispersion in several physical contexts, including fluid dynamics, plasma physics, and nonlinear optics. In addition, we developed a robust numerical scheme employing an adaptive moving mesh technique based on the MMPDE5 framework. Stability and error analyses confirmed that this method concentrated grid points around steep gradients, achieved second-order spatial convergence, and enhanced computational efficiency. By comparing numerical and exact solutions, we provided more profound insights into GBSS dynamics and facilitated future investigations of complex, multidimensional, and nonlinear wave phenomena.
- non-linear wave dynamics,
- exact solutions,
- numerical solutions,
- generalized direct algebraic method,
- adaptive moving mesh technique,
- soliton performance,
- wave interactions
Citation: Amer Ahmed, A. R. Alharbi, Haza S. Alayachi, Ishak Hashim. Exact and numerical approaches for solitary and periodic waves in a (2+1)-dimensional breaking soliton system with adaptive moving mesh[J]. AIMS Mathematics, 2025, 10(4): 8252-8276. doi: 10.3934/math.2025380

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Abstract

In this study, we examined a (2+1)-dimensional generalized breaking soliton system (GBSS) using both analytical and numerical methods. By applying a generalized direct algebraic method, we derived exact solutions that displayed a variety of solitary and periodic wave patterns. These solutions illuminated the interplay between nonlinearity and dispersion in several physical contexts, including fluid dynamics, plasma physics, and nonlinear optics. In addition, we developed a robust numerical scheme employing an adaptive moving mesh technique based on the MMPDE5 framework. Stability and error analyses confirmed that this method concentrated grid points around steep gradients, achieved second-order spatial convergence, and enhanced computational efficiency. By comparing numerical and exact solutions, we provided more profound insights into GBSS dynamics and facilitated future investigations of complex, multidimensional, and nonlinear wave phenomena.

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