Exploring solitary wave solutions of the Drinfeld-Sokolov-Wilson equation

Qinming Yao; Qinming Yao

doi:10.3934/math.2026870

AIMS Mathematics

2026, Volume 11, Issue 7: 21463-21485. doi: 10.3934/math.2026870

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Exploring solitary wave solutions of the Drinfeld-Sokolov-Wilson equation

Qinming Yao ^,

School of Statistics and Data Science, Capital University of Economics and Business, Beijing 100070, China

Received: 16 April 2026 Revised: 18 June 2026 Accepted: 03 July 2026 Published: 20 July 2026
MSC : 35C08, 35Q51

Understanding shallow water wave dynamics is essential for coastal engineering, oceanography, and nonlinear physics. In this study, I explored the nonlinear dynamics of the coupled Drinfeld-Sokolov-Wilson system, a key nonlinear partial differential equation model that plays a crucial role in understanding shallow water wave behavior. To investigate the solitary wave solutions of this model, four analytical tools were employed: the Kumar-Malik method, the generalized Arnous method, the Riccati modified extended simple equation method, and the logarithmic transformation. These techniques yielded a variety of soliton solutions, including dark, bright, mixed, kink, bright-dark, breather, multiwave, and mixed-form solitons. Moreover, the periodic, hyperbolic and the exponential function solutions were recovered. This study not only confirms known solutions but significantly enriches the literature on the Drinfeld-Sokolov-Wilson system through the application of novel analytical methods that yield an exceptionally diverse family of soliton solutions, complemented by a comprehensive graphical investigation that illustrates the profound influence of physical parameters on wave dynamics. This study affirms the practical utility of these modern methods while delivering novel insights that advance the theoretical understanding of higher-dimensional wave dynamics.
- Kumar-Malik method,
- Riccati modified extended simple equation method,
- generalized Arnous method,
- coupled Drinfeld-Sokolov-Wilson,
- solitons
Citation: Qinming Yao. Exploring solitary wave solutions of the Drinfeld-Sokolov-Wilson equation[J]. AIMS Mathematics, 2026, 11(7): 21463-21485. doi: 10.3934/math.2026870

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Abstract

Understanding shallow water wave dynamics is essential for coastal engineering, oceanography, and nonlinear physics. In this study, I explored the nonlinear dynamics of the coupled Drinfeld-Sokolov-Wilson system, a key nonlinear partial differential equation model that plays a crucial role in understanding shallow water wave behavior. To investigate the solitary wave solutions of this model, four analytical tools were employed: the Kumar-Malik method, the generalized Arnous method, the Riccati modified extended simple equation method, and the logarithmic transformation. These techniques yielded a variety of soliton solutions, including dark, bright, mixed, kink, bright-dark, breather, multiwave, and mixed-form solitons. Moreover, the periodic, hyperbolic and the exponential function solutions were recovered. This study not only confirms known solutions but significantly enriches the literature on the Drinfeld-Sokolov-Wilson system through the application of novel analytical methods that yield an exceptionally diverse family of soliton solutions, complemented by a comprehensive graphical investigation that illustrates the profound influence of physical parameters on wave dynamics. This study affirms the practical utility of these modern methods while delivering novel insights that advance the theoretical understanding of higher-dimensional wave dynamics.

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