Analytical and dynamical investigations of the Kaup–Kupershmidt model via two innovative integrability architectures

Saud Owyed; Saud Owyed

doi:10.3934/era.2026264

Electronic Research Archive

2026, Volume 34, Issue 9: 5968-5992. doi: 10.3934/era.2026264

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Analytical and dynamical investigations of the Kaup–Kupershmidt model via two innovative integrability architectures

Saud Owyed ^,

Department of Mathematics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabia

Received: 08 April 2026 Revised: 01 June 2026 Accepted: 09 June 2026 Published: 13 July 2026

This study focused on the modified F-expansion method and the $ \frac{G'}{bG' + G + a} $-expansion method to find exact solutions to the $ (1+1)- $dimensional Kaup–Kupershmidt (KK) equation, an important nonlinear model that incorporates both dispersive effects and higher-order nonlinearity. By applying these analytical techniques, several classes of exact solutions, including periodic, bright, and singular soliton solutions, were successfully obtained. The graphical representation of these solutions was evaluated by creating two-dimensional, three-dimensional, and contour plots for each solution type to explore the wide range of dynamic behaviors associated with these solutions. Additionally, modulation instability for the equation was determined with linear stability analysis. Observing the outcomes, it is clearly evident that the suggested methods are significant mathematical tools to solve nonlinear partial differential equations arising in various areas of applied mathematics and physics. The novelty of the work lies in the combined application of the $ \frac{G'}{bG' + G + a} $-expansion and modified $ F $-expansion methods to the KK equation, providing broader classes of exact solutions and deeper insight into the equation's nonlinear dynamics beyond earlier studies in the literature.
- optical soliton solutions,
- analytical techniques,
- Kaup–Kupershmidt model,
- $ \frac{G'}{bG' + G + a} $-expansion method,
- wave profile analysis,
- constraints,
- modified F-expansion method
Citation: Saud Owyed. Analytical and dynamical investigations of the Kaup–Kupershmidt model via two innovative integrability architectures[J]. Electronic Research Archive, 2026, 34(9): 5968-5992. doi: 10.3934/era.2026264

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Abstract

This study focused on the modified F-expansion method and the $ \frac{G'}{bG' + G + a} $-expansion method to find exact solutions to the $ (1+1)- $dimensional Kaup–Kupershmidt (KK) equation, an important nonlinear model that incorporates both dispersive effects and higher-order nonlinearity. By applying these analytical techniques, several classes of exact solutions, including periodic, bright, and singular soliton solutions, were successfully obtained. The graphical representation of these solutions was evaluated by creating two-dimensional, three-dimensional, and contour plots for each solution type to explore the wide range of dynamic behaviors associated with these solutions. Additionally, modulation instability for the equation was determined with linear stability analysis. Observing the outcomes, it is clearly evident that the suggested methods are significant mathematical tools to solve nonlinear partial differential equations arising in various areas of applied mathematics and physics. The novelty of the work lies in the combined application of the $ \frac{G'}{bG' + G + a} $-expansion and modified $ F $-expansion methods to the KK equation, providing broader classes of exact solutions and deeper insight into the equation's nonlinear dynamics beyond earlier studies in the literature.

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