Lie group structures and novel soliton solutions of the nonlinear mathematical model

Harun Biçer; Harun Biçer

doi:10.3934/math.2026168

AIMS Mathematics

2026, Volume 11, Issue 2: 4200-4219. doi: 10.3934/math.2026168

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Lie group structures and novel soliton solutions of the nonlinear mathematical model

Harun Biçer ^,

Genc Vocational School, Electric and Energy Programme, Bingol University, Bingol, Türkiye

Received: 11 December 2025 Revised: 22 January 2026 Accepted: 26 January 2026 Published: 10 February 2026
MSC : 35Q53, 65M60

We obtain new computational soliton solutions characterized by topological, rational, exponential, trigonometric, and hyperbolic functions for the modified Zakharov–Kuznetsov equation (mZKE). Utilizing two successful procedures, the extended rational sinh-cosh and Kudryashov expansion methods are applied to derive diverse dynamical wave structures of soliton solutions within the context of evolutionary dynamical structures of solitary wave solutions. It is permissible to choose parameters that show a solution. To improve understanding of the physical phenomena related to these dynamical models in mathematical physics, the empirical demonstration of the physical behavior of these solutions is provided. The technique of symmetry analysis is utilized to examine the governing equation. The implementation of a novel conservation theorem results in the formation of a comprehensive system of one-dimensional subalgebras. The study encompasses the Lie-Bäcklund symmetry generators. Additionally, the methodology for establishing conservation laws for nonlinear partial differential equations is clarified through the presentation of an innovative conservation theorem associated with Lie-Bäcklund symmetries. This strategy is used to come up with the conservation laws that apply to the governing equation.
- Zakharov–Kuznetsov equation,
- optimal system,
- soliton solutions,
- Lie-Bäcklund symmetries,
- conservation laws
Citation: Harun Biçer. Lie group structures and novel soliton solutions of the nonlinear mathematical model[J]. AIMS Mathematics, 2026, 11(2): 4200-4219. doi: 10.3934/math.2026168

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Abstract

We obtain new computational soliton solutions characterized by topological, rational, exponential, trigonometric, and hyperbolic functions for the modified Zakharov–Kuznetsov equation (mZKE). Utilizing two successful procedures, the extended rational sinh-cosh and Kudryashov expansion methods are applied to derive diverse dynamical wave structures of soliton solutions within the context of evolutionary dynamical structures of solitary wave solutions. It is permissible to choose parameters that show a solution. To improve understanding of the physical phenomena related to these dynamical models in mathematical physics, the empirical demonstration of the physical behavior of these solutions is provided. The technique of symmetry analysis is utilized to examine the governing equation. The implementation of a novel conservation theorem results in the formation of a comprehensive system of one-dimensional subalgebras. The study encompasses the Lie-Bäcklund symmetry generators. Additionally, the methodology for establishing conservation laws for nonlinear partial differential equations is clarified through the presentation of an innovative conservation theorem associated with Lie-Bäcklund symmetries. This strategy is used to come up with the conservation laws that apply to the governing equation.

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