Exact soliton, periodic and rational solutions of the $ M $-fractional Nurshuak-Tolkynay-Myrzakulov system

Aigul Taishiyeva; Gulgassyl Nugmanova; Aidana Azhikhan; Akbota Myrzakul; Ratbay Myrzakulov; Aigul Taishiyeva; Gulgassyl Nugmanova; Aidana Azhikhan; Akbota Myrzakul; Ratbay Myrzakulov

doi:10.3934/math.20261328

AIMS Mathematics

2026, Volume 11, Issue 10: 34063-34080. doi: 10.3934/math.20261328

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Research article

Exact soliton, periodic and rational solutions of the $ M $-fractional Nurshuak-Tolkynay-Myrzakulov system

1.
Kh. Dosmukhamedov Atyrau University, Atyrau, Kazakhstan
2.
Eurasian National University, Astana, Kazakhstan
3.
Astana IT University, Astana, Kazakhstan
4.
Ratbay Myrzakulov Eurasian International Center for Theoretical Physics

Received: 18 July 2026 Revised: 29 September 2026 Accepted: 08 October 2026 Published: 20 October 2026
MSC : 35A30,35C08

The object of this study was the Nurshuak-Tolkynay-Myrzakulov (NTM) system with $ M $-fractional in both time and space. Using appropriate traveling-wave transformations, the fractional nonlinear system was reduced to a nonlinear ordinary differential equation. Exact analytical solutions were constructed using the generalized Kudryashov method and the Sardar subequation method. The obtained solutions contained hyperbolic, trigonometric, rational, and Jacobi elliptic (cnoidal) wave structures that described localized solitons, periodic waves and algebraically decaying profiles. The influence of the fractional-order parameter $ \eta $ on the amplitude and localization width and propagation characteristics of the nonlinear waves used two- and three-dimensional graphical representations. In the limit $ \eta\rightarrow 1, $ all solutions recovered the corresponding exact solutions of the classical integer-order NTM system. This was the first study of the $ M $-fractional NTM system and its exact analytical solutions.
- $ M $-fractional,
- Nurshuak-Tolkynay-Myrzakulov equation,
- exact soliton solutions,
- Kudryashov method,
- Sardar subequation method,
- Jacobi elliptic solution solutions
Citation: Aigul Taishiyeva, Gulgassyl Nugmanova, Aidana Azhikhan, Akbota Myrzakul, Ratbay Myrzakulov. Exact soliton, periodic and rational solutions of the $ M $-fractional Nurshuak-Tolkynay-Myrzakulov system[J]. AIMS Mathematics, 2026, 11(10): 34063-34080. doi: 10.3934/math.20261328

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Abstract

The object of this study was the Nurshuak-Tolkynay-Myrzakulov (NTM) system with $ M $-fractional in both time and space. Using appropriate traveling-wave transformations, the fractional nonlinear system was reduced to a nonlinear ordinary differential equation. Exact analytical solutions were constructed using the generalized Kudryashov method and the Sardar subequation method. The obtained solutions contained hyperbolic, trigonometric, rational, and Jacobi elliptic (cnoidal) wave structures that described localized solitons, periodic waves and algebraically decaying profiles. The influence of the fractional-order parameter $ \eta $ on the amplitude and localization width and propagation characteristics of the nonlinear waves used two- and three-dimensional graphical representations. In the limit $ \eta\rightarrow 1, $ all solutions recovered the corresponding exact solutions of the classical integer-order NTM system. This was the first study of the $ M $-fractional NTM system and its exact analytical solutions.

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