Asymptotic behavior of soliton solutions of the Kairat-X model via the Hirota bilinear method: Painlevé integrability and machine learning analysis

Waseem Razzaq; Asim Zafar; Naif Almusallam; Fawaz Khaled Alarfaj; Waseem Razzaq; Asim Zafar; Naif Almusallam; Fawaz Khaled Alarfaj

doi:10.3934/math.20251320

AIMS Mathematics

2025, Volume 10, Issue 12: 30029-30052. doi: 10.3934/math.20251320

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Asymptotic behavior of soliton solutions of the Kairat-X model via the Hirota bilinear method: Painlevé integrability and machine learning analysis

1.
Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan
2.
Department of Management Information Systems, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 19 October 2025 Revised: 03 December 2025 Accepted: 11 December 2025 Published: 22 December 2025
MSC : 35A20, 35C08, 35Q55, 37K10

In this article, we investigated the integrability of the nonlinear dynamical Kairat-X model through Painlevé analysis, demonstrating that the equation satisfies the Painlevé property and is therefore integrable. We applied the bilinear Hirota method to derive several exact solutions, including breather wave, novel periodic wave, periodic cross-kink wave, kink-rogue wave interaction, and one-soliton and two-soliton solutions. A machine learning multi-layer-perceptron regressor algorithm was applied to represent the behavior of the actual, and to predict, the above solutions. Furthermore, we employed an asymptotic analysis on the gain solutions to expect the demonstration of the asymptotic behavior of these analytical solutions. The soliton solutions obtained were novel and exhibited improved reliability compared to previously reported results. These findings were further validated using symbolic computation software. A comparison with the existing literature revealed that the proposed solutions were more applicable and accurate. Several of the results were visualized using two-dimensional, three-dimensional, and contour surface plots.
- Kairat-X model,
- Painlevé analysis,
- Hirota bilinear method,
- soliton solutions,
- machine learning analysis,
- multi-layer-perceptron regressor,
- asymptotic analysis
Citation: Waseem Razzaq, Asim Zafar, Naif Almusallam, Fawaz Khaled Alarfaj. Asymptotic behavior of soliton solutions of the Kairat-X model via the Hirota bilinear method: Painlevé integrability and machine learning analysis[J]. AIMS Mathematics, 2025, 10(12): 30029-30052. doi: 10.3934/math.20251320

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Abstract

In this article, we investigated the integrability of the nonlinear dynamical Kairat-X model through Painlevé analysis, demonstrating that the equation satisfies the Painlevé property and is therefore integrable. We applied the bilinear Hirota method to derive several exact solutions, including breather wave, novel periodic wave, periodic cross-kink wave, kink-rogue wave interaction, and one-soliton and two-soliton solutions. A machine learning multi-layer-perceptron regressor algorithm was applied to represent the behavior of the actual, and to predict, the above solutions. Furthermore, we employed an asymptotic analysis on the gain solutions to expect the demonstration of the asymptotic behavior of these analytical solutions. The soliton solutions obtained were novel and exhibited improved reliability compared to previously reported results. These findings were further validated using symbolic computation software. A comparison with the existing literature revealed that the proposed solutions were more applicable and accurate. Several of the results were visualized using two-dimensional, three-dimensional, and contour surface plots.

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