Trilinearization of the Kairat-Ⅱ equation for the soliton solutions with machine learning evolution and Painlevé analysis

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

doi:10.3934/math.2026284

AIMS Mathematics

2026, Volume 11, Issue 3: 6910-6934. doi: 10.3934/math.2026284

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Research article

Trilinearization of the Kairat-Ⅱ equation for the soliton solutions with machine learning evolution and Painlevé 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 December 2025 Revised: 05 March 2026 Accepted: 10 March 2026 Published: 17 March 2026
MSC : 35A20, 35C08, 35Q55, 37K10

This research examines the integrability and soliton solution in different forms of the Kairat-Ⅱ equation. A thorough Painlevé analysis determines that the equation fulfills the criteria of compatibility conditions for integrability; thus, it is shown to exhibit soliton solutions. Applying the Hirota D-operator, we transform the Kairat-Ⅱ equation into a trilinear form. This enables the derivation of precise analytical solutions, including novel breather wave and three-wave soliton solutions by the Hirota method. These solutions reveal localized, oscillatory, and energy-exchange structures, capturing the complex nonlinear dynamics of the model. We use a machine learning analysis, the multi-layer perceptron regressor algorithm, for understanding the dynamics of these solitons, which effectively predicts the progression and dynamics of the obtained solutions. The analytical results are validated and displayed in 2D and 3D+contour plots using symbolic computation tools such as Mathematica and Python. In the end, we execute the asymptotic analysis on the gained solution and present the asymptotic behavior of the solutions. We can learn a great deal about the intricate dynamics that the Kairat-Ⅱ model governs from these visualizations.
- Kairat-Ⅱ equation,
- Painlevé analysis,
- trilinear form,
- Hirota method,
- three-wave and Breather-wave solutions,
- machine learning,
- asymptotic analysis
Citation: Waseem Razzaq, Asim Zafar, Naif Almusallam, Fawaz Khaled Alarfaj. Trilinearization of the Kairat-Ⅱ equation for the soliton solutions with machine learning evolution and Painlevé analysis[J]. AIMS Mathematics, 2026, 11(3): 6910-6934. doi: 10.3934/math.2026284

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Abstract

This research examines the integrability and soliton solution in different forms of the Kairat-Ⅱ equation. A thorough Painlevé analysis determines that the equation fulfills the criteria of compatibility conditions for integrability; thus, it is shown to exhibit soliton solutions. Applying the Hirota D-operator, we transform the Kairat-Ⅱ equation into a trilinear form. This enables the derivation of precise analytical solutions, including novel breather wave and three-wave soliton solutions by the Hirota method. These solutions reveal localized, oscillatory, and energy-exchange structures, capturing the complex nonlinear dynamics of the model. We use a machine learning analysis, the multi-layer perceptron regressor algorithm, for understanding the dynamics of these solitons, which effectively predicts the progression and dynamics of the obtained solutions. The analytical results are validated and displayed in 2D and 3D+contour plots using symbolic computation tools such as Mathematica and Python. In the end, we execute the asymptotic analysis on the gained solution and present the asymptotic behavior of the solutions. We can learn a great deal about the intricate dynamics that the Kairat-Ⅱ model governs from these visualizations.

References

[1]	S. Behera, Analytical solutions of fractional order Newell–Whitehead–Segel equation, In: Modeling, analysis and simulations of multiscale transport phenomena. ICMASMTP 2022, Singapore: Springer, 2025,207–221. https://doi.org/10.1007/978-981-96-3098-1_15
[2]	D. O'Regan, S. R. Aderyani, R. Saadati, M. Inc, Soliton solution of the nonlinear time fractional equations: Comprehensive methods to solve physical models, Axioms, 13 (2024), 92. https://doi.org/10.3390/axioms13020092 doi: 10.3390/axioms13020092
[3]	E. M. Ozkan, O. Yildirim, A. Ozkan, On the exact solutions of optical perturbed fractional Schrödinger equation, Phys. Scr., 98 (2023), 115104. https://doi.org/10.1088/1402-4896/acfa2f doi: 10.1088/1402-4896/acfa2f
[4]	K. K. Ali, A. Zafar, W. Razzaq, H. Ahmad, F. A. Awwad, E. A. A. Ismail, The kink solitary wave and numerical solutions for conformable nonlinear space–time fractional differential equations, Results Phys., 58 (2024), 107423. https://doi.org/10.1016/j.rinp.2024.107423 doi: 10.1016/j.rinp.2024.107423
[5]	H. Qawaqneh, A. Zafar, M. Raheel, A. A. Zaagan, E. H. M. Zahran, A. Cevikel, et al., New soliton solutions of M-fractional Westervelt model in ultrasound imaging via two analytical techniques, Opt. Quant. Electron., 56 (2024), 737. https://doi.org/10.1007/s11082-024-06371-1 doi: 10.1007/s11082-024-06371-1
[6]	K. S. Nisar, A. Ciancio, K. K. Ali, M. S. Osman, C. Cattani, D. Baleanu, et al., On beta-time fractional biological population model with abundant solitary wave structures, Alex. Eng. J., 61 (2022), 1996–2008. https://doi.org/10.1016/j.aej.2021.06.106 doi: 10.1016/j.aej.2021.06.106
[7]	W. Razzaq, A. Zafar, A. Akbulut, Applications of the simplest equation procedure to some fractional order differential equations in mathematical physics, Int. J. Appl. Comput. Math., 10 (2024), 56. https://doi.org/10.1007/s40819-024-01687-8 doi: 10.1007/s40819-024-01687-8
[8]	B. Hong, Abundant explicit solutions for the M-fractional coupled nonlinear Schrödinger–KdV equations, J. Low Freq. Noise Vib. Act. Control, 42 (2023), 1222–1241. https://doi.org/10.1177/14613484221148411 doi: 10.1177/14613484221148411
[9]	R. Kumar, K. S. Pandey, A. Kumar, A. Kumar, Novel traveling wave solutions of Jaulent–Miodek equations and coupled Konno–Oono systems and their dynamics, Chaos Theory Appl., 5 (2023), 281–285. https://doi.org/10.51537/chaos.1322939 doi: 10.51537/chaos.1322939
[10]	A. A. Mahmud, K. A. Muhamad, T. Tanriverdi, H. M. Baskonus, An investigation of Fokas system using two new modifications for the trigonometric and hyperbolic trigonometric function methods, Opt. Quant. Electron., 56 (2024), 717. https://doi.org/10.1007/s11082-024-06388-6 doi: 10.1007/s11082-024-06388-6
[11]	A. H. Abdel-Aty, S. Arshed, N. Raza, T. A. Alrebdi, K. S. Nisar, H. Eleuch, Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution, AIP Adv., 14 (2024), 025231. https://doi.org/10.1063/5.0191869 doi: 10.1063/5.0191869
[12]	S. Albosaily, E. M. Elsayed, M. D. Albalwi, M. Alesemi, W. W. Mohammed, The analytical stochastic solutions for the stochastic potential Yu–Toda–Sasa–Fukuyama equation with conformable derivative using different methods, Fractal Fract., 7 (2023), 787. https://doi.org/10.3390/fractalfract7110787 doi: 10.3390/fractalfract7110787
[13]	Y. Qi, Y. Tian, Y. Jiang, Existence of traveling wave solutions for the perturbed modified Gardner equation, Qual. Theory Dyn. Syst., 23 (2024), 106. https://doi.org/10.1007/s12346-024-00960-x doi: 10.1007/s12346-024-00960-x
[14]	M. N. Rafiq, M. H. Rafiq, H. Alsaud, Diversity of soliton dynamics, positive multi-complexiton solutions and modulation instability for (3+1)-dimensional extended Kairat-X equation, Mod. Phys. Lett. B, 39 (2025), 2550112. https://doi.org/10.1142/S021798492550112X doi: 10.1142/S021798492550112X
[15]	Y. Bo, D. Tian, X. Liu, Y. Jin, Discrete maximum principle and energy stability of the compact difference scheme for two‐dimensional Allen‐Cahn equation, J. Funct. Spaces, 2022 (2022), 8522231. https://doi.org/10.1155/2022/8522231 doi: 10.1155/2022/8522231
[16]	Z. Myrzakulova, S. Manukure, R. Myrzakulov, G. Nugmanova, Integrability, geometry and wave solutions of some Kairat equations, arXiv: 2307.00027, 2023. https://arXiv.org/abs/2307.00027
[17]	W. A. Faridi, A. M. Wazwaz, A. M. Mostafa, R. Myrzakulov, Z. Umurzakhova, The Lie point symmetry criteria and formation of exact analytical solutions for Kairat-Ⅱ equation: Paul–Painlevé approach, Chaos Solitons Fract., 182 (2024), 114745. https://doi.org/10.1016/j.chaos.2024.114745 doi: 10.1016/j.chaos.2024.114745
[18]	W. A. Faridi, G. H. Tipu, Z. Myrzakulova, R. Myrzakulov, S. A. AlQahtani, P. Pathak, The sensitivity demonstration and propagation of hyper-geometric soliton waves in plasma physics of Kairat-Ⅱ equation, Phys. Scr., 99 (2024), 045209. https://doi.org/10.1088/1402-4896/ad2bc2 doi: 10.1088/1402-4896/ad2bc2
[19]	M. Awadalla, A. Zafar, A. Taishiyeva, M. Raheel, R. Myrzakulov, A. Bekir, The analytical solutions to the M-fractional Kairat-Ⅱ and Kairat-X equations, Front. Phys., 11 (2023), 1335423. https://doi.org/10.3389/fphy.2023.1335423 doi: 10.3389/fphy.2023.1335423
[20]	J. Muhammad, S. U. Rehman, N. Nasreen, M. Bilal, U. Younas, Exploring the fractional effect to the optical wave propagation for the extended Kairat-Ⅱ equation, Nonlinear Dyn., 113 (2025), 1501–1512. https://doi.org/10.1007/s11071-024-10139-3 doi: 10.1007/s11071-024-10139-3
[21]	A. M. Wazwaz, Extended (3+1)-dimensional Kairat-Ⅱ and Kairat-X equations: Painlevé integrability, multiple soliton solutions, lump solutions, and breather wave solutions, Int. J. Numer. Methods Heat Fluid Flow, 34 (2024), 2177–2194. https://doi.org/10.1108/HFF-01-2024-0053 doi: 10.1108/HFF-01-2024-0053
[22]	M. Raheel, M. Inc, E. Tala-Tebue, K. H. Mahmoud, Breather, kink and rogue wave solutions of Sharma–Tasso–Olver-like equation, Opt. Quant. Electron., 54 (2022), 560. https://doi.org/10.1007/s11082-022-03933-z doi: 10.1007/s11082-022-03933-z
[23]	A. Zafar, M. Raheel, H. Rezazadeh, M. Inc, M. A. Akinlar, New chirp-free and chirped form optical solitons to the nonlinear Schrödinger equation, Opt. Quant. Electron., 53 (2021), 604. https://doi.org/10.1007/s11082-021-03254-7 doi: 10.1007/s11082-021-03254-7
[24]	T. Mathanaranjan, New optical solitons and modulation instability analysis of generalized coupled nonlinear Schrödinger–KdV system, Opt. Quant. Electron., 54 (2022), 336. https://doi.org/10.1007/s11082-022-03723-7 doi: 10.1007/s11082-022-03723-7
[25]	N. Raza, A. R. Seadawy, S. Arshed, M. H. Rafiq, A variety of soliton solutions for the Mikhailov–Novikov–Wang dynamical equation via three analytical methods, J. Geom. Phys., 176 (2022), 104515. https://doi.org/10.1016/j.geomphys.2022.104515 doi: 10.1016/j.geomphys.2022.104515
[26]	S. Akçaği, T. Aydemir, Comparison between the (G'/G)-expansion method and the modified extended tanh method, Open Phys., 14 (2016), 88–94. https://doi.org/10.1515/phys-2016-0006 doi: 10.1515/phys-2016-0006
[27]	A. Zafar, A. Bekir, M. Raheel, H. Rezazadeh, Investigation for optical soliton solutions of two nonlinear Schrödinger equations via two concrete finite series methods, Int. J. Appl. Comput. Math., 6 (2020), 65. https://doi.org/10.1007/s40819-020-00818-1 doi: 10.1007/s40819-020-00818-1
[28]	Z. Chen, J. Manafian, M. Raheel, A. Zafar, F. Alsaikhan, M. Abotaleb, Extracting the exact solitons of time-fractional three coupled nonlinear Maccari's system with complex form via four different methods, Results Phys., 36 (2022), 105400. https://doi.org/10.1016/j.rinp.2022.105400 doi: 10.1016/j.rinp.2022.105400
[29]	J. Manafian, Variety interaction solutions comprising lump solitons for a generalized BK equation by trilinear analysis, Eur. Phys. J. Plus, 136 (2021), 1097. https://doi.org/10.1140/epjp/s13360-021-02065-9 doi: 10.1140/epjp/s13360-021-02065-9
[30]	L. Cheng, Y. Zhang, W. X. Ma, Wronskian N-soliton solutions to a generalized KdV equation in (2+1)-dimensions, Nonlinear Dyn., 111 (2023), 1701–1714. https://doi.org/10.1007/s11071-022-07920-7 doi: 10.1007/s11071-022-07920-7
[31]	L. Cheng, Y. Zhang, W. X. Ma, J. Y. Ge, Wronskian and lump wave solutions to an extended second KP equation, Math. Comput. Simul., 187 (2021), 720–731. https://doi.org/10.1016/j.matcom.2021.03.024 doi: 10.1016/j.matcom.2021.03.024
[32]	H. F. Ismael, H. Bulut, Nonlinear dynamics of (2+1)-dimensional Bogoyavlenskii–Schieff equation arising in plasma physics, Math. Methods Appl. Sci., 44 (2021), 10321–10330. https://doi.org/10.1002/mma.7409 doi: 10.1002/mma.7409
[33]	C. Wang, Z. Dai, Dynamic behaviors of bright and dark rogue waves for the (2+1)-dimensional Nizhnik–Novikov–Veselov equation, Phys. Scr., 90 (2015), 065205. https://doi.org/10.1088/0031-8949/90/6/065205 doi: 10.1088/0031-8949/90/6/065205
[34]	T. Y. Zhou, B. Tian, Y. Q. Chen, Y. Shen, Painlevé analysis, auto-Bäcklund transformation and analytic solutions of a (2+1)-dimensional generalized Burgers system with variable coefficients in a fluid, Nonlinear Dyn., 108 (2022), 2417–2428. https://doi.org/10.1007/s11071-022-07211-1 doi: 10.1007/s11071-022-07211-1
[35]	S. Singh, S. S. Ray, Painlevé analysis, auto-Bäcklund transformation and new exact solutions of (2+1) and (3+1)-dimensional extended Sakovich equation with time dependent variable coefficients in ocean physics, J. Ocean Eng. Sci., 8 (2023), 246–262. https://doi.org/10.1016/j.joes.2022.01.008 doi: 10.1016/j.joes.2022.01.008
[36]	B. Mohan, S. Kumar, R. Kumar, Higher-order rogue waves and dispersive solitons of a novel P-type (3+1)-D evolution equation in soliton theory and nonlinear waves, Nonlinear Dyn., 111 (2023), 20275–20288. https://doi.org/10.1007/s11071-023-08938-1 doi: 10.1007/s11071-023-08938-1
[37]	W. Razzaq, A. Zafar, Bilinearization of generalized Bogoyavlensky–Konopelchenko equation with neural networks: Painlevé analysis, Nonlinear Dyn., 113 (2025), 25083–25096. https://doi.org/10.1007/s11071-025-11384-w doi: 10.1007/s11071-025-11384-w
[38]	J. G. Liu, J. Q. Du, Z. F. Zeng, B. Nie, New three-wave solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation, Nonlinear Dyn., 88 (2017), 655–661. https://doi.org/10.1007/s11071-016-3267-2 doi: 10.1007/s11071-016-3267-2
[39]	W. Razzaq, A. Zafar, Machine learning-enhanced soliton solutions for the Lonngren-wave equation: An integration of Painlevé analysis and Hirota bilinear method, Rend. Fis. Acc. Lincei, 36 (2025), 917–932. https://doi.org/10.1007/s12210-025-01354-0 doi: 10.1007/s12210-025-01354-0

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