Various solutions of (2+1)-dimensional sixth-order breaking soliton systems using bilinear neural network method

Abstract

This paper firstly introduces the (2+1)-dimensional sixth-order breaking soliton system (SBSS) using the bilinear neural network method, providing many types of solutions. A new structure of the SBSS is investigated, and new solutions are gathered. The solutions are expressed in terms of basic activation functions and are derived by applying these activation functions. Using the Hirota bilinear operator, the original nonlinear partial differential equation is reduced to a more tractable form. The bilinear neural network approach yields various classes of solutions, including kink, rogue, peak, breather, lump-type, and spike-type solutions. These solutions are expressed in terms of elementary functions, revealing the various dynamical behaviors inherent in the system. The results obtained not only generalize known solutions but also illustrate a deeper insight into the complex wave propagation phenomena described by the sixth-order breaking soliton equation.

Keywords

Breaking soliton system / Bilinear neural network method (BNNM) / Softmax method / High-order soliton equations / Solitary wave solutions / Analytic exact solution

References

[1] Almusawa H, Jhangeer A, Hussain Z. Observation on different dynamics of breaking soliton equation by bifurcation analysis and multistability theory. Results in Phy. 2022; 36: 105364. https://doi.org/10.1016/j.rinp.2022.105364

[2] Chen X, Xia T, Zhu L. A (2 + 1)- Dimensional integrable breaking soliton equation and its Algebro-Geometric solutions. Mathematics. 2024; 12(13): 2034. https://doi.org/10.3390/math12132034

[3] Kumar S, Almusawa H, Dhiman SK. A study of Bogoyavlenskii’s (2+1)-dimensional breaking soliton equation: Lie symmetry, dynamical behaviors and closed-form solutions. Results in Phy. 2021; 29: 104793. https://doi.org/10.1016/j.rinp.2021.104793

[4] Kumar S, Niwas M, Osman MS, Abdou MA. Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)- dimensional breaking soliton equations. Commun Theor Phys. 2021; 73(10): 105007. https://doi.org/10.1088/1572-9494/ac11ee

[5] Shang Y. Abundant explicit non-traveling wave solutions for the (2+1)-dimensional breaking soliton equation. Appl Math Lett. 2022; 131: 108029. https://doi.org/10.1016/j.aml.2022.108029

[6] Li C, Zhao M. Analytical solutions of the (2 + 1) -dimensional space-time fractional Bogoyavlenskii’s breaking soliton equation. Appl Math Lett. 2018; 84: 13-18. https://doi.org/10.1016/j.aml.2018.04.011

[7] Tariq KU-H, Seadawy AR. Bistable Bright- Dark solitary wave solutions of the (3 + 1)- dimensional Breaking soliton, Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equations and their applications. Results in Phy. 2017; 7: 1143-1149. https://doi.org/10.1016/j.rinp.2017.03.001

[8] Yan X-W, Tian S-F, Dong M-J, Zhou L, Zhang T-T. Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation. Comput Math Appl. 2018; 76(1): 179-186. https://doi.org/10.1016/j.camwa.2018.04.013

[9] Hasan FL, Abdoon MA, Saadeh R, Qazza A, Almutairi DK. Exploring analytical results for (2+1) dimensional breaking soliton equation and stochastic fractional Broer-Kaup system. AIMS Math. 2024; 9(5): 11622-11643. https://doi.org/10.3934/math.2024570

[10] Chen H, Zhu Q, Qi J. Further results about the exact solutions of conformable space-time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation. Results in Phy. 2022; 37: 105428. https://doi.org/10.1016/j.rinp.2022.105428

[11] Wazwaz AM. Generalized (2+1)-dimensional breaking soliton equation. TWMS J. App. Eng. Math. 2011; 1(1), 69-74.

[12] Xu G. Integrability of a (2+1)-dimensional generalized breaking soliton equation. Applied Mathematics Letters. 2015; 50: 16-22. https://doi.org/10.1016/j.aml.2015.05.015

[13] Hu X, Lin S, Wang L. Integrability, multiple-cosh, lumps and lump-soliton solutions to a (2+1)-dimensional generalized breaking soliton equation. Commun Nonlinear Sci Numer Simul. 2020; 91: 105447. https://doi.org/10.1016/j.cnsns.2020.105447

[14] Dong M-J, Tian S-F, Wang X-B, Zhang T-T. Lump-type solutions and interaction solutions in the (3 + 1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation. Anal Math Phys. 2019; 9(3): 1511-1523. https://doi.org/10.1007/s13324-018-0258-0

[15] Alzahrani S, Alzahrani T. Multiple solitons with bifurcations, lump waves, M-shaped and interaction solitons of three component generalized (3+1)-dimensional Breaking soliton system. AIMS Math. 2023; 8(8): 17803-17826. https://doi.org/10.3934/math.2023908

[16] Hosseini K, Seadawy AR, Mirzazadeh M, Eslami M, Radmehr S, Baleanu D. Multiwave, multicomplexiton, and positive multicomplexiton solutions to a (3 + 1)-dimensional generalized breaking soliton equation. Alex Eng J. 2020; 59(5): 3473-3479. https://doi.org/10.1016/j.aej.2020.05.027

[17] Cheng W, Li B, Chen Y. Nonlocal symmetry and exact solutions of the (2+1)- dimensional breaking soliton equation. Commun Nonlinear Sci Numer Simul. 2015; 29(1-3): 198-207. https://doi.org/10.1016/j.cnsns.2015.05.007

[18] Plaatjie K, Khalique CM. On the solutions and conservation laws of the 2D breaking soliton equation of fluid mechanics. Partial Differ Equ Appl Math. 2021; 4: 100198. https://doi.org/10.1016/j.padiff.2021.100198

[19] Osman MS. On multi-soliton solutions for the (2 + 1) -dimensional breaking soliton equation with variable coefficients in a graded-index waveguide. Comput Math Appl. 2018; 75(1): 1-6. https://doi.org/10.1016/j.camwa.2017.08.033

[20] Yao Y, Chen D, Zeng Y. Soliton solutions for a -dimensional breaking soliton equation with self-consistent sources. Nonlinear Anal Theory Methods Appl. 2010; 72(1): 57-64. https://doi.org/10.1016/j.na.2009.06.066

[21] Cerdik Yaslan H, Girgin A. SITEM for the conformable space-time fractional Boussinesq and (2 + 1)-dimensional breaking soliton equations. J Ocean Eng Sci. 2021; 6(3): 228-236. https://doi.org/10.1016/j.joes.2020.11.010

[22] Zhu X, Zuo D. Some (2 + 1) -dimensional nonlocal ’breaking soliton’-type systems. Appl Math Lett. 2019; 91: 181-187. https://doi.org/10.1016/j.aml.2018.12.011

[23] Niwas M, Kumar S, Kharbanda H. Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the generalized (3+1)- dimensional breaking soliton equation using optimal system of Lie subalgebra. J Ocean Eng Sci. 2022; 7(2): 188-201. https://doi.org/10.1016/j.joes.2021.08.002

[24] Tao Z-L. Solving the breaking soliton equation by He’s variational method. Comput Math Appl. 2009; 58(11-12): 2395-2397. https://doi.org/10.1016/j.camwa.2009.03.033

[25] Da-Quan X. Symmetry reduction and new non-traveling wave solutions of (2+1)-dimensional breaking soliton equation. Commun Nonlinear Sci Numer Simul. 2010; 15(8): 2061-2065. https://doi.org/10.1016/j.cnsns.2009.08.013

[26] Zhao Z, Dai Z, Mu G. The breather-type and periodic-type soliton solutions for the (2+1 )- dimensional breaking soliton equation. Comput Math Appl. 2011; 61(8): 2048-2052. https://doi.org/10.1016/j.camwa.2010.08.065

[27] Wang J-M, Yang X. Theta-function Solutions to the (2+l)-Dimensional Breaking Soliton Equation. Phys Lett. 2011; 28(9): 090203. https://doi.org/10.1088/0256-307X/28/9/090203

[28] Yıldırım Y, Yaşar E. A (2+1)-dimensional breaking soliton equation: Solutions and conservation laws. Chaos Sol Frac. 2018; 107: 146-155. https://doi.org/10.1016/j.chaos.2017.12.016

[29] Ting S, Xian-Guo G, Yun-Ling M. Wronskian Form of N -Soliton solution for the (2+1)- Dimensional breaking soliton equation. Phy Lett. 2007; 24(2): 305-307. https://doi.org/10.1088/0256-307X/24/2/001

[30] Tuan NM, Meesad P, Son NH. New Solutions of Breaking Soliton Equation Using Softmax Method. J Appl Math. 2025; 13. https://doi.org/10.1155/jama/8810599

[31] Gai L, Ma W-X, Sudao B. Abundant multilayer network model solutions and bright-dark solitons for a (3 + 1)-dimensional p-gBLMP equation. Nonlinear Dyn. 2021; 106(1): 867-877. https://doi.org/10.1007/s11071-021-06864-8

[32] Demiray ST, Baskonus HM, Bulut H. Application of the HPM for nonlinear (3+ 1)-dimensional breaking soliton equation. Math Eng Sci Aerosp. 2014; 5(1): 127-133.

[33] Davies Adeyemo O, Masood Khalique C. Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics. J Taibah Univ Sci. 2024; 18(1): 2331984. https://doi.org/10.1080/16583655.2024.2331984

[34] Gai L, Wu W, Ding T, Qian Y. Lump wave solutions, lump-stripe soliton inelastic collision phenomena and rogue-type wave solutions for a generalized breaking soliton system in (3+1)- dimensions. Wave Motion. 2024; 124: 103243. https://doi.org/10.1016/j.wavemoti.2023.103243

[35] Gai L, Ma W-X, Li M. Lump-type solutions, rogue wave type solutions and periodic lump-stripe interaction phenomena to a (3 + 1)- dimensional generalized breaking soliton equation. Physics Letters A. 2020; 384(8): 126178. https://doi.org/10.1016/j.physleta.2019.126178

[36] Zhou H. Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation. J Syst Sci Complex. 2024; 37(2): 511-544. https://doi.org/10.1007/s11424-024-3467-7

[37] Ma S-H, Fang J-P, Zheng C-L. New exact solutions of the (2+1)-dimensional breaking soliton system via an extended mapping method. Chaos Sol Frac. 2009; 40(1): 210-214. https://doi.org/10.1016/j.chaos.2007.07.043

[38] Gao Y-T, Tian B. New family of overturning soliton solutions for a typical breaking soliton equation. Comput Math Appl. 1995; 30(12): 97-100. https://doi.org/10.1016/0898-1221(95)00176-Y

[39] Wazwaz A-M. Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations. Phy Scri. 2010; 81(3): 035005. https://doi.org/10.1088/0031-8949/81/03/035005

[40] Tuan NM, Phayung M. A novel softmax method for solving second Benney-Luke equation. J. Comput Appl Math. 2025; 472(2025): 116791. https://doi.org/10.1016/j.cam.2025.116791

[41] Tuan NM, Meesad P. New solutions of sixth-order Benney-Luke equation using bilinear neural network method. Z. Angew Math Phys. 2025; 76(4): 133. https://doi.org/10.1007/s00033-025-02516-8

[42] Tuan NM, Thuy PTT, Cuong HHN, Hien NT. On Determining Multiple Languages through Technological Examination for Conservation Management Using Machine Learning. Forum Linguist Stud. 2025; 7(5): Article 5. https://doi.org/10.30564/fls.v7i5.9110

[43] Tuan NM, Son NH. Hirota Bilinear Performance on Hirota-Satsuma-Ito Equation Using Bilinear Neural Network Method. Int J Appl Comput Math. 2025; 11(121). https://doi.org/10.1007/s40819-025-01933-7

[44] Tuan NM. A Novel Softmax Building Method for Finding Solutions of Benney-Luke Equation. Int J Theor Phys. 2025; 64: 145. https://doi.org/10.1007/s10773-025-06002-9

[45] Tuan, N. M., & Son, N. H. A New Softmax Method Performance for Solving Chaffee-Infante Equation. Int J Math Comput Sci. 2025; 20(3): 743-749. https://doi.org/10.69793/ijmcs/03.2025/tuan

[46] Tuan NM, Alqurashi MS, Alderremy AA, Mahmoud EE. Repercussions of concentration and thermal transport of magnetized radiative ternary nanofluid over spinning sphere: Three neural layers modeling. J Radiat Res Appl Sci. 2025; 18(2): 101508. https://doi.org/10.1016/j.jrras.2025.101508

[47] Tuan NM, Meesad P. A bilinear neural network method for solving a generalized fractional (2+1)-Dimensional konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. Int J Theor Phys. 2025; 64(2025): 17. https://doi.org/10.1007/s10773-024-05855-w

[48] Tuan NM, Meesad P. Bilinear Recurrent Neural Network for a Modified Benney-Luke Equation. Int J Appl Comput Math. 2025; 11(2): 35. https://doi.org/10.1007/s40819-025-01851-8

[49] Tuan NM, Meesad P, Nguyen HHC. English-Vietnamese Machine Translation Using Deep Learning for Chatbot Applications. SN Comput Sci. 2023; 5(1): 5. https://doi.org/10.1007/s42979-023-02339-2

[50] Tuan NM, Meesad P, Hieu DV, Cuong NHH, Maliyaem M. On students’ sentiment Prediction based on deep learning: applied information literacy. SN Comput Sci. 2024; 5(7): 928. https://doi.org/10.1007/s42979-024-03281-7

[51] Wazwaz A-M. Partial differential equations and solitary waves theory. HigH Edu Press. Springer. 2009.

[52] Dai Z, Liu J, Li D. Applications of HTA and EHTA to YTSF Equation. Appl Math Comput. 2009; 207(2): 360-364. https://doi.org/10.1016/j.amc.2008.10.042

[53] Zhao Z, Dai Z, Han S. The EHTA for nonlinear evolution equations. Appl Math Comput. 2010; 217(8): 4306-4310. https://doi.org/10.1016/j.amc.2010.09.069

[54] Miao Z, Chen Y. Physics-informed neural networks method in high-dimensional integrable systems. Mod Phys Lett B. 2022; 36(01): 2150531. https://doi.org/10.1142/S021798492150531X

[55] Wang X, Bilige S. Novel interaction phenomena of the (3+1)-dimensional Jimbo-Miwa equation. Commun Theor Phys. 2020; 72(4): 045001. https://doi.org/10.1088/1572-9494/ab690c

[56] Shen J-L, Wu X-Y. Periodic-soliton and periodic-type solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation by using BNNM. Nonlinear Dynamics. 2021; 106(1): 831-840. https://doi.org/10.1007/s11071-021-06848-8

[57] Feng Y, Bilige S. Various rational solutions and rogue wave solutions of extended (2+1)- dimensional Calogero-Bogoyavlenskii-Schiff-like equation. Authorea. Preprint posted 2020. https://doi.org/10.22541/au.160407386.60382756/v1

[58] Hong B, Wang J. Exact Solutions for the Generalized Atangana-Baleanu-Riemann Fractional (3 + 1)-Dimensional Kadomtsev-Petviashvili Equation. Symmetry. 15(1), 3. https://doi.org/10.3390/sym15010003

[59] Gu Y, Malmir S, Manafian J, Ilhan OA, Alizadeh A, Othman AJ. Variety interaction between k -lump and k -kink solutions for the (3+1)- D Burger system by bilinear analysis. Res in Phy. 2022; 43: 106032. https://doi.org/10.1016/j.rinp.2022.106032

[60] Abdeljabbar A, Roshid H-O, Aldurayhim A. Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein-Gordon Equation. Symmetry. 2022; 14(6): 1223. https://doi.org/10.3390/sym14061223

[61] Zhang R-F, Li M-C, Cherraf A, Vadyala SR. The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM. Nonlinear Dynamics. 2023; 111(9): 8637-8646. https://doi.org/10.1007/s11071-023-08257-5

[62] Isah MA, Yokus A, Kaya D. A bilinear neural network method for obtaining exact analytical solutions of nonlinear evolution equations with application to the kdv equation. 2024. https://doi.org/10.22034/kjm.2024.396918.2865

[63] Isah M A, Yokus A, Kaya D. Exploring the influence of layer and neuron configurations on Boussinesq equation solutions via a bilinear neural network framework. Nonlinear Dyn. 2024; 112(15): 13361-13377. https://doi.org/10.1007/s11071-024-09708-3

[64] Ullah MS, Akter R. Soliton dynamics and stability analysis of a double-chain DNA model with various chaos detection tools. Sci Rep. 2025; 16(1): 1164. https://doi.org/10.1038/s41598-025-30779-8

[65] Usman T, Ullah MS. Analytical solutions and chaotic insights into the Hirota-Maccari system. Sci Rep. 2025; 15(1): 43540. https://doi.org/10.1038/s41598-025-27419-6