Three-wave, rogue wave and interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation

Shijie Deng; Bo Tang; Wenjing Yang; Kaili Liu; Huawei Wu; Shijie Deng; Bo Tang; Wenjing Yang; Kaili Liu; Huawei Wu

doi:10.3934/nhm.2026029

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 669-692. doi: 10.3934/nhm.2026029

Previous Article Next Article

Research article Topical Sections

Three-wave, rogue wave and interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation

1.
School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China
2.
Hubei Key Laboratory of Power System Design and Test for Electrical Vehicle, Hubei University of Arts and Science, Xiangyang 441053, China
3.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
4.
School of Automotive and Traffic Engineering, Hubei University of Arts and Science, Xiangyang 441053, China

Received: 03 January 2026 Revised: 01 March 2026 Accepted: 10 March 2026 Published: 24 April 2026

In this article, we focused on the bilinear neural network method to obtain the exact solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation. This method can also be applied to solve other nonlinear partial differential equations (NPDEs). Relying on Hirota bilinear transformation and Bell polynomial theories, we successfully constructed one-hidden-layer and multiple-hidden-layers models with assigned activation functions. Thereby, we derived many exact solutions for this equation including three-wave, rogue wave, and interaction solutions. To intuitively reflect features of these solutions, we depicted their three-dimensional, density, and curve graphs.
- bilinear neural network method,
- Hirota bilinear transformation,
- Bell polynomial,
- (2+1)-dimensional nonlinear evolution equation,
- exact solutions
Citation: Shijie Deng, Bo Tang, Wenjing Yang, Kaili Liu, Huawei Wu. Three-wave, rogue wave and interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation[J]. Networks and Heterogeneous Media, 2026, 21(2): 669-692. doi: 10.3934/nhm.2026029

Related Papers:

Abstract

In this article, we focused on the bilinear neural network method to obtain the exact solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito-Shallow-Water-Wave-like equation. This method can also be applied to solve other nonlinear partial differential equations (NPDEs). Relying on Hirota bilinear transformation and Bell polynomial theories, we successfully constructed one-hidden-layer and multiple-hidden-layers models with assigned activation functions. Thereby, we derived many exact solutions for this equation including three-wave, rogue wave, and interaction solutions. To intuitively reflect features of these solutions, we depicted their three-dimensional, density, and curve graphs.

References

[1]	R. Hirota, J. Satsuma, Nonlinear evolution equations generated from the Bäcklund transformation for the Boussinesq equation, Prog. Theor. Phys., 57 (1977), 797–807. https://doi.org/10.1143/PTP.57.797 doi: 10.1143/PTP.57.797
[2]	Y. D. Shang, Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation, Appl. Math. Comput., 187 (2007), 1286–1297. https://doi.org/10.1016/j.amc.2006.09.038 doi: 10.1016/j.amc.2006.09.038
[3]	M. Wang, G. L. He, T. Xu, Wronskian solutions and N–soliton solutions for the Hirota–Satsuma equation, Appl. Math. Lett., 159 (2025), 109279. https://doi.org/10.1016/j.aml.2024.109279 doi: 10.1016/j.aml.2024.109279
[4]	R. Hirota, The direct method in soliton theory, in Cambridge Tracts in Mathematics (eds. A. Nagai, J. Nimmo, C. Gilson), Cambridge University Press, (2004). https://doi.org/10.1017/CBO9780511543043
[5]	R. Hirota, J. Satsuma, N-Soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn., 40 (1976), 611–612. https://doi.org/10.1143/JPSJ.40.611 doi: 10.1143/JPSJ.40.611
[6]	Y. N. An, R. Guo, The mixed solutions of the (2+1)-dimensional Hirota–Satsuma–Ito equation and the analysis of nonlinear transformed waves, Nonlinear Dyn., 111 (2023), 18291–18311. https://doi.org/10.1007/s11071-023-08791-2 doi: 10.1007/s11071-023-08791-2
[7]	C. Yue, D. C. Lu, M. M. A. Khater, Abundant wave accurate analytical solutions of the fractional nonlinear Hirota–Satsuma–Shallow water wave equation, Fluids, 6 (2021), 235. https://doi.org/10.3390/fluids6070235 doi: 10.3390/fluids6070235
[8]	Z. L. Zhao, Y. F. Zhang, Periodic wave solutions and asymptotic analysis of the Hirota–Satsuma shallow water wave equation, Math. Meth. Appl. Sci., 38 (2014), 4262–4271. https://doi.org/10.1002/mma.3362 doi: 10.1002/mma.3362
[9]	W. X. Ma, J. Li, C. M. Khalique, A study on Lump solutions to a generalized Hirota-Satsuma-Ito equation in (2+1)-dimensions, Complexity, 7 (2018), 9059858. https://doi.org/10.1155/2018/9059858 doi: 10.1155/2018/9059858
[10]	Y. Q. Liu, X. Y. Wen, D. S. Wang, The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation, Comput. Math. Appl., 77 (2019), 947–966. https://doi.org/10.1016/j.camwa.2018.10.035 doi: 10.1016/j.camwa.2018.10.035
[11]	S. Khaliq, S. Ahmad, A. Ullah, H. Ahmad, S. Saifullah, T. A. Nofal, New waves solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation using a novel expansion method, Results. Phys., 50 (2023), 106450. https://doi.org/10.1016/j.rinp.2023.106450 doi: 10.1016/j.rinp.2023.106450
[12]	C. Wang, Y. Shi, W. A. Yang, X. P. Xin, Darboux transformations and exact solutions of nonlocal Kaup–Newell equations with variable coefficients, Appl. Math. Lett., 163 (2025), 109456. https://doi.org/10.1016/j.aml.2025.109456 doi: 10.1016/j.aml.2025.109456
[13]	G. H. Wang, The multi-soliton solutions of another two-component Camassa–Holm equation with Darboux transformation approach, Wave Motion, 31 (2024), 103396. https://doi.org/10.1016/j.wavemoti.2024.103396 doi: 10.1016/j.wavemoti.2024.103396
[14]	J. C. Pu, Y. Chen, Darboux transformation-based LPNN generating novel localized wave solutions, Physica D, 467 (2024), 134262. https://doi.org/10.1016/j.physd.2024.134262 doi: 10.1016/j.physd.2024.134262
[15]	M. Musette, Painlevé analysis for nonlinear partial differential equations, in The Painlevé Property (eds. R. Conte), Springer, New York, NY, (1999), 517–572. https://doi.org/10.1007/978-1-4612-1532-5
[16]	J. Satsuma, Hirota bilinear method for nonlinear evolution equations, in Direct and Inverse Methods in Nonlinear Evolution Equations: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5–12, 1999, Springer Berlin, Heidelberg, 632 (2003), 171–222. https://doi.org/10.1007/b13714
[17]	W. J. Yang, K. L. Liu, B. Tang, S. J. Deng, F. Y. Lv, Three wave solution and lump-type solution to a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation with some variable coefficients in inhomogeneous media, Networks Heterogen. Media, 20 (2025), 955–969. https://doi.org/10.3934/nhm.2025041 doi: 10.3934/nhm.2025041
[18]	Y. S. Bai, Y. N. Liu, W. X. Ma, Lie symmetry analysis, exact solutions, and conservation laws to multi-component nonlinear Schrödinger equations, Nonlinear Dyn., 111 (2023), 18439–18448. https://doi.org/10.1007/s11071-023-08833-9 doi: 10.1007/s11071-023-08833-9
[19]	U. H. M. Zaman, M. A. Arefin, M. A. Akbar, M. H. Uddin, Comprehensive dynamic-type multi-soliton solutions to the fractional order nonlinear evolution equation in ocean engineering, Ain. Shams. Eng. J., 15 (2024), 102935. https://doi.org/10.1016/j.asej.2024.102935 doi: 10.1016/j.asej.2024.102935
[20]	D. Chou, S. M. Boulaaras, H. U. Rehman, I. Iqbal, A. Akram, N. Ullah, Additional investigation of the Biswas–Arshed equation to reveal optical soliton dynamics in birefringent fiber, Opt. Quant. Electron., 56 (2024), 705. https://doi.org/10.1007/s11082-024-06366-y doi: 10.1007/s11082-024-06366-y
[21]	D. Chou, H. U. Rehman, A. Amer, A. Amer, New solitary wave solutions of generalized fractional Tzitzéica-type evolution equations using Sardar sub-equation method, Opt. Quant. Electron., 55 (2023), 1148. https://doi.org/10.1007/s11082-023-05425-0 doi: 10.1007/s11082-023-05425-0
[22]	J. Ahmad, S. Rani, N. B. Turki, N. A. Shah, Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrödinger equation in optical fiber via an analytical method, Results. Phys., 52 (2023), 106761. https://doi.org/10.1016/j.asej.2024.102935 doi: 10.1016/j.asej.2024.102935
[23]	R. F. Zhang, S. Bilige, Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation, Nonlinear Dyn., 95 (2019), 1–8. https://doi.org/10.1007/s11071-018-04739-z doi: 10.1007/s11071-018-04739-z
[24]	R. F. Zhang, M. C. Li, Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations, Nonlinear Dyn., 108 (2022), 521–531. https://doi.org/10.1007/s11071-022-07207-x doi: 10.1007/s11071-022-07207-x
[25]	M. Zeynel, E. Yaşar, A new (3 + 1) dimensional Hirota bilinear equation: Periodic, rogue, bright and dark wave solutions by bilinear neural network method, J. Ocean. Eng. Sci., (2022), 2468–0133. https://doi.org/10.1016/j.joes.2022.04.017
[26]	J. G. Liu, W. H. Zhu, Y. K. Wu, G. H. Jin, Application of multivariate bilinear neural network method to fractional partial differential equations, Results. Phys., 47 (2023), 106341. https://doi.org/10.1016/j.rinp.2023.106341 doi: 10.1016/j.rinp.2023.106341
[27]	Z. H. Zhang, J. G. Liu, A fourth-order nonlinear equation studied by using a multivariate bilinear neural network method, Nonlinear Dyn., 112 (2024), 10229–10237. https://doi.org/10.1007/s11071-024-09567-y doi: 10.1007/s11071-024-09567-y
[28]	J. W. Wang, Y. Q. Liu, L. M. Yan, K. L. Han, L. B. Feng, R. F. Zhang, Fractional sub-equation neural networks (fSENNs) method for exact solutions of space-time fractional partial differential equations, Chaos, 35 (2025), 043110. https://doi.org/10.1063/5.0259937 doi: 10.1063/5.0259937
[29]	H. W. Zhang, R. F. Zhang, Q. R. Liu, A novel Multi-Modal neurosymbolic reasoning intelligent algorithm for BLMP equation, Chinese Phys. Lett., 42 (2025), 100002. https://doi.org/10.1088/0256-307X/42/10/100002 doi: 10.1088/0256-307X/42/10/100002
[30]	X. R. Xie, R. F. Zhang, Neural network-based symbolic calculation approach for solving the Korteweg–de Vries equation, Chaos, Solitons Fractals, 194 (2025), 116232. https://doi.org/10.1016/j.chaos.2025.116232 doi: 10.1016/j.chaos.2025.116232
[31]	Y. H. Yin, X. Lü, S. K. Li, L. X. Yang, Z. Y. Gao, Graph representation learning in the ITS: Car-following informed spatiotemporal network for vehicle trajectory predictions, IEEE Trans. Intell. Veh., 10 (2025), 2642–2652. https://doi.org/10.1109/TIV.2024.3381990 doi: 10.1109/TIV.2024.3381990
[32]	Y. H. Yin, X. Lü, S. K. Li, L. X. Yang, Z. Y. Gao, Car-following informed neural networks for real-time vehicle trajectory imputation and prediction, Transportmetrica A: Transport Sci., 22 (2026), 116232. https://doi.org/10.1080/23249935.2024.2374523 doi: 10.1080/23249935.2024.2374523
[33]	Y. Chen, X. Lü, H. Tian, R. H. Li, Physics-informed neural network for barrier option pricing in coupled financial quantitative system with varying interest rate and volatility, Eng. Anal. Bound. Elem., 180 (2025), 106457. https://doi.org/10.1016/j.enganabound.2025.106457 doi: 10.1016/j.enganabound.2025.106457
[34]	Y. Chen, X. Lü, X. J. Jia, H. Tian, Barrier option pricing and volatility surface predicting with an extended physics-informed neural network, Expert. Syst. Appl., 291 (2025), 128279. https://doi.org/10.1016/j.eswa.2025.128279 doi: 10.1016/j.eswa.2025.128279
[35]	M. H. Guo, X. Lü, Y. X. Jin, Extraction and reconstruction of variable-coefficient governing equations using Res-KAN integrating sparse regression, Physica D, 481 (2025), 134689. https://doi.org/10.1016/j.physd.2025.134689 doi: 10.1016/j.physd.2025.134689
[36]	C. Han, X. Lü, Variable coefficient-informed neural network for PDE inverse problem in fluid dynamics, Physica D, 472 (2025), 134362. https://doi.org/10.1016/j.physd.2024.134362 doi: 10.1016/j.physd.2024.134362
[37]	Y. B. Su, X. Lü, S. K. Li, L. X. Yang, Z. Y. Gao, Self-adaptive equation embedded neural networks for traffic flow state estimation with sparse data, Phys. Fluids., 36 (2024), 104127. https://doi.org/10.1063/5.0230757 doi: 10.1063/5.0230757
[38]	W. X. Ma, Generalized bilinear diﬀerential equations, Stud. Nonlinear Sci., 2 (2011), 140–144.
[39]	W. X. Ma, Bilinear equations, Bell polynomials and linear superposition principle, J. Phys.: Conf. Ser., 411 (2013), 012021. https://doi.org/10.1088/1742-6596/411/1/012021 doi: 10.1088/1742-6596/411/1/012021
[40]	Y. Guo, Y. Chen, J. Manafian, S. Malmir, K. H. Mahmoud, A. SA. Alsubaie, Exploring N-soliton solutions, multiple rogue wave and the linear superposition principle to the generalized hirota satsuma-ito equation, Sci. Rep., 14 (2024), 26171. https://doi.org/10.1038/s41598-024-74333-4 doi: 10.1038/s41598-024-74333-4
[41]	K. L. Liu, W. J. Yang, B. Tang, New solutions of the (3+1)-dimensional combined pKP-BKP equation, Phys. Lett. A, 560 (2025), 130941. https://doi.org/10.1016/j.physleta.2025.130941 doi: 10.1016/j.physleta.2025.130941
[42]	R. F. Zhang, M. C. Li, M. Albishari, F. C. Zheng, Z. Z. Lan, Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like equation, Appl. Math. Comput., 403 (2021), 126201. https://doi.org/10.1016/j.amc.2021.126201 doi: 10.1016/j.amc.2021.126201
[43]	D. P. Rijnsdorp, P. B. Smit, R. T. Guza, A nonlinear, non-dispersive energy balance for surfzone waves: Infragravity wave dynamics on a sloping beach, J. Fluid. Mech., 944 (2022), A45. https://doi.org/10.1017/jfm.2022.512 doi: 10.1017/jfm.2022.512

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)