Novel solutions and stability analysis for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation using an improved generalized Riccati equation mapping technique

Noha M. Kamel; Hamdy M. Ahmed; Wafaa B. Rabie; Noha M. Kamel; Hamdy M. Ahmed; Wafaa B. Rabie

doi:10.3934/math.2026851

AIMS Mathematics

2026, Volume 11, Issue 7: 20956-20985. doi: 10.3934/math.2026851

Previous Article Next Article

Research article Special Issues

Novel solutions and stability analysis for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation using an improved generalized Riccati equation mapping technique

1.
Department of Physics and Engineering Mathematics, Faculty of Engineering, Ain Shams University, Cairo, Egypt
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El-Shorouk Academy, Cairo, Egypt
3.
Department of Mathematics, Faculty of Science, Luxor University, Taiba Luxor, Egypt

Received: 27 April 2026 Revised: 05 July 2026 Accepted: 10 July 2026 Published: 16 July 2026
MSC : 35C07, 35C08, 35C09

This study retrieved novel abundant exact solutions for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation using the improved generalized Riccati equation mapping technique. The B-type Kadomtsev–Petviashvili equation is of high importance in modeling nonlinear wave propagation in weakly dispersive media in fluid dynamics and plasmas. The derived solutions included singular, bright, and bright-dark solitons, as well as hyperbolic, rational, exponential, and singular periodic solutions. The validation of the derived solutions was confirmed by direct substitution into the B-type Kadomtsev–Petviashvili equation under study. A detailed stability analysis was performed utilizing the linear stability analysis concept. The obtained stability results indicated the presence of neutral stability and the non-existence of modulation instability for any choice of the system parameters. The wave dynamics of some derived exact solutions are illustrated by two-dimensional, three-dimensional, and density plot graphics. Additionally, the results of stability analysis are illustrated by two-dimensional and three-dimensional graphics.
Citation: Noha M. Kamel, Hamdy M. Ahmed, Wafaa B. Rabie. Novel solutions and stability analysis for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation using an improved generalized Riccati equation mapping technique[J]. AIMS Mathematics, 2026, 11(7): 20956-20985. doi: 10.3934/math.2026851

Related Papers:

Abstract

This study retrieved novel abundant exact solutions for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation using the improved generalized Riccati equation mapping technique. The B-type Kadomtsev–Petviashvili equation is of high importance in modeling nonlinear wave propagation in weakly dispersive media in fluid dynamics and plasmas. The derived solutions included singular, bright, and bright-dark solitons, as well as hyperbolic, rational, exponential, and singular periodic solutions. The validation of the derived solutions was confirmed by direct substitution into the B-type Kadomtsev–Petviashvili equation under study. A detailed stability analysis was performed utilizing the linear stability analysis concept. The obtained stability results indicated the presence of neutral stability and the non-existence of modulation instability for any choice of the system parameters. The wave dynamics of some derived exact solutions are illustrated by two-dimensional, three-dimensional, and density plot graphics. Additionally, the results of stability analysis are illustrated by two-dimensional and three-dimensional graphics.

References

[1]	H. Shao, S. Bilige, Dynamical asymptotic analysis to a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation: The superposition formulas of rational solutions and interaction solutions under the bilinear vector method, Math. Comput. Simulat., 240 (2026), 938–952. http://doi.org/10.1016/j.matcom.2025.07.058 doi: 10.1016/j.matcom.2025.07.058
[2]	H. Riaz, A. Farooq, Exact solutions and nonlinear wave interactions in a non-commutative coupled dispersionless system with variable coefficients, Nonlinear Dyn., 113 (2025), 35125–35139. https://doi.org/10.1007/s11071-025-11833-6 doi: 10.1007/s11071-025-11833-6
[3]	H. W. A. Riaz, Quasi-Wronskian solitons in the non-commutative kuralay-IIA equation: analysis and simulations, Phys. Scr., 100 (2025), 025239. http://doi.org/10.1088/1402-4896/ada4ee doi: 10.1088/1402-4896/ada4ee
[4]	H. W. A. Riaz, J. Lin, Darboux transformation for a semi-discrete matrix coupled dispersionless system, Appl. Math. Lett., 158 (2024), 109217. http://doi.org/10.1016/j.aml.2024.109217 doi: 10.1016/j.aml.2024.109217
[5]	H. W. A. Riaz, A. Farooq, Hybrid analytical and neural-network approaches to the non-local short pulse equation, Eur. Phys. J. C, 85 (2025), 895. https://doi.org/10.1140/epjc/s10052-025-14634-8 doi: 10.1140/epjc/s10052-025-14634-8
[6]	W. Hereman, Shallow water waves and solitary waves, In: Mathematics of complexity and dynamical systems, New York: Springer, 2012, 1520–1532. http://doi.org/10.1007/978-1-4614-1806-1_96
[7]	W. B. Rabie, H. M. Ahmed, Diverse exact and solitary wave solutions to new extended KdV6 equation using IM extended tanh-function technique, Pramana, 98 (2024), 81. http://doi.org/10.1007/s12043-024-02767-6 doi: 10.1007/s12043-024-02767-6
[8]	S. A. Alkhateeb, S. Hussain, W. Albalawi, S. A. El-Tantawy, E. I. El-Awady, Dissipative Kawahara ion-acoustic solitary and cnoidal waves in a degenerate magnetorotating plasma, J. Taibah Univ. Sci., 17 (2023), 2187606. http://doi.org/10.1080/16583655.2023.2187606 doi: 10.1080/16583655.2023.2187606
[9]	M. S. Tariq, W. Masood, M. Siddiq, S. Asghar, B. M. Alotaibi, S. M. E. Ismaeel, et al., Bäcklund transformation for analyzing a cylindrical Korteweg–de Vries equation and investigating multiple soliton solutions in a plasma, Phys. Fluids, 35 (2023), 103105. http://doi.org/10.1063/5.0166075 doi: 10.1063/5.0166075
[10]	B. B. Kadomtsev, V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Dokl. Akad. Nauk SSSR, 192 (1970), 753–756.
[11]	Y. Zhang, L. Gui, Inverse scattering transform for the two-dimensional B-type Kadomtsev–Petviashvili equation, J. Comput. Appl. Math., 474 (2026), 117011. http://doi.org/10.1016/j.cam.2025.117011 doi: 10.1016/j.cam.2025.117011
[12]	A. Rahman, S. A. Khan, A. Qamar, Kadomtsev–Petviashvili equation for solitary waves in warm dense astrophysical electron-positron-ion plasmas, Astrophys. Space Sci., 347 (2013), 119–127. http://doi.org/10.1007/s10509-013-1501-7 doi: 10.1007/s10509-013-1501-7
[13]	B. Sahu, N. K. Ghosh, Kadomstev-Petviashvili solitons in quantum plasmas, Astrophys. Space Sci., 343 (2013), 289–292. http://doi.org/10.1007/s10509-012-1246-8 doi: 10.1007/s10509-012-1246-8
[14]	H. Berjamin, M. Destrade, G. Saccomandi, The KP equation of plane elastodynamics, SIAM J. Appl. Math., 85 (2025), 1458–1474. http://doi.org/10.1137/24M1710036 doi: 10.1137/24M1710036
[15]	S. V. Bulanov, P. V. Sasorov, F. Pegoraro, H. Kadlecová, S. S. Bulanov, T. Z. Esirkepov, et al., Electromagnetic solitons in quantum vacuum, Phys. Rev. D, 101 (2020), 016016. https://doi.org/10.1103/physrevd.101.016016 doi: 10.1103/physrevd.101.016016
[16]	K. K. Ali, M. S. Mohamed, M. Maneea, A novel approach to $q$-fractional partial differential equations: Unraveling solutions through semi-analytical methods, AIMS Mathematics, 9 (2024), 33442–33466. http://doi.org/10.3934/math.20241596 doi: 10.3934/math.20241596
[17]	Z. Li, M. Wu, E. Hussain, Y. Yildirim, Exactly explicit solutions of (2+1)-dimensional conformable fractional diffusive Predator–Prey model via neural networks method, Front. Phys., 14 (2026), 1824530. http://doi.org/10.3389/fphy.2026.1824530 doi: 10.3389/fphy.2026.1824530
[18]	Z. Li, E. Hussain, Bifurcation analysis and chaotic behaviors of and a traveling-wave solution to the Zhiber–Shabat equation with a truncated M-fractional derivative, Fractal Fract., 10 (2026), 335. https://doi.org/10.3390/fractalfract10050335 doi: 10.3390/fractalfract10050335
[19]	K. K. Ali, A. Wazwaz, M. Maneea, Efficient solutions for fractional Tsunami shallow-water mathematical model: A comparative study via semi analytical techniques, Chaos Soliton. Fract., 178 (2024), 114347. http://doi.org/10.1016/j.chaos.2023.114347 doi: 10.1016/j.chaos.2023.114347
[20]	N. Raza, A. Jhangeer, Z. Amjad, B. Rani, D. Baleanu, Exploring soliton dynamics and wave interactions in an extended Kadomtsev-Petviashvili-Boussinesq equation, Ain Shams Eng. J., 16 (2025), 103395. http://doi.org/10.1016/j.asej.2025.103395 doi: 10.1016/j.asej.2025.103395
[21]	E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Operator approach to the Kadomtsev–Petviashvili equation-transformation groups for soliton equations Ⅲ, J. Phys. Soc. Jap., 50 (1981), 3806–3812. https://doi.org/10.1143/JPSJ.50.3806 doi: 10.1143/JPSJ.50.3806
[22]	E. Date, M. Kashiwara, T. Miwa, Vertex operators and $\tau$ functions transformation groups for soliton equations, Ⅱ, Proc. Japan Acad. Ser. A Math. Sci., 57 (1981), 387–392. http://doi.org/10.3792/pjaa.57.387 doi: 10.3792/pjaa.57.387
[23]	E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations: Ⅳ. A new hierarchy of soliton equations of KP-type, Physica D, 4 (1982), 343–365. http://doi.org/10.1016/0167-2789(82)90041-0 doi: 10.1016/0167-2789(82)90041-0
[24]	X. Zhang, L. Wang, W. Chen, X. Yao, X. Wang, Y. Zhao, Dynamics of transformed nonlinear waves in the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation Ⅰ: Transitions mechanisms, Commun. Nonlinear Sci., 105 (2022), 106070. http://doi.org/10.1016/j.cnsns.2021.106070 doi: 10.1016/j.cnsns.2021.106070
[25]	L. Feng, S. Tian, X. Wang, T. Zhang, Rogue waves, homoclinic breather waves and soliton waves for the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation, Appl. Math. Lett., 65 (2017), 90–97. http://doi.org/10.1016/j.aml.2016.10.009 doi: 10.1016/j.aml.2016.10.009
[26]	A. Wazwaz, Breather wave solutions for an integrable (3+1)-dimensional combined pKP–BKP equation, Chaos Soliton. Fract., 182 (2024), 114886. http://doi.org/10.1016/j.chaos.2024.114886 doi: 10.1016/j.chaos.2024.114886
[27]	Y. Yuan, X. Zhao, Resonant solitons of the B-type Kadomtsev-Petviashvili equation, Phys. Lett. A, 458 (2023), 128592. https://doi.org/10.1016/j.physleta.2022.128592 doi: 10.1016/j.physleta.2022.128592
[28]	M. H. Bashar, M. A. Mannaf, A. Rahman, M. S. Islam, H. Z. Mawa, P. Akter, Exploration of soliton solutions and bifurcation analysis in fluid dynamics governed by M fractional (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation, Eng. Rep., 8 (2026), e70642. http://doi.org/10.1002/eng2.70642 doi: 10.1002/eng2.70642
[29]	K. Wang, J. Liu, J. Si, G. Wang, Nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid mechanics, Axioms, 12 (2023), 95. http://doi.org/10.3390/axioms12010095 doi: 10.3390/axioms12010095
[30]	C. Hu, B. Tian, X. Wu, Z. Du, X. Zhao, Lump wave-soliton and rogue wave-soliton interactions for a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in a fluid, Chinese J. Phys., 56 (2018), 2395–2403. https://doi.org/10.1016/j.cjph.2018.06.021 doi: 10.1016/j.cjph.2018.06.021
[31]	A. Wazwaz, Two B-type Kadomtsev–Petviashvili equations of (2+1) and (3+1) dimensions: Multiple soliton solutions, rational solutions and periodic solutions, Comput. Fluids, 86 (2013), 357–362. http://doi.org/10.1016/j.compfluid.2013.07.028 doi: 10.1016/j.compfluid.2013.07.028
[32]	J. Tu, S. Tian, M. Xu, P. Ma, T. Zhang, On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics, Comput. Math. Appl., 72 (2016), 2486–2504. http://doi.org/10.1016/j.camwa.2016.09.003 doi: 10.1016/j.camwa.2016.09.003
[33]	F. Yuan, J. He, Y. Cheng, The degeneration of the breathers for the BKP equation, Chinese J. Phys., 71 (2021), 190–201. http://doi.org/10.1016/j.cjph.2020.02.009 doi: 10.1016/j.cjph.2020.02.009
[34]	X. Wu, B. Tian, H. Chai, Y. Sun, Rogue waves and lump solutions for a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid mechanics, Mod. Phys. Lett. B, 31 (2017), 1750122. http://doi.org/10.1142/S0217984917501226 doi: 10.1142/S0217984917501226
[35]	X. Yan, S. Tian, X. Wang, T. Zhang, Solitons to rogue waves transition, lump solutions and interaction solutions for the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics, Int. J. Comput. Math., 96 (2018), 1839–1848. https://doi.org/10.1080/00207160.2018.1535708 doi: 10.1080/00207160.2018.1535708
[36]	C. Hu, B. Tian, X. Du, C. Zhang, Bright/dark breather-soliton, lump wave-soliton and rogue wave-soliton interactions for a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in fluid mechanics, Nonlinear Dyn., 108 (2022), 1585–1598. http://doi.org/10.1007/s11071-022-07204-0 doi: 10.1007/s11071-022-07204-0
[37]	A. M. Wazwaz, Study on a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in nonlinear physics: Multiple soliton solutions, lump solutions, and breather wave solutions, Chaos Soliton. Fract., 189 (2024), 115668. http://doi.org/10.1016/j.chaos.2024.115668 doi: 10.1016/j.chaos.2024.115668
[38]	M. Ekici, On solving the (3+1)-dimensional B-type Kadomtsev-Petviashvili equation by using two efficient method, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 29 (2025), 54–61. http://doi.org/10.19113/sdufenbed.1611725 doi: 10.19113/sdufenbed.1611725
[39]	H. Alsaud, M. N. Rafiq, M. H. Rafiq, A novel investigation of the extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation: Analysis and simulations, Int. J. Theor. Phys., 64 (2025), 216. http://doi.org/10.1007/s10773-025-06081-8 doi: 10.1007/s10773-025-06081-8
[40]	H. Ur Rehman, A. R. Seadawy, S. Razzaq, S. T. R. Rizvi, Optical fiber application of the improved generalized Riccati equation mapping method to the perturbed nonlinear Chen-Lee-Liu dynamical equation, Optik, 290 (2023), 171309. https://doi.org/10.1016/j.ijleo.2023.171309 doi: 10.1016/j.ijleo.2023.171309
[41]	M. Rani, N. Ahmed, S. S. Dragomir, S. T. Mohyud-Din, I. Khan, K. S. Nisar, Some newly explored exact solitary wave solutions to nonlinear inhomogeneous Murnaghan's rod equation of fractional order, J. Taibah Univ. Sci., 15 (2021), 97–110. https://doi.org/10.1080/16583655.2020.1841472 doi: 10.1080/16583655.2020.1841472
[42]	B. Li, Y. Chen, H. Xuan, H. Zhang, Generalized Riccati equation expansion method and its application to the $(3+1)$-dimensional Jumbo–Miwa equation, Appl. Math. Comput., 152 (2004), 581–595. http://doi.org/10.1016/S0096-3003(03)00578-2 doi: 10.1016/S0096-3003(03)00578-2
[43]	N. M. Kamel, H. M. Ahmed, W. B. Rabie, Exploring novel soliton, wave solutions, and modulation instability analysis for the (3+1)-dimensional KP-SKR equation using the improved generalized Riccati equation mapping approach, Sci. Rep., 15 (2025), 37741. http://doi.org/10.1038/s41598-025-22030-1 doi: 10.1038/s41598-025-22030-1
[44]	Y. Salathiel, Y. Amadou, G. Betchewe, S. Y. Doka, K. T. Crepin, Soliton solutions and traveling wave solutions for a discrete electrical lattice with nonlinear dispersion through the generalized Riccati equation mapping method, Nonlinear Dyn., 87 (2017), 2435–2443. http://doi.org/10.1007/s11071-016-3201-7 doi: 10.1007/s11071-016-3201-7
[45]	S. Bibi, N. Ahmed, I. Faisal, S. T. Mohyud-Din, M. Rafiq, U. Khan, Some new solutions of the Caudrey–Dodd–Gibbon (CDG) equation using the conformable derivative, Adv. Differ. Equ., 2019 (2019), 89. http://doi.org/10.1186/s13662-019-2030-7 doi: 10.1186/s13662-019-2030-7
[46]	M. Bilal, Y. M. Alawaideh, S. Ur Rehman, I. Popa, B. M. Al-khamiseh, H. Ghannam, Complex soliton wave patterns of Gross–Pitaevskii systems: Application in quantum and optical engineering, Sci. Rep., 15 (2025), 44265. http://doi.org/10.1038/s41598-025-27902-0 doi: 10.1038/s41598-025-27902-0
[47]	A. Ankiewicz, M. Bokaeeyan, N. Akhmediev, Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation, Phys. Rev. E, 99 (2019), 050201. http://doi.org/10.1103/PhysRevE.99.050201 doi: 10.1103/PhysRevE.99.050201
[48]	N. Alam, M. S. Ullah, T. A. Nofal, H. M. Ahmed, K. K. Ahmed, M. A. Al-Nahhas, Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis, Nonlinear Eng., 13 (2024), 20240034. https://doi.org/10.1515/nleng-2024-0034 doi: 10.1515/nleng-2024-0034
[49]	N. M. Kamel, H. M. Ahmed, W. B. Rabie, Solitons unveilings and modulation instability analysis for sixth-order coupled nonlinear Schrödinger equations in fiber bragg gratings, AIMS Mathematics, 10 (2025), 6952–6980. http://doi.org/10.3934/math.2025318 doi: 10.3934/math.2025318
[50]	N. M. Kamel, H. M. Ahmed, W. B. Rabie, Unveiling solitons and traveling wave solutions with modulation instability analysis for extended (3+1)-dimensional seventh-order standard Ito and modified Ito equations, Ain Shams Eng. J., 16 (2025), 103475. http://doi.org/10.1016/j.asej.2025.103475 doi: 10.1016/j.asej.2025.103475
[51]	M. S. Ghayad, H. M. Ahmed, N. M. Badra, W. B. Rabie, Generation of multi-form exact wave solutions and linear stability analysis in the generalized (3+1)-D P-type plasma system using a modified extended mapping technique, Sci. Rep., 16 (2026), 15173. http://doi.org/10.1038/s41598-026-49817-0 doi: 10.1038/s41598-026-49817-0
[52]	A. E. Rateb, H. M. Ahmed, A. Darwish, M. Ammar, W. B. Rabie, Analytical wave families and stability dynamics in a modified complex Ginzburg–Landau model via the modified extended direct algebraic method, Sci. Rep., 16 (2026), 7485. http://doi.org/10.1038/s41598-026-37824-0 doi: 10.1038/s41598-026-37824-0
[53]	F. M. Omar, M. A. S. Murad, S. S. Mahmood, S. Malik, T. Radwan, Optical solitons, bifurcation, and chaos in the nonlinear conformable Schrödinger equation with group velocity dispersion coefficients and second-order spatiotemporal terms, Sci. Rep., 15 (2025), 20073. http://doi.org/10.1038/s41598-025-04387-5 doi: 10.1038/s41598-025-04387-5
[54]	T. Li, L. Shen, Evolution of wave characteristics during wind-wave generation, 2022, arXiv: 2208.02489. https://doi.org/10.48550/arXiv.2208.02489
[55]	R. Wu, A. Guo, S. Zhu, J. Liu, Growth of wind-driven waves under uniform currents, Coast. Eng., 198 (2025), 104704. http://doi.org/10.1016/j.coastaleng.2025.104704 doi: 10.1016/j.coastaleng.2025.104704
[56]	B. D. V. Campos, Characterization of rational solutions of a KdV-like equation, Math. Comput. Simulat., 201 (2022), 396–416. http://doi.org/10.1016/j.matcom.2022.05.022 doi: 10.1016/j.matcom.2022.05.022

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)