Localized wave solutions of the second-order Alice-Bob Benjamin-Ono equation via KP-hierarchy reduction

Majid Madadi; Lanre Akinyemi; Majid Madadi; Lanre Akinyemi

doi:10.3934/math.2026437

AIMS Mathematics

2026, Volume 11, Issue 4: 10611-10637. doi: 10.3934/math.2026437

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Localized wave solutions of the second-order Alice-Bob Benjamin-Ono equation via KP-hierarchy reduction

Majid Madadi ^1,2,
Lanre Akinyemi ^{3
,
,}

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
2.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3.
Department of Mathematics, Prairie View A & M University, Texas, USA

Received: 15 February 2026 Revised: 22 March 2026 Accepted: 27 March 2026 Published: 17 April 2026
MSC : 35Q51, 37K10, 37K40

Alice-Bob systems offer a powerful framework for describing nonlocal correlations between spatially separated physical events. In this paper, we investigate nonlinear wave structures of the second-order Alice-Bob Benjamin-Ono system, an integrable nonlocal model for two-place correlated phenomena. Using the Kadomtsev-Petviashvili hierarchy reduction method, we derive explicit Gram-determinant breather solutions and construct higher-order breather and breather-soliton interactions. By combining block-determinant formulations with long-wave limits, we obtain semi-rational solutions that describe interactions among breathers, solitons, and lump waves. In addition, through bilinear theory and Schur polynomial techniques, we derive general higher-order rogue wave solutions in compact determinant form. These solutions exhibit rich geometric patterns, including triangular and polygonal rogue wave configurations, whose orientations are controlled by free complex parameters while preserving peak amplitudes. The results reveal diverse interaction dynamics and complex localized wave structures in the second-order system. This unified determinant framework significantly enriches the family of exact solutions and provides new insights into nonlinear wave dynamics in nonlocal integrable systems.
- Alice-Bob systems,
- Benjamin-Ono equation,
- Kadomtsev-Petviashvili-hierarchy reduction,
- rational solutions,
- rogue wave
Citation: Majid Madadi, Lanre Akinyemi. Localized wave solutions of the second-order Alice-Bob Benjamin-Ono equation via KP-hierarchy reduction[J]. AIMS Mathematics, 2026, 11(4): 10611-10637. doi: 10.3934/math.2026437

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Abstract

Alice-Bob systems offer a powerful framework for describing nonlocal correlations between spatially separated physical events. In this paper, we investigate nonlinear wave structures of the second-order Alice-Bob Benjamin-Ono system, an integrable nonlocal model for two-place correlated phenomena. Using the Kadomtsev-Petviashvili hierarchy reduction method, we derive explicit Gram-determinant breather solutions and construct higher-order breather and breather-soliton interactions. By combining block-determinant formulations with long-wave limits, we obtain semi-rational solutions that describe interactions among breathers, solitons, and lump waves. In addition, through bilinear theory and Schur polynomial techniques, we derive general higher-order rogue wave solutions in compact determinant form. These solutions exhibit rich geometric patterns, including triangular and polygonal rogue wave configurations, whose orientations are controlled by free complex parameters while preserving peak amplitudes. The results reveal diverse interaction dynamics and complex localized wave structures in the second-order system. This unified determinant framework significantly enriches the family of exact solutions and provides new insights into nonlinear wave dynamics in nonlocal integrable systems.

References

[1]	J. Cohen, J. A. Screen, J. C. Furtado, M. Barlow, D. Whittleston, D. Coumou, et al., Recent Arctic amplification and extreme mid-latitude weather, Nature Geosci., 7 (2014), 627–637. https://doi.org/10.1038/ngeo2234 doi: 10.1038/ngeo2234
[2]	J. A. Screen, Arctic amplification decreases temperature variance in northern mid-to high-latitudes, Nature Clim. Change, 4 (2014), 577–582. https://doi.org/10.1038/nclimate2268 doi: 10.1038/nclimate2268
[3]	J. M. Wallace, D. S. Gutzler, Teleconnections in the geopotential height field during the Northern Hemisphere winter, Mon. Weather Rev., 109 (1981), 784–812.
[4]	B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett., 116 (2016), 061102. https://doi.org/10.1103/PhysRevLett.116.061102 doi: 10.1103/PhysRevLett.116.061102
[5]	A. Aspect, P. Grangier, G. Roger, Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell's inequalities, Phys. Rev. Lett., 49 (1982), 91. https://doi.org/10.1103/PhysRevLett.49.91 doi: 10.1103/PhysRevLett.49.91
[6]	C. H. Bennett, G. Brassard, Proceedings of the IEEE international conference on computers, systems and signal processing, New York, 1984.
[7]	M. J. Ablowitz, Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013), 064105. https://doi.org/10.1103/PhysRevLett.110.064105 doi: 10.1103/PhysRevLett.110.064105
[8]	S. Y. Lou, F. Huang, Alice-Bob physics: coherent solutions of nonlocal KdV systems, Sci. Rep., 7 (2017), 869. https://doi.org/10.1038/s41598-017-00844-y doi: 10.1038/s41598-017-00844-y
[9]	C. C. Li, S. Y. Lou, M. Jia, Coherent structure of Alice-Bob modified Korteweg-de Vries equation, Nonlinear Dyn., 93 (2018), 1799–1808. https://doi.org/10.1007/s11071-017-3895-1 doi: 10.1007/s11071-017-3895-1
[10]	W. B. Wu, S. Y. Lou, Exact solutions of an Alice-Bob KP equation, Commun. Theor. Phys., 7 (2017), 629. https://doi.org/10.1088/0253-6102/71/6/629 doi: 10.1088/0253-6102/71/6/629
[11]	S. Y. Lou, Alice-Bob systems, Ps-Td-C principles and multi-soliton solutions, 2016, arXiv: 1603.03975.
[12]	S. Y. Lou, Alice-Bob systems, $\hat{\mathrm{P}}\text{-}\hat{\mathrm{T}}\text{-}\hat{\mathrm{C}}$ symmetry invariant and symmetry breaking soliton solutions, J. Math. Phys., 59 (2018), 083507. https://doi.org/10.1063/1.5051989 doi: 10.1063/1.5051989
[13]	T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559–592. https://doi.org/10.1017/S002211206700103X doi: 10.1017/S002211206700103X
[14]	H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Jpn., 39 (1975), 1082-1091. https://doi.org/10.1143/JPSJ.39.1082 doi: 10.1143/JPSJ.39.1082
[15]	A. S. Fokas, B. Fuchssteiner, The hierarchy of the Benjamin-Ono equation, Phys. Lett. A, 86 (1981), 341–345. https://doi.org/10.1016/0375-9601(81)90551-X doi: 10.1016/0375-9601(81)90551-X
[16]	H. Leblond, D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep., 523 (2013), 61–126. https://doi.org/10.1016/j.physrep.2012.10.006 doi: 10.1016/j.physrep.2012.10.006
[17]	W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. Van Immerzeele, A. Meerpoel, Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, J. Phys. A: Math. Gen., 19 (1986), 607. https://doi.org/10.1088/0305-4470/19/5/016 doi: 10.1088/0305-4470/19/5/016
[18]	A. Korpel, P. P. Banerjee, A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions, P. IEEE, 72 (1984), 1109–1130. https://doi.org/10.1109/PROC.1984.12992 doi: 10.1109/PROC.1984.12992
[19]	J. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris, 72 (1871), 755–759.
[20]	Y. Y. Li, H. C. Hu, Nonlocal symmetries and interaction solutions of the Benjamin-Ono equation, Appl. Math. Lett., 75 (2018), 18–23. https://doi.org/10.1016/j.aml.2017.06.012 doi: 10.1016/j.aml.2017.06.012
[21]	N. Taghizadeh, M. Mirzazadeh, F. Farahrooz, Exact soliton solutions for second-order Benjamin-Ono equation, Appl. Appl. Math., 6 (2011), 31.
[22]	Y. S. Özkan, Double reduction of second order Benjamin-Ono equation via conservation laws and the exact solutions, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 23 (2021), 210–223.
[23]	G. L. He, L. He, The application of trigonal curve theory to the second-order Benjamin-Ono hierarchy, Adv. Differ. Equ., 2014 (2014), 195. https://doi.org/10.1186/1687-1847-2014-195 doi: 10.1186/1687-1847-2014-195
[24]	W. Liu, High-order rogue waves of the Benjamin-Ono equation and the nonlocal nonlinear Schrödinger equation, Mod. Phys. Lett. B, 31 (2017), 1750269. https://doi.org/10.1142/S0217984917502694 doi: 10.1142/S0217984917502694
[25]	W. Tan, Z. D. Dai, Spatiotemporal dynamics of lump solution to the (1+1)-dimensional Benjamin-Ono equation, Nonlinear Dyn., 89 (2017), 2723–2728. https://doi.org/10.1007/s11071-017-3620-0 doi: 10.1007/s11071-017-3620-0
[26]	W. Shen, Z. Y. Ma, J. X. Fei, Q. Y. Zhu, J. C. Chen, Abundant symmetry-breaking solutions of the nonlocal Alice-Bob Benjamin-Ono system, Complexity, 2020 (2020), 2370970. https://doi.org/10.1155/2020/2370970 doi: 10.1155/2020/2370970
[27]	C. Gilson, F. Lambert, J. Nimmo, R. Willox, On the combinatorics of the Hirota D-operators, Proc. A, 452 (1996), 223–234. https://doi.org/10.1098/rspa.1996.0013 doi: 10.1098/rspa.1996.0013
[28]	L. Akinyemi, Robustness of localized waves under temporal modulation in higher-dimensional nonlinear evolution equations, Phys. Lett. A, 576 (2026), 131411. https://doi.org/10.1016/j.physleta.2026.131411 doi: 10.1016/j.physleta.2026.131411
[29]	M. Madadi, L. Akinyemi, K. Hosseini, Wronskian and rational solutions of a (2+1)-dimensional nonlinear evolution equation with time-dependent coefficients, Nonlinear Dyn., 114 (2026), 274. https://doi.org/10.1007/s11071-025-12151-7 doi: 10.1007/s11071-025-12151-7
[30]	P. Chatterjee, L. Mandi, The separation of one-soliton-shock to multi-soliton-shock of dustion acoustic wave using Lax pair and Darboux transformation of Burgers'equation, Phys. Fluids, 35 (2023), 087131. https://doi.org/10.1063/5.0160542 doi: 10.1063/5.0160542
[31]	F. J. Yu, L. Li, Soliton robustness, interaction and stability for a variable coefficients Schrödinger (VCNLS) equation with inverse scattering transformation, Chaos Soliton. Fract., 185 (2024), 115185. https://doi.org/10.1016/j.chaos.2024.115185 doi: 10.1016/j.chaos.2024.115185
[32]	M. Z. Raza, M. A. B. Iqbal, A. Khan, D. K. Almutairi, T. Abdeljawad, Soliton solutions of the (2+1)-dimensional Jaulent-Miodek evolution equation via effective analytical techniques, Sci. Rep., 15 (2025), 3495. https://doi.org/10.1038/s41598-025-87785-z doi: 10.1038/s41598-025-87785-z
[33]	L. Akinyemi, H. Rezazadeh, Q. H. Shi, M. Inc, M. M. Khater, H. Ahmad, et al., New optical solitons of perturbed nonlinear Schrödinger-Hirota equation with spatio-temporal dispersion, Res. Phys., 29 (2021), 104656. https://doi.org/10.1016/j.rinp.2021.104656 doi: 10.1016/j.rinp.2021.104656
[34]	M. Madadi, E. Asadi, Rogue waves from Kadomtsev-Petviashvili reduction: systematic derivation and dynamics in an extended Kadomtsev-Petviashvili equation, J. Appl. Math. Comput., 72 (2026), 72. https://doi.org/10.1007/s12190-025-02733-4 doi: 10.1007/s12190-025-02733-4
[35]	E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations-Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci., 18 (1982), 1077–1110. https://doi.org/10.2977/prims/1195183297 doi: 10.2977/prims/1195183297
[36]	M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, North-Holland Mathematics Studies, 81 (1983), 259–271. https://doi.org/10.1016/S0304-0208(08)72096-6 doi: 10.1016/S0304-0208(08)72096-6
[37]	R. Hirota, The direct method in soliton theory, Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543043
[38]	W. Liu, A.-M. Wazwaz, X. X. Zheng, Families of semi-rational solutions to the Kadomtsev-Petviashvili Ⅰ equation, Commun. Nonlinear Sci., 67 (2019), 480–491. https://doi.org/10.1016/j.cnsns.2018.07.020 doi: 10.1016/j.cnsns.2018.07.020
[39]	M. J. Ablowitz, J. Satsuma, Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys., 19 (1978), 2180–2186. https://doi.org/10.1063/1.523550 doi: 10.1063/1.523550
[40]	Y. Ohta, J. K. Yang, General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation, Proc. A, 468 (2012), 1716–1740. https://doi.org/10.1098/rspa.2011.0640 doi: 10.1098/rspa.2011.0640
[41]	B. Yang, J. K. Yang, General rogue waves in the Boussinesq equation, J. Phys. Soc. Jpn., 89 (2020), 024003. https://doi.org/10.7566/JPSJ.89.024003 doi: 10.7566/JPSJ.89.024003
[42]	M. Madadi, M. Inc, Determinantal solutions to the (3+1)-dimensional Painlevé-type evolution equation; higher-order rogue and soliton waves, Wave Motion, 139 (2025), 103624. https://doi.org/10.1016/j.wavemoti.2025.103624 doi: 10.1016/j.wavemoti.2025.103624
[43]	Y. K. Liu, B. Li, Dynamics of rogue waves on multisoliton background in the Benjamin-Ono equation, Pramana, 88 (2017), 57. https://doi.org/10.1007/s12043-016-1361-0 doi: 10.1007/s12043-016-1361-0
[44]	W. B. Ma, B. Sudao, H. B. Shao, Multiple rogue wave solutions of the (1+1)-dimensional Benjamin-Ono equation, Phys. Scr., 99 (2024), 065219. https://doi.org/10.1088/1402-4896/ad40d9 doi: 10.1088/1402-4896/ad40d9
[45]	P. A. Clarkson, E. L. Mansfield, The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity, 16 (2003), R1. https://doi.org/10.1088/0951-7715/16/3/201 doi: 10.1088/0951-7715/16/3/201
[46]	B. Yang, J. K. Yang, Rogue wave patterns in the nonlinear Schrödinger equation, Physica D, 419 (2021), 132850. https://doi.org/10.1016/j.physd.2021.132850 doi: 10.1016/j.physd.2021.132850
[47]	P. Huang, Y. K. Wang, D. Zhou, Rogue wave patterns of Newell type long-wave-short-wave model, Chaos Soliton. Fract., 175 (2023), 114038. https://doi.org/10.1016/j.chaos.2023.114038 doi: 10.1016/j.chaos.2023.114038
[48]	F. Z. Zheng, Z. Y. Qin, G. Mu, T. Y. Wang, On the Role of AM Polynomial Roots in the Determination of Rogue Wave Patterns in Benjamin-ono Equation, Nonlinear Dyn., 113 (2025), 21703–21723. https://doi.org/10.1007/s11071-025-11273-2 doi: 10.1007/s11071-025-11273-2
[49]	L. K. Yunkai, L. Biao, A.-M. Wazwaz, Novel high-order breathers and rogue waves in the Boussinesq equation via determinants, Math. Meth. Appl. Sci., 43 (2020), 3701–3715. https://doi.org/10.1002/mma.6148 doi: 10.1002/mma.6148
[50]	P. Dong, Z.-Y. Ma, J.-X. Fei, W.-P. Cao, The shifted parity and delayed time-reversal symmetry-breaking solutions for the (1+1)-dimensional Alice–Bob Boussinesq equation, Front. Phys., 11 (2023), 1137999. https://doi.org/10.3389/fphy.2023.1137999 doi: 10.3389/fphy.2023.1137999

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