Rogue waves are rare, high-amplitude events that can appear unexpectedly in many physical settings, yet the mechanisms behind their complex structures, especially at higher orders, are not fully understood. In this work, we study higher-order rogue waves in the $ (3+1) $-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH–KP) equation. Using the Kadomtsev–Petviashvili (KP) hierarchy reduction method, we obtain general rational solutions and reveal how their internal parameters shape a wide range of wave patterns. A central result of our study is an analytical characterization of the higher-order rogue-wave patterns through the root geometry of the Yablonskii–Vorob'ev (YV) polynomial hierarchy, which traditionally arise in the theory of Painlevé equations. In the regime where one internal parameter becomes large, we derive an asymptotic decomposition of the exact solution into fundamental rogue-wave components and obtain explicit formulas that map the relevant YV roots to their physical locations. This provides a quantified extension of the previously known YV pattern mechanism to the reduced (3 + 1)-dimensional gCH–KP equation and avoids the direct evaluation of increasingly large determinant representations. Numerical tests confirm these predictions and demonstrate a unified algebraic–geometric mechanism for the formation of higher-order rogue wave patterns.
Citation: Majid Madadi, Kamyar Hosseini, Evren Hincal, Sekson Sirisubtawee. Rogue-wave patterns in the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili equation: true vs. predicted[J]. AIMS Mathematics, 2026, 11(10): 33994-34038. doi: 10.3934/math.20261326
Rogue waves are rare, high-amplitude events that can appear unexpectedly in many physical settings, yet the mechanisms behind their complex structures, especially at higher orders, are not fully understood. In this work, we study higher-order rogue waves in the $ (3+1) $-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH–KP) equation. Using the Kadomtsev–Petviashvili (KP) hierarchy reduction method, we obtain general rational solutions and reveal how their internal parameters shape a wide range of wave patterns. A central result of our study is an analytical characterization of the higher-order rogue-wave patterns through the root geometry of the Yablonskii–Vorob'ev (YV) polynomial hierarchy, which traditionally arise in the theory of Painlevé equations. In the regime where one internal parameter becomes large, we derive an asymptotic decomposition of the exact solution into fundamental rogue-wave components and obtain explicit formulas that map the relevant YV roots to their physical locations. This provides a quantified extension of the previously known YV pattern mechanism to the reduced (3 + 1)-dimensional gCH–KP equation and avoids the direct evaluation of increasingly large determinant representations. Numerical tests confirm these predictions and demonstrate a unified algebraic–geometric mechanism for the formation of higher-order rogue wave patterns.
| [1] |
E. A. Az-Zo'bi, An approximate analytic solution for isentropic flow by an inviscid gas model, Arch. Mech., 66 (2014), 203–212. http://doi.org/10.24423/aom.1209 doi: 10.24423/aom.1209
|
| [2] |
X. Li, Y. Wang, X. Ma, L. Chen, M. D. V. Francisco, X. Zhang, PIC-MCC simulation of single pulse DBD discharge evolution process, Plasma Sources Sci. Technol., 35 (2026), 015021. http://doi.org/10.1088/1361-6595/ae33ed doi: 10.1088/1361-6595/ae33ed
|
| [3] |
D. Guo, Z. Zhang, J. Liu, J. Zhang, Y. Lin, Multi-horizon flight trajectory prediction enabled by time-frequency wavelet transform, Nat. Commun., 17 (2026), 633. https://doi.org/10.1038/s41467-025-67399-9 doi: 10.1038/s41467-025-67399-9
|
| [4] |
K. K. Ahmed, S. Z. Majid, M. I. Asjad, M. S. Alqarni, T. Radwan, Diverse soliton solutions for nonlinear higher-order Sasa-Satsuma equation with its dynamical analysis, Bound. Value Probl., 2026 (2026), 34. https://doi.org/10.1186/s13661-025-02215-3 doi: 10.1186/s13661-025-02215-3
|
| [5] |
H. F. Ismael, S. T. Abdulkadir, H. R. Nabi, S. F. Ahmed, Novel nonlinear dynamical solutions to the (2+1)-dimensional variable coefficients equation arise in oceanography, Eng. Rep., 7 (2025), e70240. https://doi.org/10.1002/eng2.70240 doi: 10.1002/eng2.70240
|
| [6] |
R. Su, J. Zhao, S. Bao, H. Fu, H. Wang, X. Zhang, et al., Deep learning approach for internal tide prediction based on China Ocean Reanalysis data, Ocean Eng., 340 (2025), 122348. https://doi.org/10.1016/j.oceaneng.2025.122348 doi: 10.1016/j.oceaneng.2025.122348
|
| [7] |
Y. Xia, B. Zhang, C. Tao, L. Wang, Fully nonlinear simulation of the hydrodynamic performance of a submerged cylindrical wave energy converter in the presence of current, J. Mar. Sci. Eng., 13 (2025), 1763. https://doi.org/10.3390/jmse13091763 doi: 10.3390/jmse13091763
|
| [8] |
J. Shi, Y. Chen, X. Pang, M. R. Belić, D. Mihalache, Y. Liu, et al., Gap solitons and truncated nonlinear Bloch waves in combined linear and quintic nonlinear lattices, Opt. Express, 34 (2026), 15864–15876. https://doi.org/10.1364/OE.587649 doi: 10.1364/OE.587649
|
| [9] |
E. Hussain, S. A. A. Shah, A. Bariq, Z. Li, M. R. Ahmad, A. E. Ragab, et al., Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques, Sci. Rep., 14 (2024), 13520. https://doi.org/10.1038/s41598-024-60732-0 doi: 10.1038/s41598-024-60732-0
|
| [10] |
Z. Z. Lan, $N$-soliton solutions, Bäcklund transformation and Lax Pair for a generalized variable-coefficient cylindrical Kadomtsev–Petviashvili equation, Appl. Math. Lett., 158 (2024), 109239. https://doi.org/10.1016/j.aml.2024.109239 doi: 10.1016/j.aml.2024.109239
|
| [11] |
L. Cheng, H. Lu, B. Guo, Q. Liang, P. Han, Z. Li, et al., Fast prediction of underwater vehicle motion under internal solitary waves, Phys. Fluids, 38 (2026), 027115. https://doi.org/10.1063/5.0315969 doi: 10.1063/5.0315969
|
| [12] |
Z. Li, H. Hu, C. Chen, Z. Wang, Z. Xie, P. Du, Internal solitary waves within background shear currents in a three-layer fluid system, Acta Mech. Sin., 42 (2026), 351039. https://doi.org/10.1007/s10409-026-51039-x doi: 10.1007/s10409-026-51039-x
|
| [13] |
E. A. Az-Zo'bi, New kink solutions for the van der Waals p-system, Math. Meth. Appl. Sci., 42 (2019), 6216–6226. https://doi.org/10.1002/mma.5717 doi: 10.1002/mma.5717
|
| [14] |
I. Siddique, R. M. Zulqarnain, M. A. Akbar, S. Ali, Lump collision dynamics in the generalized $(3+1)$-dimensional variable coefficient B-type Kadomtsev-Petviashvili equation, Phys. Scr., 100 (2025), 015231. https://doi.org/10.1088/1402-4896/ad986c doi: 10.1088/1402-4896/ad986c
|
| [15] |
A. Waheed, M. Awais, M. A. Z. Raja, M. Y. Malik, A. S. Alqahtani, Peakon and solitary wave solutions of the LAX equation: neuro computing procedure, Int. Commun. Heat Mass Transf., 152 (2024), 107321. https://doi.org/10.1016/j.icheatmasstransfer.2024.107321 doi: 10.1016/j.icheatmasstransfer.2024.107321
|
| [16] | S. Hayer, O. J. Andersen, Freak waves: rare realizations of a typical population or typical realizations of a rare population, The Tenth International Offshore and Polar Engineering Conference, 2000. Available from: https://onepetro.org/ISOPEIOPEC/proceedings-abstract/ISOPE00/All-ISOPE00/ISOPE-I-00-233/7014. |
| [17] |
A. Ankiewicz, N. Devine, N. Akhmediev, Are rogue waves robust against perturbations, Phys. Lett. A, 373 (2009), 3997–4000. https://doi.org/10.1016/j.physleta.2009.08.053 doi: 10.1016/j.physleta.2009.08.053
|
| [18] | P. A. E. M. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr., 33 (2003), 863–884. |
| [19] |
S. Khondaker, A. Mannan, N. A. Chowdhury, A. A. Mamun, Rogue waves in multi-pair plasma medium, Contrib. Plasma Phys., 59 (2019), e201800125. https://doi.org/10.1002/ctpp.201800125 doi: 10.1002/ctpp.201800125
|
| [20] |
D. H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, ANZIAM J., 25 (1983), 16–43. https://doi.org/10.1017/S0334270000003891 doi: 10.1017/S0334270000003891
|
| [21] |
N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E, 80 (2009), 026601. https://doi.org/10.1103/PhysRevE.80.026601 doi: 10.1103/PhysRevE.80.026601
|
| [22] |
W. Q. Peng, Y. Chen, Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schrödinger equation via inverse scattering transform method, Math. Methods Appl. Sci., 46 (2023), 126–141. https://doi.org/10.1002/mma.8498 doi: 10.1002/mma.8498
|
| [23] |
G. H. Yang, X. H. Zhao, Z. Z. Lan, Bäcklund transformations and nonlinear wave solutions for an extended (2+1)-dimensional Kadomtsev–Petviashvili equation, Eur. Phys. J. Spec. Top., 234 (2025), 5429–5440. https://doi.org/10.1140/epjs/s11734-025-01853-z doi: 10.1140/epjs/s11734-025-01853-z
|
| [24] |
P. A. Clarkson, E. Dowie, Rational solutions of the Boussinesq equation and applications to rogue waves, Trans. Math. Appl., 1 (2017), tnx003. https://doi.org/10.1093/imatrm/tnx003 doi: 10.1093/imatrm/tnx003
|
| [25] |
M. Madadi, M. Inc, A study of a $(3+1)$-dimensional Hirota bilinear model with variable coefficients: generalized higher-order rogue waves and Wronskian solutions, Qual. Theory Dyn. Syst., 24 (2025), 138. https://doi.org/10.1007/s12346-025-01292-0 doi: 10.1007/s12346-025-01292-0
|
| [26] |
Y. Ohta, J. Yang, General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation, Proc. R. Soc. A, 468 (2012), 1716–1740. https://doi.org/10.1098/rspa.2011.0640 doi: 10.1098/rspa.2011.0640
|
| [27] |
M. J. Ablowitz, B. M. Herbst, On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation, SIAM J. Appl. Math., 50 (1990), 339–351. https://doi.org/10.1137/0150021 doi: 10.1137/0150021
|
| [28] |
P. Painleve, Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math., 25 (1902), 1–85. https://doi.org/10.1007/BF02419020 doi: 10.1007/BF02419020
|
| [29] |
B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critiques fixes, Acta Math., 33 (1910), 1–55. https://doi.org/10.1007/BF02393211 doi: 10.1007/BF02393211
|
| [30] |
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
|
| [31] |
D. J. Kedziora, A. Ankiewicz, N. Akhmediev, Circular rogue wave clusters, Phys. Rev. E, 84 (2011), 056611. https://doi.org/10.1103/PhysRevE.84.056611 doi: 10.1103/PhysRevE.84.056611
|
| [32] |
D. J. Kedziora, A. Ankiewicz, N. Akhmediev, Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions, Phys. Rev. E, 88 (2013), 013207. https://doi.org/10.1103/PhysRevE.88.013207 doi: 10.1103/PhysRevE.88.013207
|
| [33] |
Z. Duan, X. Tao, B. Yang, Patterns of rogue waves in the sharp-line Maxwell–Bloch system, Chaos Solitons Fract., 187 (2024), 115407. https://doi.org/10.1016/j.chaos.2024.115407 doi: 10.1016/j.chaos.2024.115407
|
| [34] |
K. Kajiwara, Y. Ohta, Determinant structure of the rational solutions for the Painlevé Ⅱ equation, J. Math. Phys., 37 (1996), 4693–4704. https://doi.org/10.1063/1.531648 doi: 10.1063/1.531648
|
| [35] |
J. He, L. Wang, L. Li, K. Porsezian, R. Erdélyi, Few-cycle optical rogue waves: complex modified Korteweg–de Vries equation, Phys. Rev. E, 89 (2014), 062917. https://doi.org/10.1103/PhysRevE.89.062917 doi: 10.1103/PhysRevE.89.062917
|
| [36] |
T. Breunung, B. Balachandran, Prediction of freak waves from buoy measurements, Sci. Rep., 14 (2024), 16048. https://doi.org/10.1038/s41598-024-66315-3 doi: 10.1038/s41598-024-66315-3
|
| [37] |
P. Huang, Y. Wang, D. Zhou, Rogue wave patterns of Newell type long-wave–short-wave model, Chaos Solitons Fract., 175 (2023), 114038. https://doi.org/10.1016/j.chaos.2023.114038 doi: 10.1016/j.chaos.2023.114038
|
| [38] |
R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661. https://doi.org/10.1103/PhysRevLett.71.1661 doi: 10.1103/PhysRevLett.71.1661
|
| [39] |
Z. Liu, T. Qian, Peakons of the Camassa–Holm equation, Appl. Math. Model., 26 (2002), 473–480. https://doi.org/10.1016/S0307-904X(01)00086-5 doi: 10.1016/S0307-904X(01)00086-5
|
| [40] |
A. M. Wazwaz, The Camassa–Holm–KP equations with compact and noncompact travelling wave solutions, Appl. Math. Comput., 170 (2005), 347–360. https://doi.org/10.1016/j.amc.2004.12.002 doi: 10.1016/j.amc.2004.12.002
|
| [41] |
J. G. Liu, H. Zhao, Multiple rogue wave solutions for the generalized $(2+ 1)$-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation, Chin. J. Phys., 77 (2022), 985–991. https://doi.org/10.1016/j.cjph.2021.10.010 doi: 10.1016/j.cjph.2021.10.010
|
| [42] |
Z. Ji, Y. Nie, L. Li, Y. Xie, M. Wang, Rational solutions of an extended $(2+1)$-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation in liquid drop, AIMS Math., 8 (2023), 3163–3184. https://doi.org/10.3934/math.2023162 doi: 10.3934/math.2023162
|
| [43] |
C. Lu, L. Xie, H. Yang, Analysis of Lie symmetries with conservation laws and solutions for the generalized $(3+1)$-dimensional time fractional Camassa–Holm–Kadomtsev–Petviashvili equation, Comput. Math. Appl., 77 (2019), 3154–3171. https://doi.org/10.1016/j.camwa.2019.01.022 doi: 10.1016/j.camwa.2019.01.022
|
| [44] |
Y. Liu, Y. Yang, Rogue wave solutions for the $(3+1)$-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation, Chin. J. Phys., 86 (2023), 508–514. https://doi.org/10.1016/j.cjph.2023.11.008 doi: 10.1016/j.cjph.2023.11.008
|
| [45] |
L. Cheng, W. X. Ma, Soliton and lump solutions to a fourth-order nonlinear wave equation in $(2+1)$-dimensions, Qual. Theory Dyn. Syst., 24 (2025), 237. https://doi.org/10.1007/s12346-025-01384-x doi: 10.1007/s12346-025-01384-x
|
| [46] |
W. Chen, L. Tang, L. Tian, Lump, breather and interaction solutions to the $(3+ 1)$-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili equation, J. Math. Anal. Appl., 526 (2023), 127275. https://doi.org/10.1016/j.jmaa.2023.127275 doi: 10.1016/j.jmaa.2023.127275
|
| [47] |
L. Li, B. Cheng, Z. Dai, Novel evolutionary behaviors of soliton solutions for the $(3+1)$-dimensional generalized Camassa–Holm–Kadomtsev–Petciashvili equation, Nonlinear Dyn., 112 (2024), 2157–2173. https://doi.org/10.1007/s11071-023-09122-1 doi: 10.1007/s11071-023-09122-1
|
| [48] |
C. Gilson, F. Lambert, J. Nimmo, R. Willox, On the combinatorics of the Hirota D-operators, Proc. R. Soc. Lond. A, 452 (1996), 223–234. https://doi.org/10.1098/rspa.1996.0013 doi: 10.1098/rspa.1996.0013
|
| [49] |
Y. Wang, Y. Chen, Binary Bell polynomial manipulations on the integrability of a generalized (2+1)-dimensional Korteweg–de Vries equation, J. Math. Anal. Appl., 400 (2013), 624–634. https://doi.org/10.1016/j.jmaa.2012.11.028 doi: 10.1016/j.jmaa.2012.11.028
|
| [50] |
G. Dolgikh, S. Dolgikh, V. Chupin, V. Ovcharenko, V. Shvets, S. Yakovenko, Registration of nonlinear hydrophysical disturbances-rogue waves in full-scale conditions, J. Mar. Sci. Eng., 10 (2022), 1997. https://doi.org/10.3390/jmse10121997 doi: 10.3390/jmse10121997
|
| [51] |
B. Yang, J. 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
|
| [52] |
M. Taneda, Remarks on the Yablonskii-Vorob'ev polynomials, Nagoya Math. J., 159 (2000), 87–111. https://doi.org/10.1017/S0027763000007431 doi: 10.1017/S0027763000007431
|
| [53] |
S. Fukutani, K. Okamoto, H. Umemura, Special polynomials and the Hirota bilinear relations of the second and the fourth Painlevé equations, Nagoya Math. J., 159 (2000), 179–200. https://doi.org/10.1017/S0027763000007479 doi: 10.1017/S0027763000007479
|
| [54] | A. Ankiewicz, N. Akhmediev, Multi-rogue waves and triangular numbers, Rom. Rep. Phys., 69 (2017), 104. |
| [55] |
B. Yang, J. Yang, Universal rogue wave patterns associated with the Yablonskii–Vorob'ev polynomial hierarchy, Physica D, 425 (2021), 132958. https://doi.org/10.1016/j.physd.2021.132958 doi: 10.1016/j.physd.2021.132958
|
| [56] |
F. Zheng, Z. Qin, Z. Mu, T. 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
|
| [57] |
B. Yang, J. Yang, Rogue wave patterns associated with Adler–Moser polynomials in the nonlinear Schrödinger equation, Appl. Math. Lett., 148 (2024), 108871. https://doi.org/10.1016/j.aml.2023.108871 doi: 10.1016/j.aml.2023.108871
|
| [58] |
H. Lin, L. Ling, Rogue wave patterns associated with Adler–Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation, Stud. Appl. Math., 154 (2025), e12782. https://doi.org/10.1111/sapm.12782 doi: 10.1111/sapm.12782
|
| [59] |
B. Yang, J. Yang, Rogue wave patterns associated with Okamoto polynomial hierarchies, Stud. Appl. Math., 151 (2023), 60–115. https://doi.org/10.1111/sapm.12573 doi: 10.1111/sapm.12573
|
| [60] |
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
|
| [61] |
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
|
Figures(12) / Tables(3)
Majid Madadi, Kamyar Hosseini, Evren Hincal, Sekson Sirisubtawee. Rogue-wave patterns in the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili equation: true vs. predicted[J]. AIMS Mathematics, 2026, 11(10): 33994-34038. doi: 10.3934/math.20261326
DownLoad: