Integrability, Hirota <i>D</i>-operator expression, multi solitons, breather wave, and complexiton of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation

Kamyar Hosseini; Farzaneh Alizadeh; Sekson Sirisubtawee; Chaiyod Kamthorncharoen; Samad Kheybari; Kaushik Dehingia; Kamyar Hosseini; Farzaneh Alizadeh; Sekson Sirisubtawee; Chaiyod Kamthorncharoen; Samad Kheybari; Kaushik Dehingia

doi:10.3934/math.2025242

AIMS Mathematics

2025, Volume 10, Issue 3: 5248-5263. doi: 10.3934/math.2025242

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Integrability, Hirota D-operator expression, multi solitons, breather wave, and complexiton of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation

1.
Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
2.
Research Center of Applied Mathematics, Khazar University, Baku, Azerbaijan
3.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand
4.
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand
5.
Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, Kyrenia 99320, Turkey
6.
Department of Mathematics, Sonari College, 785690, Sonari, Assam, India

Received: 12 December 2024 Revised: 12 February 2025 Accepted: 20 February 2025 Published: 10 March 2025
MSC : 35R10, 74J30, 74J35

In this paper, we conducted an in-depth study of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon (gKdV–CDG) equation modeling specific oceanic waves. Through the Bell polynomial approach (BPA), the Hirota $ D $-operator expression of the gKdV–CDG equation was first constructed. An integrability test of the governing model was then carried out, and consequently, multi solitons were constructed using the Hirota method. Ultimately, using symbolic computations, breather and complexiton waves were derived from the gKdV–CDG equation by serving distinct ansatzes. A few representations positioned two- and three-dimensionally were provided to characterize the nonlinear wave's physical features. Based on the results, suitable methods were suggested to assess the height and width of nonlinear waves in the ocean.
- gKdV–CDG equation,
- Hirota D-operator expression,
- integrability,
- multi solitons,
- breather and complexiton waves
Citation: Kamyar Hosseini, Farzaneh Alizadeh, Sekson Sirisubtawee, Chaiyod Kamthorncharoen, Samad Kheybari, Kaushik Dehingia. Integrability, Hirota D-operator expression, multi solitons, breather wave, and complexiton of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon equation[J]. AIMS Mathematics, 2025, 10(3): 5248-5263. doi: 10.3934/math.2025242

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Abstract

In this paper, we conducted an in-depth study of a generalized Korteweg-de Vries–Caudrey Dodd Gibbon (gKdV–CDG) equation modeling specific oceanic waves. Through the Bell polynomial approach (BPA), the Hirota $ D $-operator expression of the gKdV–CDG equation was first constructed. An integrability test of the governing model was then carried out, and consequently, multi solitons were constructed using the Hirota method. Ultimately, using symbolic computations, breather and complexiton waves were derived from the gKdV–CDG equation by serving distinct ansatzes. A few representations positioned two- and three-dimensionally were provided to characterize the nonlinear wave's physical features. Based on the results, suitable methods were suggested to assess the height and width of nonlinear waves in the ocean.

References

[1]	A. M. Wazwaz, Painlevé integrability and lump solutions for two extended (3+1)- and (2+1)-dimensional Kadomtsev–Petviashvili equations, Nonlinear Dyn., 111 (2023), 3623–3632. https://doi.org/10.1007/s11071-022-08074-2 doi: 10.1007/s11071-022-08074-2
[2]	A. M. Wazwaz, Extended (3+1)-dimensional Kairat-Ⅱ and Kairat-X equations: Painleve integrability, multiple soliton solutions, lump solutions, and breather wave solutions, Int. J. Numer. Method. H., 34 (2024), 2177–2194. https://doi.org/10.1108/HFF-01-2024-0053 doi: 10.1108/HFF-01-2024-0053
[3]	K. Hosseini, S. Salahshour, D. Baleanu, M. Mirzazadeh, K. Dehingia, A new generalized KdV equation: Its lump-type, complexiton and soliton solutions, Int. J. Mod. Phys. B, 36 (2022), 2250229. https://doi.org/10.1142/S0217979222502290 doi: 10.1142/S0217979222502290
[4]	W. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simulat., 43 (1997), 13–27. https://doi.org/10.1016/S0378-4754(96)00053-5 doi: 10.1016/S0378-4754(96)00053-5
[5]	A. M. Wazwaz, S. A. El-Tantawy, Solving the (3+1)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota's method, Nonlinear Dyn., 88 (2017), 3017–3021. https://doi.org/10.1007/s11071-017-3429-x doi: 10.1007/s11071-017-3429-x
[6]	K. Hosseini, R. Ansari, R. Pouyanmehr, F. Samadani, M. Aligoli, Kinky breather-wave and lump solutions to the (2+1)-dimensional Burgers equations, Anal. Math. Phys., 10 (2020), 65. https://doi.org/10.1007/s13324-020-00405-z doi: 10.1007/s13324-020-00405-z
[7]	Y. Zhou, W. X. Ma, Complexiton solutions to soliton equations by the Hirota method, J. Math. Phys., 58 (2017), 101511. https://doi.org/10.1063/1.4996358 doi: 10.1063/1.4996358
[8]	J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys., 24 (1983), 522–526. https://doi.org/10.1063/1.525721 doi: 10.1063/1.525721
[9]	Y. L. Ma, A. M. Wazwaz, B. Q. Li, A new (3+1)-dimensional Sakovich equation in nonlinear wave motion: Painlevé integrability, multiple solitons and soliton molecules, Qual. Theory Dyn. Syst., 21 (2022), 158. https://doi.org/10.1007/s12346-022-00689-5 doi: 10.1007/s12346-022-00689-5
[10]	J. Chu, X. Chen, Y. Liu, Integrability, lump solutions, breather solutions and hybrid solutions for the (2+1)-dimensional variable coefficient Korteweg-de Vries equation, Nonlinear Dyn., 112 (2024), 619–634. https://doi.org/10.1007/s11071-023-09062-w doi: 10.1007/s11071-023-09062-w
[11]	L.L. Zhang, X. Lü, S.Z. Zhu, Painlevé analysis, Bäcklund transformation and soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq Equation, Int. J. Theor. Phys., 63 (2024), 160. https://doi.org/10.1007/s10773-024-05670-3 doi: 10.1007/s10773-024-05670-3
[12]	E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258–277. https://doi.org/10.2307/1968431 doi: 10.2307/1968431
[13]	F. Lambert, I. Loris, J. Springael, R. Willox, On a direct bilinearization method: Kaup's higher-order water wave equation as a modified nonlocal Boussinesq equation, J. Phys. A: Math. Gen., 27 (1994), 5325. https://doi.org/10.1088/0305-4470/27/15/028 doi: 10.1088/0305-4470/27/15/028
[14]	F. Lambert, J. Springael, Construction of Bäcklund transformations with binary Bell polynomials, J. Phys. Soc. Jpn., 66 (1997), 2211–2213. https://doi.org/10.1143/JPSJ.66.2211 doi: 10.1143/JPSJ.66.2211
[15]	Y. Zhang, W. W. Wei, T. F. Cheng, Y. Song, Binary Bell polynomial application in generalized (2+1)-dimensional KdV equation with variable coefficients, Chinese Phys. B, 20 (2011), 110204. https://doi.org/10.1088/1674-1056/20/11/110204 doi: 10.1088/1674-1056/20/11/110204
[16]	Y. H. Wang, C. Temuer, Y. Q. Yang, Integrability for the generalised variable-coefficient fifth-order Korteweg-de Vries equation with Bell polynomials, Appl. Math. Lett., 29 (2014), 13–19. https://doi.org/10.1016/j.aml.2013.10.007 doi: 10.1016/j.aml.2013.10.007
[17]	U. K. Mandal, A. Das, W. X. Ma, Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3+1)-dimensional nonlinear evolution equation, Phys. Fluids, 36 (2024), 037151. https://doi.org/10.1063/5.0195378 doi: 10.1063/5.0195378
[18]	K. Hosseini, F. Alizadeh, E. Hinçal, M. Ilie, M. S. Osman, Bilinear Bäcklund transformation, Lax pair, Painlevé integrability, and different wave structures of a 3D generalized KdV equation, Nonlinear Dyn., 112 (2024), 18397–18411. https://doi.org/10.1007/s11071-024-09944-7 doi: 10.1007/s11071-024-09944-7
[19]	T. Umar, K. Hosseini, B. Kaymakamzade, S. Boulaaras, M. S. Osman, Hirota D-operator forms, multiple soliton waves, and other nonlinear patterns of a 2D generalized Kadomtsev–Petviashvili equation, Alex. Eng. J., 108 (2024), 999–1010. https://doi.org/10.1016/j.aej.2024.09.070 doi: 10.1016/j.aej.2024.09.070
[20]	E. Asadi, K. Hosseini, M. Madadi, Superposition of soliton, breather and lump waves in a non-Painleve integrabale extension of the Boiti–Leon–Manna–Pempinelli equation, Phys. Scr., 99 (2024), 125242. https://doi.org/10.1088/1402-4896/ad8f74 doi: 10.1088/1402-4896/ad8f74
[21]	A. M. Wazwaz, N-soliton solutions for the combined KdV–CDG equation and the KdV–Lax equation, Appl. Math. Comput., 203 (2008), 402–407. https://doi.org/10.1016/j.amc.2008.04.047 doi: 10.1016/j.amc.2008.04.047
[22]	A. Biswas, G. Ebadi, H. Triki, A. Yildirim, N. Yousefzadeh, Topological soliton and other exact solutions to KdV–Caudrey–Dodd–Gibbon equation, Results Math., 63 (2013), 687–703. https://doi.org/10.1007/s00025-011-0226-6 doi: 10.1007/s00025-011-0226-6
[23]	H. Ma, H. Huang, A. Deng, Soliton molecules, asymmetric solitons and hybrid solutions for KdV–CDG equation, Partial Differ. Equ. Appl. Math., 5 (2022), 100214. https://doi.org/10.1016/j.padiff.2021.100214 doi: 10.1016/j.padiff.2021.100214
[24]	K. Hosseini, A. Akbulut, D. Baleanu, S. Salahshour, M. Mirzazadehh, K. Dehingia, The Korteweg-de Vries–Caudrey–Dodd–Gibbon dynamical model: Its conservation laws, solitons, and complexiton, J. Ocean Eng. Sci., 2022. In press. https://doi.org/10.1016/j.joes.2022.06.003
[25]	H. Almusawa, A. Jhangeer, Exploring wave interactions and conserved quantities of KdV–Caudrey–Dodd–Gibbon equation using Lie theory, Mathematics, 12 (2024), 2242. https://doi.org/10.3390/math12142242 doi: 10.3390/math12142242
[26]	M. D. Kruskal, N. Joshi, R. Halburd, Analytic and asymptotic methods for nonlinear singularity analysis: A review and extensions of tests for the Painlevé property, In: Integrability of nonlinear systems, Berlin, Heidelberg: Springer, 1997,171–205. https://doi.org/10.1007/BFb0113696
[27]	D. Baldwin, W. Hereman, Symbolic software for the Painlevé test of nonlinear ordinary and partial differential equations, J. Nonlinear Math. Phys., 13 (2006), 90–110. https://doi.org/10.2991/jnmp.2006.13.1.8 doi: 10.2991/jnmp.2006.13.1.8
[28]	J. Hietarinta, A search for bilinear equations passing Hirota's three soliton condition. Ⅰ. KdV type bilinear equations, J. Math. Phys., 28 (1987), 1732–1742. https://doi.org/10.1063/1.527815 doi: 10.1063/1.527815
[29]	W. X. Ma, Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations, Commun. Nonlinear Sci., 16 (2011), 2663–2666. https://doi.org/10.1016/j.cnsns.2010.10.003 doi: 10.1016/j.cnsns.2010.10.003
[30]	W. X. Ma, N-soliton solution of a combined pKP–BKP equation, J. Geom. Phys., 165 (2021), 104191. https://doi.org/10.1016/j.geomphys.2021.104191 doi: 10.1016/j.geomphys.2021.104191
[31]	W. X. Ma, Complexiton solutions to the Korteweg-de Vries equation, Phys. Lett. A, 301 (2002), 35–44. https://doi.org/10.1016/S0375-9601(02)00971-4 doi: 10.1016/S0375-9601(02)00971-4
[32]	F. Alizadeh, K. Hosseini, S. Sirisubtawee, E. Hinçal, Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles, Bound. Value Probl., 2024 (2024), 111. https://doi.org/10.1186/s13661-024-01921-8 doi: 10.1186/s13661-024-01921-8

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