Bäcklund transformations, kink soliton, breather- and travelling-wave solutions for a \((3+1)\)-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics. (English) Zbl 07837814
Summary: In this paper, we investigate a \((3+1)\)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation in fluid dynamics. Based on the Hirota method, we give a bilinear auto-Bäcklund transformation. Via the truncated Painlevé expansion, we get a Painlevé-type auto-Bäcklund transformation. With the aid of the symbolic computation, we derive some one- and two-kink soliton solutions. We present the oblique and parallel elastic interactions between the two-kink solitons. Via the extended homoclinic test technique, we construct some breather-wave solutions. Besides, we derive some lump solutions with the periods of the breather-wave solutions to the infinity. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. Based on the polynomial-expansion method, travelling-wave solutions are constructed.
MSC:
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 35Cxx | Representations of solutions to partial differential equations |
| 37Kxx | Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
fluid dynamics; \((3+1)\)-dimensional BKP equation; Hirota method; extended homoclinic test technique; polynomial-expansion methodReferences:
| [1] | Shen, Y.; Tian, B.; Zhou, T. Y., In nonlinear optics, fluid dynamics and plasma physics: symbolic computation on a (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff system, Eur. Phys. J. Plus, 136, 572, (2021) |
| [2] | Wu, H. Y.; Jiang, L. H., Instruction on the construction of coherent structures based on variable separation solutions of (2+1)-dimensional nonlinear evolution equations in fluid mechanics, Nonlinear Dyn., 97, 403-412, (2019) · Zbl 1430.35202 |
| [3] | Yu, W. T.; Zhou, Q.; Mirzazadeh, M.; Liu, W. J.; Biswas, A., Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics, J. Adv. Res., 15, 69-76, (2019) |
| [4] | Tamang, J.; Saha, A., Bifurcations of small-amplitude supernonlinear waves of the mKdV and modified Gardner equations in a three-component electron-ion plasma, Phys. Plasmas, 27, Article 012105 pp., (2020) |
| [5] | He, Z. M.; Wen, L.; Wang, Y. J.; Chen, G. P.; Tan, R. B.; Dai, C. Q.; Zhang, X. F., Dynamics and pattern formation of ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable interactions, Phys. Rev. E, 99, Article 062216 pp., (2019) |
| [6] | Paliathanasis, A., Lie symmetry analysis and one-dimensional optimal system for the generalized 2+1 Kadomtsev-Petviashvili equation, Phys. Scr., 95, Article 055223 pp., (2020) |
| [7] | Yang, H. Z.; Liu, W.; Zhao, Y. M., Lie symmetry analysis, traveling wave solutions, and conservation laws to the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation, Complexity, 2020, Article 3465860 pp., (2020) · Zbl 1456.35071 |
| [8] | Zhang, Y. N.; Pang, J., Lump and lump-type solutions of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, J. Appl. Math., 2019, 1-5, (2019) · Zbl 1442.35409 |
| [9] | David, D.; Kamran, N.; Levi, D. A.; Winternitz, P., Subalgebras of loop algebras and symmetries of the Kadomtsev-Petviashvili equation, Phys. Rev. Lett., 55, 2111-2113, (1985) |
| [10] | Guo, D.; Tian, S. F.; Wang, X. B.; Zhang, T. T., Dynamics of lump solutions, rogue wave solutions and traveling wave solutions for a (3+1)-dimensional VC-BKP equation, East Asian J. Appl. Math., 9, 780-796, (2019) · Zbl 1464.35283 |
| [11] | Gao, X. Y., Bäcklund transformation and shock-wave-type solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid mechanics, Ocean Eng., 96, 245-247, (2015) |
| [12] | Hirota, R., The Direct Method in Soliton Theory, (2004), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 1099.35111 |
| [13] | Jia, T. T.; Chai, Y. Z.; Hao, H. Q., Multi-soliton solutions and breathers for the generalized coupled nonlinear Hirota equations via the Hirota method, Superlattice. Microstruct., 105, 172-182, (2017) |
| [14] | Luo, L., Bäcklund transformation of variable-coefficient Boiti-Leon-Manna-Pempinelli equation, Appl. Math. Lett., 94, 94-98, (2019) · Zbl 1412.35055 |
| [15] | Guan, X.; Liu, W. J.; Zhou, Q.; Biswas, A., Darboux transformation and analytic solutions for a generalized super-NLS-mKdV equation, Nonlinear Dyn., 98, 1491-1500, (2019) |
| [16] | Zhang, X. E.; Chen, Y., Inverse scattering transformation for generalized nonlinear Schrödinger equation, Appl. Math. Lett., 98, 306-313, (2019) · Zbl 1428.35547 |
| [17] | Kumar, S.; Kumar, D., Solitary wave solutions of (3+1)-dimensional extended Zakharov-Kuznetsov equation by Lie symmetry approach, Comput. Math. Appl., 77, 2096-2113, (2019) · Zbl 1442.35382 |
| [18] | Tahir, M.; Awan, A. U.; Osman, M. S.; Baleanu, D.; Alqurashi, M. M., Abundant periodic wave solutions for fifth-order Sawada-Kotera equations, Results Phys., 17, Article 103105 pp., (2020) |
| [19] | Osman, M. S.; Inc, M.; Liu, J. G.; Hosseini, K.; Yusuf, A., Different wave structures and stability analysis for the generalized (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation, Phys. Scr., 95, Article 035229 pp., (2020) |
| [20] | Osman, M. S.; Baleanu, D.; Adem, A. R.; Hosseini, K.; Mirzazadeh, M.; Eslami, M., Double-wave solutions and Lie symmetry analysis to the (2+1)-dimensional coupled Burgers equations, Chin. J. Phys., 63, 122-129, (2020) · Zbl 07829558 |
| [21] | Ismael, H. F.; Bulut, H.; Park, C.; Osman, M. S., M-lump, N-soliton solutions, and the collision phenomena for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation, Results Phys., 19, Article 103329 pp., (2020) |
| [22] | Osman, M. S.; Wazwaz, A. M., A general bilinear form to generate different wave structures of solitons for a (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Math. Methods Appl. Sci., 42, 6277-6283, (2019) · Zbl 1434.35144 |
| [23] | Osman, M. S., One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation, Nonlinear Dyn., 96, 1491-1496, (2019) · Zbl 1437.35603 |
| [24] | Liu, J. G.; Osman, M. S.; Zhu, W. H.; Zhou, L.; Baleanu, D., The general bilinear techniques for studying the propagation of mixed-type periodic and lump-type solutions in a homogenous-dispersive medium, AIP Adv., 10, Article 105325 pp., (2020) |
| [25] | Javid, A.; Raza, N.; Osman, M. S., Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets, Commun. Theor. Phys., 71, 362-366, (2019) |
| [26] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations, Appl. Math. Lett., 104, Article 106170 pp., (2020) · Zbl 1437.86001 |
| [27] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system, Chaos Solitons Fract., 138, Article 109950 pp., (2020) · Zbl 1490.35314 |
| [28] | Feng, Y. J.; Gao, Y. T.; Li, L. Q.; Jia, T. T., Bilinear form, solitons, breathers and lumps of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics, Eur. Phys. J. Plus, 135, 272, (2020) |
| [29] | Feng, Y. J.; Gao, Y. T.; Jia, T. T.; Li, L. Q., Soliton interactions of a variable-coefficient three-component AB system for the geophysical flows, Mod. Phys. Lett. B, 33, Article 1950354 pp., (2019) |
| [30] | Su, J. J.; Gao, Y. T.; Deng, G. F.; Jia, T. T., Solitary waves, breathers, and rogue waves modulated by long waves for a model of a baroclinic shear flow, Phys. Rev. E, 100, Article 042210 pp., (2019) |
| [31] | Su, J. J.; Gao, Y. T.; Ding, C. C., Darboux transformations and rogue wave solutions of a generalized AB system for the geophysical flows, Appl. Math. Lett., 88, 201-208, (2019) · Zbl 1448.76087 |
| [32] | Hu, L.; Gao, Y. T.; Jia, S. L.; Su, J. J.; Deng, G. F., Solitons for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique, Mod. Phys. Lett. B, 33, Article 1950376 pp., (2019) |
| [33] | Rezazadeh, H., New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity, Optik, 167, 218-227, (2018) |
| [34] | Hu, L.; Gao, Y. T.; Jia, T. T.; Deng, G. F.; Li, L. Q., Higher-order hybrid waves for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for an irrotational incompressible fluid via the modified Pfaffian technique, Z. Angew. Math. Phys., 72, 75, (2021) · Zbl 1467.76022 |
| [35] | Yuan, F.; Cheng, Y.; He, J. S., Degeneration of breathers in the Kadomttsev-Petviashvili I equation, Commun. Nonlinear Sci. Numer. Simul., 83, Article 105027 pp., (2020) · Zbl 1456.35183 |
| [36] | Zhao, Z. L.; He, L. C., Multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation, Appl. Math. Lett., 95, 114-121, (2020) · Zbl 1448.35418 |
| [37] | Seadawy, A. R., Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves, Eur. Phys. J. Plus, 132, 29, (2017) |
| [38] | Rizvi, S. T.R.; Younis, M.; Baleanu, D.; Iqbal, H., Lump and rogue wave solutions for the Broer-Kaup-Kupershmidt system, Chin. J. Phys., 68, 19-27, (2020) · Zbl 07848579 |
| [39] | Chen, S. S.; Tian, B.; Liu, L.; Yuan, Y. Q.; Du, X. X., Breathers, multi-peak solitons, breather-to-soliton transitions and modulation instability of the variable-coefficient fourth-order nonlinear Schrödinger system for an inhomogeneous optical fiber, Chin. J. Phys., 62, 274-283, (2019) · Zbl 07828458 |
| [40] | Liu, J. G.; Zhu, W. H., Multiple rogue wave solutions for (2+1)-dimensional Boussinesq equation, Chin. J. Phys., 67, 492-500, (2020) · Zbl 07833201 |
| [41] | Liu, F. Y.; Gao, Y. T.; Yu, X.; Li, L. Q.; Ding, C. C.; Wang, D., Lie group analysis and analytic solutions for a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation in fluid mechanics and plasma physics, Eur. Phys. J. Plus, 136, 656, (2021) |
| [42] | Deng, G. F.; Gao, Y. T.; Ding, C. C.; Su, J. J., Solitons and breather waves for the generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics, ocean dynamics and plasma physics, Chaos Solitons Fract., 140, Article 110085 pp., (2020) · Zbl 1495.35147 |
| [43] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Hetero-Backlund Transformation and similarity reduction of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics, Phys. Lett. A, 384, Article 126788 pp., (2020) · Zbl 1448.37084 |
| [44] | Deng, G. F.; Gao, Y. T.; Su, J. J.; Ding, C. C.; Jia, T. T., Solitons and periodic waves for the (2+1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics, Nonlinear Dyn., 99, 1039-1052, (2020) · Zbl 1459.35076 |
| [45] | Li, L. Q.; Gao, Y. T.; Hu, L.; Jia, T. T.; Ding, C. C.; Feng, Y. J., Bilinear form, soliton, breather, lump and hybrid solutions for a (2+1)-dimensional Sawada-Kotera equation, Nonlinear Dyn., 100, 2729-2738, (2020) · Zbl 1516.37117 |
| [46] | Jia, T. T.; Gao, Y. T.; Yu, X.; Li, L. Q., Lax pairs, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation, Appl. Math. Lett., 114, Article 106702 pp., (2021) · Zbl 1458.35371 |
| [47] | Jia, T. T.; Gao, Y. T.; Deng, G. F.; Hu, L., Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons, Nonlinear Dyn., 98, 269-282, (2019) · Zbl 1430.74073 |
| [48] | Liu, F. Y.; Gao, Y. T.; Yu, X.; Ding, C. C.; Deng, G. F.; Jia, T. T., Painlevé analysis, Lie group analysis and soliton-cnoidal, resonant, hyperbolic function and rational solutions for the modified Korteweg-de-Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics, Chaos Solitons Fract., 144, Article 110559 pp., (2021) · Zbl 1498.35479 |
| [49] | Li, M.; Xu, T., Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Phys. Rev. E, 91, Article 033202 pp., (2015) |
| [50] | Wang, L.; Zhang, J. H.; Wang, Z. Q.; Liu, C.; Li, M.; Qi, F. H.; Guo, R., Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation, Phys. Rev. E, 93, Article 012214 pp., (2016) |
| [51] | Ma, W. X.; Zhou, Y., Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differ. Equ., 264, 2633-2659, (2018) · Zbl 1387.35532 |
| [52] | Gao, X. Y.; Guo, Y. J.; Shan, W. R., Cosmic dusty plasmas via a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili-Burgers-type equation: auto-Backlund transformations, solitons and similarity reductions plus observational/experimental supports, Wave. Random Complex, (2021), in press |
| [53] | Lü, X.; Hua, Y. F.; Chen, S. J.; Tang, X. F., Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws, Commun. Nonlinear Sci. Numer. Simul., 95, Article 105612 pp., (2021) · Zbl 1456.35073 |
| [54] | Tu, J. M.; Tian, S. F.; Xu, M. J.; Song, X. Q.; Zhang, T. T., Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles, Nonlinear Dyn., 83, 1199-1215, (2016) · Zbl 1351.37249 |
| [55] | Weiss, J.; tabor, M.; Carnevale, G., The Painlevé property for partial differential equations, J. Math. Phys., 24, 522-526, (1983) · Zbl 0514.35083 |
| [56] | Lan, Z. Z., Periodic, breather and rogue wave solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation in fluid dynamics, Appl. Math. Lett., 94, 126-132, (2019) · Zbl 1412.35050 |
| [57] | Hu, W. Q.; Gao, Y. T.; Zhao, C.; Jia, S. L.; Lan, Z. Z., Breathers, quasi-periodic and travelling waves for a generalized-dimensional Yu-Toda-Sasa-Fukayama equation in fluids, Wave. Random Complex, 27, 458-481, (2017) · Zbl 07659352 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
