A symbolic computation approach and its application to the Kadomtsev-Petviashvili equation in two (3+1)-dimensional extensions

Weaam Alhejaili; Mohammed. K. Elboree; Abdelraheem M. Aly; Weaam Alhejaili; Mohammed. K. Elboree; Abdelraheem M. Aly

doi:10.3934/math.20221099

AIMS Mathematics

2022, Volume 7, Issue 11: 20085-20104. doi: 10.3934/math.20221099

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Research article

A symbolic computation approach and its application to the Kadomtsev-Petviashvili equation in two (3+1)-dimensional extensions

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
3.
Department of Mathematics, Faculty of Science, King Khalid University, Abha 62529, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

Received: 05 June 2022 Revised: 14 July 2022 Accepted: 02 August 2022 Published: 13 September 2022
MSC : 35Q58, 35Q99, 34A34

This work examines the multi-rogue-wave solutions for the Kadomtsev-Petviashvili (KP) equation in form of two (3+1)-dimensional extensions, which are soliton equations, using a symbolic computation approach. This approach is stated in terms of the special polynomials developed through a Hirota bilinear equation. The first, second, and third-order rogue wave solutions are derived for these equations. The interaction of many rogue waves is illustrated by the multi-rogue waves. The physical explanations and properties of the obtained results are plotted for specific values of the parameters $ \alpha $ and $ \beta $ to understand the physics behind the huge (rogue) wave appearance. The figures are represented in three-dimensional, and the contour plots and the density are shown at different values of parameters. The obtained results are significant for showing the dynamic actions of higher-rogue waves in the deep ocean and nonlinear optical fibers.
- (3+1)-KP equation,
- bilinear form,
- symbolic computation approach,
- rogue waves
Citation: Weaam Alhejaili, Mohammed. K. Elboree, Abdelraheem M. Aly. A symbolic computation approach and its application to the Kadomtsev-Petviashvili equation in two (3+1)-dimensional extensions[J]. AIMS Mathematics, 2022, 7(11): 20085-20104. doi: 10.3934/math.20221099

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Abstract

This work examines the multi-rogue-wave solutions for the Kadomtsev-Petviashvili (KP) equation in form of two (3+1)-dimensional extensions, which are soliton equations, using a symbolic computation approach. This approach is stated in terms of the special polynomials developed through a Hirota bilinear equation. The first, second, and third-order rogue wave solutions are derived for these equations. The interaction of many rogue waves is illustrated by the multi-rogue waves. The physical explanations and properties of the obtained results are plotted for specific values of the parameters $ \alpha $ and $ \beta $ to understand the physics behind the huge (rogue) wave appearance. The figures are represented in three-dimensional, and the contour plots and the density are shown at different values of parameters. The obtained results are significant for showing the dynamic actions of higher-rogue waves in the deep ocean and nonlinear optical fibers.

References

[1]	W. R. Sun, B. Tian, H. L. Zhen, Y. Sun, Breathers and rogue waves of the fifth-order nonlinear Schr$\ddot{o}$dinger equation in the Heisenberg ferromagnetic spin chain, Nonlinear Dyn., 81 (2015), 725–732. https://doi.org/10.1007/s11071-015-2022-4 doi: 10.1007/s11071-015-2022-4
[2]	X. Y. Xie, B. Tian, Y. F. Wang, Y. Sun, Y. Jiang, Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber, Ann. Phys., 362 (2015), 884–892. https://doi.org/10.1016/j.aop.2015.09.001 doi: 10.1016/j.aop.2015.09.001
[3]	C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue waves in the ocean: Observations, theories and modeling, Advances in Geophysical and Environmental Mechanics and Mathematics Series, Springer, Berlin, 2009.
[4]	A. Osborne, Nonlinear ocean waves and the inverse scattering transform, Elsevier, New York, 2010.
[5]	N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A, 373 (2009), 675–678. https://doi.org/10.1016/j.physleta.2008.12.036 doi: 10.1016/j.physleta.2008.12.036
[6]	N. Akhmediev, J. M. Soto-Crespo, A. Ankiewicz, Extreme waves that appear from nowhere: on the nature of rogue waves, Phys. Lett. A, 373 (2009), 2137–2145. https://doi.org/10.1016/j.physleta.2009.04.023 doi: 10.1016/j.physleta.2009.04.023
[7]	E. Pelinovsky, C. Kharif, Extreme ocean waves, Springer, Berlin 2008. https://doi.org/10.1007/978-1-4020-8314-3
[8]	P. A. Clarkson, E. Dowie, Rational solutions of the Boussinesq equation and applications to rogue waves, Trans. Math. Appl., 1 (2017), 1–26. https://doi.org/10.1093/imatrm/tnx003 doi: 10.1093/imatrm/tnx003
[9]	A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, P. V. E. McClintock, Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium, Phys. Rev. Lett., 101 (2008), 065303. https://doi.org/10.1103/PhysRevLett.101.065303 doi: 10.1103/PhysRevLett.101.065303
[10]	B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, et al., The Peregrine soliton in nonlinear fibre optics, Nature Phys., 6 (2010), 790–795. https://doi.org/10.1038/nphys1740 doi: 10.1038/nphys1740
[11]	J. He, L. Guo, Y. Zhang, A. Chabchoub, Theoretical and experimental evidence of non-symmetric doubly localized rogue waves, Proc. Math. Phys. Eng. Sci., 470 (2014), 20140318. https://doi.org/10.1098/rspa.2014.0318 doi: 10.1098/rspa.2014.0318
[12]	A. Chabchoub, N. P. Hoffmann, N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106 (2011), 204502. https://doi.org/10.1103/PhysRevLett.106.204502 doi: 10.1103/PhysRevLett.106.204502
[13]	F. Demontis, B. Prinari, C. van der Mee, F. Vitale, The inverse scattering transform for the focusing nonlinear Schrodinger equation with asymmetric boundary conditions, J. Math. Phys., 55 (2014), 101505. https://doi.org/10.1063/1.4898768 doi: 10.1063/1.4898768
[14]	W. Liu, Y. Zhang, Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation, Mod. Phys. Lett. B, 32 (2018), 1850359. https://doi.org/10.1142/S0217984918503591 doi: 10.1142/S0217984918503591
[15]	B. Guo, L. Ling, Q. P. Liu, Nonlinear Schr$\ddot{o}$dinger equation: Generalized Darboux transformation and rogue wave solutions, Phys. Rev. E, 85 (2012), 026607. https://doi.org/10.1103/PhysRevE.85.026607 doi: 10.1103/PhysRevE.85.026607
[16]	X. W. Yan, S. F. Tian, M. J. Dong, L. Zou, B$\ddot{a}$cklund transformation, rogue wave solutions and interaction phenomena for a (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation, Nonlinear Dyn., 92 (2018), 709–720. https://doi.org/10.1007/s11071-018-4085-5 doi: 10.1007/s11071-018-4085-5
[17]	K. J. Wang, Periodic solution of the time-space fractional complex nonlinear Fokas-Lenells equation by an ancient Chinese algorithm, Optik, 243 (2021), 167461. https://doi.org/10.1016/j.ijleo.2021.167461 doi: 10.1016/j.ijleo.2021.167461
[18]	K. J. Wang, G. D. Wang, Variational theory and new abundant solutions to the (1+2)-dimensional chiral nonlinear Schr$\ddot{o}$dinger equation in optics, Phys. Lett. A, 412 (2021), 127588. https://doi.org/10.1016/j.physleta.2021.127588 doi: 10.1016/j.physleta.2021.127588
[19]	K. J. Wang, G. D. Wang, Study on the explicit solutions of the Benney-Luke equation via the variational direct method, Math. Methods Appl. Sci., 44 (2021), 14173–14183. https://doi.org/10.1002/mma.7683 doi: 10.1002/mma.7683
[20]	B. B. Kadomtsev, V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15 (1970), 539–541.
[21]	M. K. Elboree, Higher order rogue waves for the (3+1)-dimensional Jimbo-Miwa equation, Int. J. Nonlinear Sci. Numer. Simul., 2021. https://doi.org/10.1515/ijnsns-2020-0065
[22]	W. Liu, Y. Zhang, Multiple rogue wave solutions for a (3+1)-dimensional Hirota bilinear equation, Appl. Math. Lett., 98 (2019), 184–190. https://doi.org/10.1016/j.aml.2019.05.047 doi: 10.1016/j.aml.2019.05.047
[23]	M. S. Ullah, H. O. Roshid, F. S. Alshammari, M. Z. Ali, Collision phenomena among the solitons, periodic and Jacobi elliptic functions to a (3+1)-dimensional Sharma-Tasso-Olver-like model, Results Phys., 36 (2022), 105412. https://doi.org/10.1016/j.rinp.2022.105412 doi: 10.1016/j.rinp.2022.105412
[24]	H. O. Roshid, N. F. M. Noor, M. S. Khatun, H. M. Baskonus, F. B. M. Belgacem, Breather, multi-shock waves and localized excitation structure solutions to the extended BKP-Boussinesq equation, Commun. Nonlinear Sci. Numer. Simul., 101 (2021), 105867. https://doi.org/10.1016/j.cnsns.2021.105867 doi: 10.1016/j.cnsns.2021.105867
[25]	R. Li, X. Geng, Rogue periodic waves of the sine-Gordon equation, Appl. Math. Lett., 102 (2020), 106147. https://doi.org/10.1016/j.aml.2019.106147 doi: 10.1016/j.aml.2019.106147
[26]	M. Zheng, X. Dong, C. Chen, M. Li, Multiple-order rogue wave solutions to a (2+1)-dimensional Boussinesq type equation, Commun. Theor. Phys., 74 (2022), 085002.
[27]	J. G. Liu, W. H. Zhu, Multiple rogue wave solutions for (2+1)-dimensional Boussinesq equation, Chin. J. Phys., 67 (2020), 492–500. https://doi.org/10.1016/j.cjph.2020.08.008 doi: 10.1016/j.cjph.2020.08.008
[28]	J. G. Rao, Y. B. Liu, C. Qian, J. S. He, Rogue waves and Hybrid solutions of the Boussinesq equation, Z. Naturforsch. A, 72 (2017), 307–314.
[29]	Z. Zhao, L. He, Multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation, Appl. Math. Lett., 95 (2019), 114–121. https://doi.org/10.1016/j.aml.2019.03.031 doi: 10.1016/j.aml.2019.03.031
[30]	Z. Zhao, L. He, Resonance Y-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation, Appl. Math. Lett., 122 (2021), 107497. https://doi.org/10.1016/j.aml.2021.107497 doi: 10.1016/j.aml.2021.107497
[31]	L. He, Z. Zhao, Multiple lump solutions and dynamics of the generalized (3+1)-dimensional KP equation, Mod. Phys. Lett. B, 34 (2020), 2050167. https://doi.org/10.1142/S0217984920501675 doi: 10.1142/S0217984920501675
[32]	Zhaqilao, A symbolic computation approach to constructing rogue waves with a controllable center in the nonlinear systems, Comput. Math. Appl., 75 (2018), 3331–3342. https://doi.org/10.1016/j.camwa.2018.02.001 doi: 10.1016/j.camwa.2018.02.001
[33]	R. Hirota, Direct method in soliton theory, In: R. K. Bullough, P. J. Caudrey, Solitons, Springer, Berlin, 1980. https://doi.org/10.1007/978-3-642-81448-8_5
[34]	A. M. Wazwaz, Multiple soliton solutions for two (3+1)-dimensional extensions of the KP equation, Int. J. Nonlinear Sci., 12 (2011), 471–477.

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