Lump and travelling wave solutions of a (3 + 1)-dimensional nonlinear evolution equation

doi:10.1016/j.joes.2022.04.018

Abstract

Abstract:

In this paper, the (3+1 )-dimensional nonlinear evolution equation is studied analytically. The bilinear form of given model is achieved by using the Hirota bilinear method. As a result, the lump waves and collisions between lumps and periodic waves, the collision among lump wave and single, double-kink soliton solutions as well as the collision between lump, periodic, and single, double-kink soliton solutions for the given model are constructed. Furthermore, some new traveling wave solutions are developed by applying the exp(âˆ’Ï•(Î¾)) expansion method. The 3D, 2D and contours plots are drawn to demonstrate the nature of the nonlinear model for setting appropriate set of parameters. As a result, a collection of bright, dark, periodic, rational function and elliptic function solutions are established. The applied strategies appear to be more powerful and efficient approaches to construct some new traveling wave structures for various contemporary models of recent era.

Highlights

● The Hirota bilinear method and the $\exp (- ϕ (ξ))$ expansion technique are used to generate some new solitary wave solutions.

● The collision among lump wave and single, double-kink soliton solutions as well as the collision between lump, periodic, and single, double-kink soliton solutions of the given model are also constructed.

Key words: The Hirota bilinear method, Lump wave solution, Travelling wave solution, Exp(−φ(ξ)) expansion technique

Kalim U. Tariq, Raja Nadir Tufail. Lump and travelling wave solutions of a (3 + 1)-dimensional nonlinear evolution equation[J]. Journal of Ocean Engineering and Science, 2024, 9(2): 164-172.

Kalim U. Tariq, Raja Nadir Tufail. [J]. Journal of Ocean Engineering and Science, 2024, 9(2): 164-172.

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URL: https://www.qk.sjtu.edu.cn/joes/EN/10.1016/j.joes.2022.04.018

https://www.qk.sjtu.edu.cn/joes/EN/Y2024/V9/I2/164

Figures/Tables 14

Fig.1 Profiles of the solution

| g_{1} (x, y, z, t) |

of (1) for

γ_{0} = 0.7

γ_{1} = 0.5

γ_{2} = 0.75

γ_{3} = - 0.25

μ_{0} = 0.8

μ_{3} = 0.6

Fig.1 Profiles of the solution $| g_{1} (x, y, z, t) |$ of (1) for $γ_{0} = 0.7$ , $γ_{1} = 0.5$ , $γ_{2} = 0.75$ , $γ_{3} = - 0.25$ , $μ_{0} = 0.8$ , $μ_{3} = 0.6$ .

Fig.2 Profiles of the solution

| g_{2} (x, y, z, t) |

of (1) for

α_{0} = - 0.5

α_{1} = 0.55

α_{2} = 0.75

α_{3} = 0.5

β_{0} = 0.6

β_{1} = - 0.5

β_{2} = 0.4

β_{3} = 0.25

μ_{0} = 0.5, μ_{1} = 0.6

μ_{2} = 0.7

Fig.2 Profiles of the solution $| g_{2} (x, y, z, t) |$ of (1) for $α_{0} = - 0.5$ , $α_{1} = 0.55$ , $α_{2} = 0.75$ , $α_{3} = 0.5$ , $β_{0} = 0.6$ , $β_{1} = - 0.5$ , $β_{2} = 0.4$ , $β_{3} = 0.25$ , $μ_{0} = 0.5, μ_{1} = 0.6$ , $μ_{2} = 0.7$ .

Fig.3 Profiles of the solution

| g_{4} (x, y, z, t) |

of (1) for

α_{0} = 0.5

α_{1} = 0.4

α_{3} = - 0.6

β_{0} = 0.55

β_{1} = 0.7

β_{2} = 0.56

β_{3} = 0.4

β_{4} = 0.3

μ_{1} = 0.5

Fig.3 Profiles of the solution $| g_{4} (x, y, z, t) |$ of (1) for $α_{0} = 0.5$ , $α_{1} = 0.4$ , $α_{3} = - 0.6$ , $β_{0} = 0.55$ , $β_{1} = 0.7$ , $β_{2} = 0.56$ , $β_{3} = 0.4$ , $β_{4} = 0.3$ , $μ_{1} = 0.5$ .

Fig.4 Profiles of the solution

| g_{5} (x, y, z, t) |

of (1) for

γ_{0} = 0.25

γ_{1} = 0.55

γ_{2} = - 0.3

γ_{3} = 0.025

Fig.4 Profiles of the solution $| g_{5} (x, y, z, t) |$ of (1) for $γ_{0} = 0.25$ , $γ_{1} = 0.55$ , $γ_{2} = - 0.3$ , $γ_{3} = 0.025$ .

Fig.5 Profiles of the solution

| g_{7} (x, y, z, t) |

of (1) for

β_{0} = 0.15

β_{1} = 0.12

β_{3} = 0.15

α_{0} = 0.1

α_{1} = 0.12

α_{3} = 0.15

μ_{0} = 0.4

μ_{1} = 0.5

μ_{2} = 0.6

Fig.5 Profiles of the solution $| g_{7} (x, y, z, t) |$ of (1) for $β_{0} = 0.15$ , $β_{1} = 0.12$ , $β_{3} = 0.15$ , $α_{0} = 0.1$ , $α_{1} = 0.12$ , $α_{3} = 0.15$ , $μ_{0} = 0.4$ , $μ_{1} = 0.5$ , $μ_{2} = 0.6$ .

Fig.6 Profiles of the solution

| g_{8} (x, y, z, t) |

of (1) for

γ_{0} = 1.55

γ_{1} = 0.75

γ_{2} = - 0.8

γ_{4} = - 0.25

Fig.6 Profiles of the solution $| g_{8} (x, y, z, t) |$ of (1) for $γ_{0} = 1.55$ , $γ_{1} = 0.75$ , $γ_{2} = - 0.8$ , $γ_{4} = - 0.25$ .

Fig.7 Profiles of the solution

| g_{9} (x, y, z, t) |

of (1) for

γ_{0} = 0.1

γ_{1} = - 2.25

γ_{2} = 2.55

γ_{3} = 2.25

μ_{3} = 2.6

Fig.7 Profiles of the solution $| g_{9} (x, y, z, t) |$ of (1) for $γ_{0} = 0.1$ , $γ_{1} = - 2.25$ , $γ_{2} = 2.55$ , $γ_{3} = 2.25$ , $μ_{3} = 2.6$ .

Fig.8 Profiles of the solution

| g_{10} (x, y, z, t) |

of (1) for

β_{0} = 0.255

β_{1} = - 0.5

β_{3} = 0.255

γ_{0} = 0.25

γ_{1} = 0.225

γ_{3} = - 0.125

μ_{2} = 2.6

μ_{3} = 0.25

Fig.8 Profiles of the solution $| g_{10} (x, y, z, t) |$ of (1) for $β_{0} = 0.255$ , $β_{1} = - 0.5$ , $β_{3} = 0.255$ , $γ_{0} = 0.25$ , $γ_{1} = 0.225$ , $γ_{3} = - 0.125$ , $μ_{2} = 2.6$ , $μ_{3} = 0.25$ .

Fig.9 Profiles of the solution

| g_{11} (x, y, z, t) |

of (1) for

α = 1.25

β = 1.2

γ = 1.125

C = 1.35

P = 0.125

Q = - 1.1

Fig.9 Profiles of the solution $| g_{11} (x, y, z, t) |$ of (1) for $α = 1.25$ , $β = 1.2$ , $γ = 1.125$ , $C = 1.35$ , $P = 0.125$ , $Q = - 1.1$ .

Fig.10 Profiles of the solution

| g_{12} (x, y, z, t) |

of (1) for

α = 1.7

β = 1.8

γ = 1.9

C = 1.35

P = 1.5

Q = 0.5

Fig.10 Profiles of the solution $| g_{12} (x, y, z, t) |$ of (1) for $α = 1.7$ , $β = 1.8$ , $γ = 1.9$ , $C = 1.35$ , $P = 1.5$ , $Q = 0.5$ .

Fig.11 Profiles of the solution

| g_{13} (x, y, z, t) |

of (1) for

α = - 0.7

β = 0.85

γ = 0.95

C = 0.8

P = - 0.56

Q = 0.55

Fig.11 Profiles of the solution $| g_{13} (x, y, z, t) |$ of (1) for $α = - 0.7$ , $β = 0.85$ , $γ = 0.95$ , $C = 0.8$ , $P = - 0.56$ , $Q = 0.55$ .

Fig.12 Profiles of the solution

| g_{14} (x, y, z, t) |

of (1) for

α = 0.6

β = 0.5

γ = 0.5

C = 0.6

P = - 0.4

Q = 0.55

Fig.12 Profiles of the solution $| g_{14} (x, y, z, t) |$ of (1) for $α = 0.6$ , $β = 0.5$ , $γ = 0.5$ , $C = 0.6$ , $P = - 0.4$ , $Q = 0.55$ .

Fig.13 Profiles of the solution

| g_{15} (x, y, z, t) |

of (1) for

α = - 0.7

β = 0.85

γ = 0.95

C = 0.8

P = 1.56

Q = 0.55

Fig.13 Profiles of the solution $| g_{15} (x, y, z, t) |$ of (1) for $α = - 0.7$ , $β = 0.85$ , $γ = 0.95$ , $C = 0.8$ , $P = 1.56$ , $Q = 0.55$ .

Fig.14 Profiles of the solution

| g_{17} (x, y, z, t) |

of (1) for

α = - 0.2

β = 0.1

γ = 0.3

C = 0.9

P = 0.5

Q = 0.75

Fig.14 Profiles of the solution $| g_{17} (x, y, z, t) |$ of (1) for $α = - 0.2$ , $β = 0.1$ , $γ = 0.3$ , $C = 0.9$ , $P = 0.5$ , $Q = 0.75$ .

References 73

[1]	A.R. Seadawy, N. Cheemaa. Mod. Phys. Lett. B,33 (18) (2019), p. 1950203 DOI: 10.1142/s0217984919502038
[2]	H. Naher, F.A. Abdullah. Appl. Math. Sci., 6 (111) (2012), pp. 5495-5512
[3]	A.M. Wazwaz. Math. Comput. Model., 40 (5-6) (2004), pp. 499-508
[4]	I. Ahmed, A.R. Seadawy, D. Lu. Phys. Scr., 94 (5) (2019), p. 055205 DOI: 10.1088/1402-4896/ab0455
[5]	N.A. Kudryashov. Optik, 183 (2019), pp. 642-649
[6]	A.M. Wazwaz. Appl. Math. Comput., 187 (2) (2007), pp. 1131-1142
[7]	Y.-L. Sun, W.-X. Ma, J.P. Yu. Appl. Math. Lett., 120 (2021), p. 107224
[8]	A.R. Seadawy, S.T.R. Rizvi, S. Ahmad, M. Younis, D. Baleanu. Open Phys., 19 (1) (2021), pp. 1-10 DOI: 10.1515/phys-2020-0224
[9]	S.T.R. Rizvi, I. Bibi, M. Younis, A. Bekir. Chin. Phys. B, 30 (1) (2021), p. 010502
[10]	M. Higazy, S. Muhammad, A. Al-Ghamdi, M.M. Khater. J. Ocean Eng. Sci. (2022)
[11]	M.M. Miah, A.R. Seadawy, H.S. Ali, M.A. Akbar. J. Ocean Eng. Sci., 5 (3) (2020), pp. 269-278
[12]	M. Durand, D. Langevin. Eur. Phys. J. E, 7 (1) (2002), pp. 35-44
[13]	N. Kudryashov. Phys. Lett. A, 155 (4-5) (1991), pp. 269-275
[14]	L.-x. Li, E.-q. Li, M.l. Wang. Appl. Math., 25 (4) (2010), pp. 454-462 DOI: 10.1007/s11766-010-2128-x
[15]	K. Hosseini, P. Mayeli, D. Kumar. J. Mod. Opt., 65 (3) (2018), pp. 361-364 DOI: 10.1080/09500340.2017.1380857
[16]	A.M. Wazwaz. Phys. Scr., 89 (9) (2014), p. 095206 DOI: 10.1088/0031-8949/89/9/095206
[17]	Z. Ivezic, S.M. Kahn, J.A. Tyson, B. Abel, E. Acosta, R. Allsman, D. Alonso, Y. AlSayyad, S.F. Anderson, J. Andrew, et al.. Astrophys. J., 873 (2) (2019), p. 111 DOI: 10.3847/1538-4357/ab042c
[18]	A. Bekir, A. Boz. Phys. Lett. A, 372 (10) (2008), pp. 1619-1625
[19]	A.D. Alruwaili, A.R. Seadawy, S.T. Rizvi, S.A.O. Beinane. Mathematics, 10 (2) (2022), p. 200 DOI: 10.3390/math10020200
[20]	A. Bekir. Phys. Lett. A, 372 (19) (2008), pp. 3400-3406
[21]	M. Osman. Waves Random Complex Media, 26 (4) (2016), pp. 434-443 DOI: 10.1080/17455030.2016.1166288
[22]	M. Khater, A. Jhangeer, H. Rezazadeh, L. Akinyemi, M.A. Akbar, M. Inc, H. Ahmad. Opt. Quantum Electron., 53 (11) (2021), pp. 1-27 DOI: 10. 3138/diaspora.21.1.2020-06-15
[23]	Z. Pinar, H. Rezazadeh, M. Eslami. Opt. Quantum Electron., 52 (12) (2020), pp. 1-16 DOI: 10.23834/isrjournal.839937
[24]	A. Zafar, M. Shakeel, A. Ali, L. Akinyemi, H. Rezazadeh. Opt. Quantum Electron., 54 (1) (2022), pp. 1-15 DOI: 10.53963/pjmr.2022.006.4
[25]	Y.-Q. Chen, Y.-H. Tang, J. Manafian, H. Rezazadeh, M. Osman. Nonlinear Dyn., 105 (3) (2021), pp. 2539-2548 DOI: 10.1007/s11071-021-06642-6
[26]	U. Akram, A.R. Seadawy, S. Rizvi, M. Younis, S. Althobaiti, S. Sayed. Results Phys., 20 (2021), p. 103725
[27]	S.T.R. Rizvi, S. Ahmad, M.F. Nadeem, M. Awais. Optik, 226 (2021), p. 165955
[28]	M.J. Ablowitz, H. Segur. Solitons and the Inverse Scattering Transform. SIAM (1981)
[29]	M.J. Ablowitz, M. Ablowitz, P. Clarkson, P.A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. vol. 149, Cambridge University Press (1991)
[30]	A.R. Seadawy, M. Bilal, M. Younis, S. Rizvi, S. Althobaiti, M. Makhlouf. Chaos, Solitons Fractals, 144 (2021), p. 110669
[31]	A.R. Seadawy, S. Rehman, M. Younis, S. Rizvi, S. Althobaiti, M. Makhlouf. Phys. Scr., 96 (4) (2021), p. 045202 DOI: 10.1088/1402-4896/abdcf7
[32]	A.R. Seadawy, D. Lu, M.M. Khater. J. Ocean Eng. Sci., 2 (2) (2017), pp. 137-142
[33]	M. Abdelkawy, A. Bhrawy, E. Zerrad, A. Biswas. Acta Phys. Pol. A, 129 (3) (2016), pp. 278-283 DOI: 10.12693/APhysPolA.129.278
[34]	A. Bekir. Commun. Nonlinear Sci. Numer. Simul., 13 (9) (2008), pp. 1748-1757
[35]	R. Hirota. Direct Methods in Soliton Theory, Solitons, Springer (1980), pp. 157-176 DOI: 10.1007/978-3-642-81448-8_5
[36]	J. Satsuma. Hirota bilinear method for nonlinear evolution equations. Direct and Inverse Methods in Nonlinear Evolution Equations, Springer (2003), pp. 171-222 DOI: 10.1007/978-3-540-39808-0_4
[37]	M. Younis, S. Ali, S.T.R. Rizvi, M. Tantawy, K.U. Tariq, A. Bekir. Commun. Nonlinear Sci. Numer. Simul., 94 (2021), p. 105544
[38]	Y. Gurefe, E. Misirli. Comput. Math. Appl., 61 (8) (2011), pp. 2025-2030
[39]	E. Misirli, Y. Gurefe. Math. Comput. Appl.,16 (1) (2011), pp. 258-266 DOI: 10.3390/mca16010258
[40]	U. Younas, M. Younis, A.R. Seadawy, S. Rizvi, S. Althobaiti, S. Sayed. Results Phys., 20 (2021), p. 103766
[41]	F.S. Khodadad, S. Mirhosseini-Alizamini, B. GÃ¼nay, L. Akinyemi, H. Rezazadeh, M. Inc. Opt. Quantum Electron., 53 (12) (2021), pp. 1-17
[42]	X. Geng, G. He. J. Math. Phys., 51 (3) (2010), p. 033514
[43]	S. Carillo, M.L. Schiavo, C. Schiebold, et al.. SIGMA, 12 (2016), p. 087
[44]	R. Hirota, J. Satsuma. Prog. Theor. Phys.,57 (3) (1977), pp. 797-807 DOI: 10.1143/PTP.57.797
[45]	S. Rizvi, A.R. Seadawy, S. Abbas, S. Latif, S. Althobaiti. Comput. Appl. Math.,41 (1) (2022), pp. 1-18 DOI: 10.1504/ijmor.2022.10052805
[46]	E. Zayed, S.H. Ibrahim. Chin. Phys. Lett., 29 (6) (2012), p. 060201
[47]	H. Jafari, N. Kadkhoda, C.M. Khalique. Comput. Math. Appl., 64 (6) (2012), pp. 2084-2088
[48]	H. Rehman, A.R. Seadawy, M. Younis, S. Rizvi, I. Anwar, M. Baber, A. Althobaiti. Results Phys., 33 (2022), p. 105069
[49]	G.W. Bluman, S. Kumei. Symmetries and Differential Equations. vol. 81, Springer Science and Business Media (2013)
[50]	B. Bira, T.R. Sekhar, D. Zeidan. Math. Methods Appl. Sci., 41 (16) (2018), pp. 6717-6725 DOI: 10.1002/mma.5186
[51]	S.-y. Lou, L.L. Chen. J. Math. Phys., 40 (12) (1999), pp. 6491-6500
[52]	C.-Q. Dai, Y.Y. Wang. Commun. Nonlinear Sci. Numer. Simul., 19 (1) (2014), pp. 19-28
[53]	X. Geng. J. Phys. A, 36 (9) (2003), p. 2289
[54]	Zhaqilao. Phys. Lett. A, 377 (42) (2013), pp. 3021-3026
[55]	Zhaqilao Z.B. Li. Mod. Phys. Lett. B, 22 (30) (2008), pp. 2945-2966
[56]	X. Geng, Y. Ma. Phys. Lett. A, 369 (4) (2007), pp. 285-289
[57]	S.T.R. Rizvi, K. Ali, A. Bekir, B. Nawaz, M. Younis. Qual. Theory Dyn. Syst.,21 (1) (2022), pp. 1-14 DOI: 10.1504/ijmor.2022.10052805
[58]	M.-D. Chen, X. Li, Y. Wang, B. Li. Commun. Theor. Phys., 67 (6) (2017), p. 595 DOI: 10.1088/0253-6102/67/6/595
[59]	Y. Tang, S. Tao, M. Zhou, Q. Guan. Nonlinear Dyn., 89 (1) (2017), pp. 429-442 DOI: 10.1007/s11071-017-3462-9
[60]	A.M. Wazwaz. Appl. Math. Comput., 215 (4) (2009), pp. 1548-1552
[61]	A.M. Wazwaz. Math. hods Appl. Sci., 36 (3) (2013), pp. 349-357 DOI: 10.1002/mma.2600
[62]	A.M. Wazwaz. Cent. Eur. J. Eng., 4 (4) (2014), pp. 352-356
[63]	N. Liu, Y. Liu. Comput. Math. Appl., 71 (8) (2016), pp. 1645-1654
[64]	M. Niwas, S. Kumar, H. Kharbanda. J. Ocean Eng. Sci., 7 (2) (2022), pp. 188-201
[65]	M.M. Miah, H.S. Ali, M.A. Akbar, A.R. Seadawy. J. Ocean Eng. Sci., 4 (2) (2019), pp. 132-143
[66]	S. Singh, S.S. Ray. J. Ocean Eng. Sci. (2022)
[67]	A. Akbulut, M. Kaplan, M.K. Kaabar. J. Ocean Eng. Sci. (2021)
[68]	H. Esen, N. Ozdemir, A. Secer, M. Bayram. J. Ocean Eng. Sci. (2021)
[69]	S. Kumar, A. Kumar, B. Mohan. J. Ocean Eng. Sci. (2021)
[70]	S. Kumar, S. Rani. J. Ocean Eng. Sci. (2021)
[71]	L. Ravi, S.S. Ray, S. Sahoo. J. Ocean Eng. Sci., 2 (1) (2017), pp. 34-46
[72]	R. Hirota. The Direct Method in Soliton Theory. vol. 155, Cambridge University Press (2004)
[73]	B. Ghanbari. Results Phys., 29 (2021), p. 104689

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