Abstract
In this article, we discuss the wave behavior to the \((3+1)\)-dimensional dynamical nonlinear model which models the waves in the shallow water. Some natural issues, including tides, storms, atmospheric flows, and tsunamis, are associated with shallow water waves. Long water waves, also known as shallow water waves, are water waves that have a large water wavelength in relation to their depth. The Hirota bilinear method (HBM) together with different test functions is used to secure the diversity of wave structures. The Hirota method is a well-known and reliable mathematical tool for finding soliton solutions of nonlinear partial differential equations (NLPDEs) in many fields, such as mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and others. However, it demands bilinearization of nonlinear PDEs. The solutions in different kinds such as breather-type, lump-periodic, rouge waves and two wave solutions are extracted. NLPDEs are well-explained by the applied technique since it offers previously derived solutions and also extracts new exact solutions by incorporating the results of multiple procedures. Moreover, in explaining the physical representation of certain solutions, we also plot 3D, 2D, and contour graphs using the corresponding parameter values. This paper’s findings can enhance the nonlinear dynamical behavior of a given system and demonstrate the efficacy of the employed methodology. We believe that a large number of specialists in engineering models will benefit from this research. The results indicate that the employed algorithm is effective, swift, concise, and applicable to complex systems.
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References
J. Manafian, O.A. Ilhan, A. Alizadeh, Periodic wave solutions and stability analysis for the KP-BBM equation with abundant novel interaction solutions. Phys. Scr. 95, 065203 (2020)
O.A. Ilhan, J. Manafian, A. Alizadeh, S.A. Mohammed, M lump and interaction between M lump and N stripe for the third-order evolution equation arising in the shallow water. Adv. Differ. Equ. 2020, 207 (2020)
J. Manafian, O.A. Ilhan, A. Alizadeh, S.A. Mohammed, Multiple rogue wave and solitary solutions for the generalized BK equation via Hirota bilinear and SIVP schemes arising in fluid mechanics. Commun. Theor. Phys. 72, 075002 (2020)
H.M. Baskonus, T.A. Sulaiman, H. Bulut, Bright, dark optical and other solitons to the generalized higher-order NLSE in optical fibers. Opt. Quant. Electron. 50(6), 1–12 (2018)
U. Younas, J. Ren, Investigation of exact soliton solutions in magneto-optic waveguides and its stability analysis. Results Phys. 21, 103816 (2021)
O.A. Ilhan, J. Manafian, A. Alizadeh, H.M. Baskonus, New exact solutions for nematicons in liquid crystals by the tan \((\frac{\phi }{2})\) expansion method arising in fluid mechanics. Eur. Phys. J. Plus 135, 313 (2020)
E. Alimirzaluo, M. Nadjafikhah, J. Manafian, Some new exact solutions of \((3+1)\)-dimensional Burgers system via Lie symmetry analysis. Adv. Differ. Equ. 2021, 60 (2021)
M. Dehghan, J. Manafian, The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method. Z. Naturforsch. A 64a, 420–30 (2009)
H.M. Baskonus, W. Gao, Investigation of optical solitons to the nonlinear complex Kundu–Eckhaus and Zakharov–Kuznetsov–Benjamin–Bona–Mahony equations in conformable. Opt. Quant. Electron. 54, 388 (2022)
H.M. Baskonus, J.L.G. Guirao, A. Kumar, F.S.V. Causanilles, G.R. Bermudez, Complex mixed dark-bright wave patterns to the modified \(\alpha\) and modified Vakhnenko–Parkes equations. Alex. Eng. J. 59(4), 2149–2160 (2020)
G. Yel, H.M. Baskonus, H. Bulut, Regarding some novel exponential travelling wave solutions to the Wu–Zhang system arising in nonlinear water wave model. Indian J. Phys. 93(8), 1031–1039 (2019)
D. Guo, S.F. Tian, T.T. Zhang, J. Li, Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrodinger system. Non-linear Dyn. 94(4), 2749–2761 (2018)
H.F. Ismael, H.M. Baskonus, H. Bulut, Abundant novel solutions of the conformable Lakshmanan–Porsezian–Daniel model. Discrete Contin. Dyn. Syst. 14(7), 2311 (2021)
Y.M. Li, H.M. Baskonus, A.M. Khudhur, Investigations of the complex wave patterns to the generalized CalogeroBogoyavlenskii–Schiff equation. Soft. Comput. 25(10), 6999–7008 (2021)
G. Yel, H.M. Baskonus, Solitons in conformable time-fractional Wu–Zhang system arising in coastal design. Pramana 93(4), 1–10 (2019)
H.M. Baskonus, C. Cattani, A. Ciancio, Periodic, complex and kink-type solitons for the nonlinear model in microtubules. Appl. Sci. 21, 34–45 (2019)
M. Eslami, Trial solution technique to chiral nonlinear Schrodinger’s equation in \((1+2)\)-dimensions. Nonlinear Dyn. 85(2), 813–816 (2016)
H. Bulut, H.A. Isik, T.A. Sulaiman, On some complex aspects of the \((2+1)\)-dimensional Broer–Kaup–Kupershmidt system. ITM Web Confer. 13, 01019 (2017)
M.M.A. Khater, A.R. Seadawy, D. Lu, Optical soliton and rogue wave solutions of the ultra-short femto-second pulses in an optical fiber via two different methods and its applications. Optik 158, 434–450 (2018)
A. Jhangeer, H.M. Baskonus, G. Yel, W. Gao, New exact solitary wave solutions, bifurcation analysis and first order conserved quantities of resonance nonlinear Schrödinger’s equation with Kerr law nonlinearity. J. King Saud Univer.—Sci. 33, 101180 (2021)
M. Younis, U. Younas, S.U. Rehman, M. Bilal, A. Waheed, Optical bright-dark and Gaussian soliton with third order dispersion. Optik 134, 233–238 (2017)
D. Lu, A.R. Seadawy, M.M.A. Khater, Dispersive optical soliton solutions of the generalized Radhakrishnan–Kundu–Lakshmanan dynamical equation with power law nonlinearity and its applications. Optik 164, 54–64 (2018)
W. Gao, M. Senel, G. Yel, H.M. Baskonus, B. Senel, New complex wave patterns to the electrical transmission line model arising in network system. Aims Math. 5(3), 1881–1892 (2020)
U. Younas, T.A. Sulaiman, J. Ren, A. Yusuf, Lump interaction phenomena to the nonlinear ill-posed Boussinesq dynamical wave equation. J. Geom. Phys. 178, 104586 (2022)
U. Younas, H. Rezazadeh, J. Ren, M. Bilal, Propagation of diverse exact solitary wave solutions in separation phase of iron \(Fe-Cr-X(X=Mo, Cu)\) for the ternary alloys. Int. J. Mod. Phys. B 36(04), 2250039 (2022)
U. Younas, J. Ren, T.A. Sulaiman, M. Bilal, A. Yusuf, On the lump solutions, breather waves, two-wave solutions of \((2+1)\)-dimensional Pavlov equation and stability analysis. Mod. Phys. Lett. B 36, 2250084 (2022)
U. Younas, T.A. Sulaiman, J. Ren, On the optical soliton structures in the magneto electro-elastic circular rod modeled by nonlinear dynamical longitudinal wave equation. Opt. Quant. Electron. 54, 688 (2022)
W.X. Ma, Z. Qin, X. Lu, Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–931 (2016)
X. Yong, W.X. Ma, Y. Huang, Y. Liu, Lump solutions to the Kadomtsev–Petviashvili I equation with a self-consistent source. Comput. Math. Appl. 75, 3414–3419 (2018)
J.Y. Yang, W.X. Ma, Abundant lump-type solutions of the Jimbo–Miwa equation in \((3+1)\)-dimensions. Comput. Math. Appl. 73, 220–225 (2017)
Y. Tang, S. Tao, Q. Guan, Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput. Math. Appl. 72, 2334–2342 (2016)
D.J. Kaup, The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction. J. Math. Phys. 22, 1176–1181 (1981)
H.Q. Zhang, W.X. Ma, Lump solutions to the \((2+1)\)-dimensional Sawada–Kotera equation. Nonlinear Dyn. 87, 2305–2310 (2017)
S.T. Chen, W.X. Ma, Lump solutions to a generalized Bogoyavlensky–Konopelchenko equation. Front. Math. China 13, 525–534 (2018)
Y. Shen, B. Tian, S.H. Liu, Solitonic fusion and fission for a \((3 +1)\)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Phys. Lett. A 405, 1544–1556 (2019)
Y. Feng, Y. Gao, L. Li, T. Jia, Bilinear form and solutions of a \((3+1)\)-dimensional generalized nonlinear evolution equation for the shallow-water waves. Appl. Anal. 100, 127429 (2021)
W. Liu, X. Zheng, C. Wang, S. Li, Fission and fusion collision of high-order lumps and solitons in a \((3 + 1)\)-dimensional nonlinear evolution equation. Nonlinear Dyn. 96, 2463–2473 (2019)
M.T. Darvishi, L. Kavitha, M. Najafi, V.S. Kumar, Elastic collision of mobile solitons of a \((3+1)\)-dimensional soliton equation. Nonlinear Dyn. 86, 765–778 (2016)
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The authors would like to acknowledge the financial support provided for this research via the National Natural Science Foundation of China (52071298).
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Younas, U., Sulaiman, T.A. & Ren, J. On the collision phenomena to the \((3+1)\)-dimensional generalized nonlinear evolution equation: Applications in the shallow water waves. Eur. Phys. J. Plus 137, 1166 (2022). https://doi.org/10.1140/epjp/s13360-022-03401-3
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DOI: https://doi.org/10.1140/epjp/s13360-022-03401-3