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Study of W-shaped, V-shaped, and other type of surfaces of the ZK-BBM and GZD-BBM equations

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Abstract

The Zakharov-Kuznetsov Benjamin-Bona-Mahony equation and its generalized form, considered in this study are two notable models for describing the magneto-acoustic waves in plasma, acoustic-gravity waves, the acoustic waves in harmonic crystals, long-wavelength in liquids, hydro-magnetic waves, shallow water waves etc. The sine-Gordon expansion (SGE) approach is put to use to acquire the broad-spectral typical solitary wave solutions from the exact solutions and to establish new shape of surfaces, namely the W-shaped, V-shaped, parabolic, compacton, bright and dark soliton for specific parameter values. Different types of solitons in terms of hyperbolic, and trigonometric functions are achieved. We present three-dimensional, two-dimensional, and contour plots of the results obtained through setting different parametric values to objectify the facts modulated by the formerly acknowledged models by computerized software Matlab. The solutions achieved prove that the SGE approach is a powerful and effective technique in physical sciences and engineering for analyzing nonlinear evolutionary equations.

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References

  • Abdel-Gawad, H.I., Tantawy, M., Inc, M., Yusuf, A.: Construction of rogue waves and conservation laws of the complex coupled Kadomtsev-Petviashvili equation. Int. J. Mod. Phys. B 34(12), 2050115 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Akbar, M.A., Ali, N.H.M., Roy, R.: Closed form solutions of two time fractional nonlinear wave equations. Results Phys. 9, 1031–1039 (2018)

    Article  ADS  Google Scholar 

  • Akbar, M.A., Kayum, M.A., Osman, M.S., Abdel-Aty, A.H., Eleuch, H.: Analysis of voltage and current flow of electrical transmission lines through mZK equation. Results in Phy. 20, 103696 (2021)

    Article  Google Scholar 

  • Akbulut, A., Kaplan, M., Tascan, F.: The investigation of exact solutions of nonlinear partial differential equations by using method. Optik 132, 382–387 (2017)

    Article  ADS  Google Scholar 

  • Ali, K.K., Osman, M.S., Abdel-Aty, M.: New optical solitary wave solutions of Fokas-Lenells equation in optical fiber via Sine-Gordon expansion method. Alex. Eng. J. 59(3), 1191–1196 (2020)

    Article  Google Scholar 

  • Ali, K.K., Wazwaz, A.M., Osman, M.S.: Optical soliton solutions to the generalized nonautonomous nonlinear Schrödinger equations in optical fibers via the sine-Gordon expansion method. Optik 208, 164132 (2020)

    Article  ADS  Google Scholar 

  • Ali, K.K., Cattani, C., Gómez-Aguilar, J.F., Baleanu, D., Osman, M.S.: Analytical and numerical study of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model. Chaos Solitons & Fractals 139, 110089 (2020)

    Article  MathSciNet  Google Scholar 

  • Aliyu, A.I., Li, Y., Inc, M., Baleanu, D., Alshomrani, A.S.: Solitons and complexitons to the (2+1)-dimensional Heisenberg ferromagnetic spin chain model. Int. J. Mod. Phys. B 33(30), 1950368 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Aliyu, A.I., Alshomrani, A.S., Baleanu, D., Inc, M.: Optical solitons for Triki-Biswas equation by two analytic approaches. AIMS Math. 5(2), 1001 (2020)

    Article  MathSciNet  Google Scholar 

  • Alquran, M., Al-Khaled, K., Ananbeh, H.: New soliton solutions for systems of nonlinear evolution equations by the rational sine-cosine method. Stud. Math. Sci. 3(1), 1–9 (2011)

    Google Scholar 

  • Arnous, A.H., Biswas, A., Ekici, M., Alzahrani, A.K., Belic, M.R.: Optical solitons and conservation laws of Kudryashov’s equation with improved modified extended tanh-function. Optik 225, 165406 (2021)

    Article  ADS  Google Scholar 

  • Arshed, S., Biswas, A., Guggilla, P., Alshomrani, A.S.: Optical solitons for Radhakrishnan–Kundu–Lakshmanan equation with full nonlinearity. Physics Letters A 384(26), 126191 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Barman, H.K., Seadawy, A.R., Akbar, M.A., Baleanu, D.: Competent closed form soliton solutions to the Riemann wave equation and the Novikov-Veselov equation. Results in Phy. 17, 103131 (2020)

    Article  Google Scholar 

  • Barman, H.K., Roy, R., Mahmud, F., Akbar, M.A., Osman, M.S.: Harmonizing wave solutions to the Fokas-Lenells model through the generalized Kudryashov method. Optik 229, 166294 (2021)

    Article  ADS  Google Scholar 

  • Baskonus, H.M.: New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlinear Dyn. 86(1), 177–183 (2016)

    Article  MathSciNet  Google Scholar 

  • Baskonus, H.M., Bulut, H.: On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method. Wave Random Complex 25(4), 720–728 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Baskonus, H.M., Eskitascioglu, E.I.: Complex wave surfaces to the extended shallow water wave model with (2+1)-dimensional. Comput. Methods Diff. Equ. 8(3), 585–596 (2020)

    MathSciNet  MATH  Google Scholar 

  • Baskonus, H.M., Bulut, H., Sulaiman, T.A.: New complex hyperbolic structures to the Lonngren-wave equation by using sine-Gordon expansion method. Appl. Math. Nonlinear Sci. 4(1), 129–138 (2019)

    Article  MathSciNet  Google Scholar 

  • Baskonus, H.M., Guirao, J.L.G., Kumar, A., Causanilles, F.S.V., Bermudez, G.R.: Complex mixed dark-bright wave patterns to the modified α and modified Vakhnenko-Parkes equations. Alex. Eng. J. 59(4), 2149–2160 (2020)

    Article  Google Scholar 

  • Bayones, F.S., Nisar, K.S., Khan, K.A., Raza, N., Hussien, N.S., Osman, M.S., Abualnaja, K.M.: Magneto-hydrodynamics (MHD) flow analysis with mixed convection moves through a stretching surface. AIP Advances 11(4), 045001 (2021)

    Article  ADS  Google Scholar 

  • Bibi, S., Mohyud-Din, S.T.: Traveling wave solutions of ZK-BBM equation sine-cosine method. Commun. Numer. Anal. 2014, 1–9 (2014). https://doi.org/10.5899/2014/cna-00154

    Article  MathSciNet  Google Scholar 

  • Chen, Y., Khater, M.M.A., Inc, M., Attia, R.A.M., Lu, D.: Abundant analytical solutions of the fractional nonlinear (2+1)-dimensional BLMP equation arising in incompressible fluid. Int. J. Mod. Phys. B 34(09), 2050084 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Chowdhury, M.A., Miah, M.M., Ali, H.S., Chu, Y.M., Osman, M.S.: An investigation to the nonlinear (2+ 1)-dimensional soliton equation for discovering explicit and periodic wave solutions. Results in Phy. 23, 104013 (2021)

    Article  Google Scholar 

  • Dhawan, S., Machado, J.A.T., Brzeziński, D.W., Osman, M.S.: A chebyshev wavelet collocation method for some types of differential problems. Symmetry 13(4), 536 (2021)

    Article  Google Scholar 

  • Gao, W., Senel, M., Yel, G., Baskonus, H.M., Senel, B.: New complex wave patterns to the electrical transmission line model arising in network system. AIMS Math 5(3), 1881–1892 (2020)

    Article  MathSciNet  Google Scholar 

  • Ghanbari, B., Inc, M., Yusuf, A., Baleanu, D.: New solitary wave solutions and stability analysis of the Benney-Luke and the Phi-4 equations in mathematical physics. AIMS Math. 4(6), 1523–1539 (2019)

    Article  MathSciNet  Google Scholar 

  • Guirao, J.L.G., Baskonus, H.M., Kumar, A.: Regarding new wave patterns of the newly extended nonlinear (2+1)-dimensional Boussinesq equation with fourth order. Math. 8(3), 341 (2020)

    Article  MathSciNet  Google Scholar 

  • Güner, Ö., Bekir, A., Moraru, L., Biswas, A.: Bright and dark soliton solutions of the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony nonlinear evolution equation. Proc. Rom. Acad. Ser. A 16, 422–429 (2015)

    MathSciNet  Google Scholar 

  • Guo, M., Dong, H., Liu, J., Yang, H.: The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basic function method. Nonlinear Anal-Model. 24(1), 1–19 (2019)

    MathSciNet  MATH  Google Scholar 

  • Houwe, A., Abbagari, S., Salathiel, Y., Inc, M., Doka, S.Y., Crepin, K.T., Baleanu, D.: Complex traveling-wave and solitons solutions to the Klein-Gordon-Zakharov equations. Results in Phy. 17, 103127 (2020)

    Article  Google Scholar 

  • Inc, M., Aliyu, A.I., Yusuf, A., Baleanu, D.: Optical solitons for Biswas-Milovic model in nonlinear optics by sine-Gordon equation method. Optik 157, 267–274 (2018)

    Article  ADS  Google Scholar 

  • Iqbal, A.A., Mohyud-Din, S.T.: Solitary wave solutions Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation. J. Egyptian Math. Soc. 24(1), 44–48 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Islam, M.E., Barman, H.K., Akbar, M.A.: Search for interactions of phenomena described by the coupled Higgs field equation through analytical solutions. Opt. Quant. Electron. 52(468), 1–19 (2020)

    Google Scholar 

  • Kadkhoda, N., Jafari, H.: Analytical solutions of the Gerdjikov-Ivanov equation by using -expansion method. Optik 139, 72–76 (2017)

    Article  ADS  Google Scholar 

  • Kayum, M.A., Ara, S., Barman, H.K., Akbar, M.A.: Soliton solutions to voltage analysis in nonlinear electrical transmission lines and electric signals in telegraph lines. Results in Phy. 18, 103269 (2020)

    Article  Google Scholar 

  • Kayum, M.A., Akbar, M.A., Osman, M.S.: Competent closed form soliton solutions to the nonlinear transmission and the low-pass electrical transmission lines. Eur. Phys. J. Plus 135(7), 1–20 (2020)

    Article  Google Scholar 

  • Kayum, M.A., Akbar, M.A., Osman, M.S.: Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma. Wave Random Complex (2020). https://doi.org/10.1080/17455030.2020.1831711

    Article  Google Scholar 

  • Kayum, M.A., Seadawy, A.R., Akbar, A.M., Sugati, T.G.: Stable solutions to the nonlinear RLC transmission line equation and the Sinh-Poisson equation arising in mathematical physics. Open Phys. 18(1), 710–725 (2020)

    Article  Google Scholar 

  • Kayum, M.A., Ara, S., Osman, M.S., Akbar, M.A., Gepreel, K.A.: Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics. Results in Phys. 20, 103762 (2021)

    Article  Google Scholar 

  • Kayum, M.A., Ara, S., Osman, M.S., Akbar, M.A., Gepreel, K.A.: Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics. Results in Phy. 20, 103762 (2021)

    Article  Google Scholar 

  • Korpinar, Z., Tchier, F., Inc, M.: On optical solitons of the fractional (3+1)-dimensional NLSE with conformable derivatives. Front. Phys. 8, 87 (2020)

    Article  Google Scholar 

  • Korpinar, Z., Inc, M., Bayram, M., Hashemi, M.S.: New optical solitons for Biswas-Arshed equation with higher order dispersions and full nonlinearity. Optik 206, 163332 (2020)

    Article  ADS  Google Scholar 

  • Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Soliton Fract. 24(5), 1217–1231 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kumar, R., Kumar, M., Kumar, A.: Some soliton solutions of nonlinear partial differential equations by Tan-Cot method. IOSR J. Math. 6(6), 23–28 (2013)

    Article  Google Scholar 

  • Kumar, D., Seadawy, A.R., Chowdhury, R.: On new complex soliton structures of the nonlinear partial differential equation describing the pulse narrowing nonlinear transmission lines. Opt. Quant. Electron. 50(2), 1–14 (2018)

    Article  Google Scholar 

  • Li, C., Chen, L., Li, G.: Optical solitons of space-time fractional Sasa-Satsuma equation by F-expansion method. Optik 224, 165527 (2020)

    Article  ADS  Google Scholar 

  • Lu, B.: The first integral method for some time fractional differential equations. J. Math. Anal. Appl. 395(2), 684–693 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, W.X.: Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations. Appl. Math. Lett. 102, 106161 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, W.X., Osman, M.S., Arshed, S., Raza, N., Srivastava, H.M.: Practical analytical approaches for finding novel optical solitons in the single-mode fibers. Chin. J. Phys. 72, 475–486 (2021)

    Article  MathSciNet  Google Scholar 

  • Mahmoud, A.E., Ammar, A.S., Abualnaja, K.M., Inc, M.: New solutions for the unstable nonlinear Schrodinger equation arising in natural science. AIMS Math. 5(3), 1893–1912 (2020)

    Article  MathSciNet  Google Scholar 

  • Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu-Eckhaus equation via -expansion method. Optik 127(14), 5543–5551 (2016)

    Article  ADS  Google Scholar 

  • Naher, H.: New approach of -expansion method and new approach of generalized -expansion method for ZKBBM equation. J. Egyptian Math. Soc. 23(1), 42–48 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Osman, M.S., Rezazadeh, H., Eslami, M., Neirameh, A., Mirzazadeh, M.: Analytical study of solitons to benjamin-bona-mahony-peregrine equation with power law nonlinearity by using three methods. U. Politeh. Buch. Ser. A 80(4), 267–278 (2018)

    MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Roshid, H.O., Roshid, M.M., Rahman, N., Pervin, M.R.: New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation. Propuls. Power Res. 6(1), 49–57 (2017)

    Article  Google Scholar 

  • Roy, R., Akbar, M.A.: A new approach to study nonlinear space-time fractional sine-Gordon and Burgers equations. IOP SciNotes 1(3), 035003 (2020)

    Article  ADS  Google Scholar 

  • Roy, R., Akbar, M.A., Wazwaz, A.M.: Exact wave solutions for the nonlinear time fractional Sharma-Tasso-Olver equation and the fractional Klein-Gordon equation in mathematical physics. Opt. Quant. Electron. 50(1), 25 (2018)

    Article  Google Scholar 

  • Roy, R., Roy, S., Hossain, M.N., Alam, M.Z.: Study on nonlinear partial differential equation by implementing MSE method. Glob Scientific J 8(1), 1651–1665 (2020)

    Google Scholar 

  • Roy, R., Akbar, M.A., Seadawy, A.R., Baleanu, D.: Search for adequate closed form wave solutions to space–time fractional nonlinear equations. Partial Differ. Equ. Appl. Math. 3, 100025 (2021)

    Article  Google Scholar 

  • Shakeel, M., Mohyud-Din, S.T.: New -expansion method and its application to the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation. J. Assoc. Arab Univ. Basic Appl. Sci. 18, 66–81 (2015)

    Google Scholar 

  • Song, M., Liu, Z.: Periodic wave solutions and their limits for the ZK-BBM equation. Appl. Math. Comput. 232, 9–26 (2014)

    MathSciNet  MATH  Google Scholar 

  • Song, M., Yang, C.: Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. Appl. Math. Comput. 216(11), 3234–3243 (2010)

    MathSciNet  MATH  Google Scholar 

  • Tasbozan, O., Kurt, A.: Approximate analytical solution of ZK-BBM equation. Sohag J. Math 2(2), 57–60 (2015)

    Google Scholar 

  • Wael, W.M., Abdelrahman, M.A.E., Inc, M., Hamza, A.E., Akinlar, M.A.: Soliton solutions for system of ion sound and Langmuir waves. Opt. Quant. Electron. 52(460), 1–10 (2020)

    Google Scholar 

  • Wazwaz, A.M.: Compact and noncompact physical structures for the ZK-BBM equation. Appl. Math. Comput. 169(1), 713–725 (2005)

    MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh-coth method. Appl. Math. Comput. 190(1), 633–640 (2007)

    MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188(2), 1467–1475 (2007)

    MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations. Chaos Soliton Fract. 38(5), 1505–1516 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: Partial differential equations and solitary waves theory. Springer Science & Business Media, New York, USA (2010)

    Google Scholar 

  • Wazwaz, A.M.: Bright and dark optical solitons for (2+ 1)-dimensional Schrödinger (NLS) equations in the anomalous dispersion regimes and the normal dispersive regimes. Optik 192, 162948 (2019)

    Article  ADS  Google Scholar 

  • Yang, L.: Application of classification of traveling wave solutions to the Zakhrov-Kuznetsov-Benjamin-Bona-Mahony equation. Appl. Math. 5(10), 1432 (2014)

    Article  Google Scholar 

  • Yang, D.Y., Tian, B., Qu, Q.X., Yuan, Y.Q., Zhang, C.R., Tian, H.Y.: Generalized Darboux transformation and the higher-order semirational solutions for a non-linear Schrödinger system in a birefringent fiber. Modern Phy. Lett. B 34(supp01), 2150013 (2020)

    Article  ADS  Google Scholar 

  • Yel, G.: New wave patterns to the doubly dispersive equation in nonlinear dynamic elasticity. Pramana 94, 1–8 (2020)

    Article  ADS  Google Scholar 

  • Yel, G., Baskonus, H.M., Bulut, H.: Novel archetypes of new coupled Konno-Oono equation by using sine-Gordon expansion method. Opt. Quant. Electron. 49(9), 1–10 (2017)

    Article  Google Scholar 

  • Yıldırım, Y.: Optical solitons with Biswas-Arshed equation by sine–Gordon equation method. Optik 223, 165622 (2020)

    Article  ADS  Google Scholar 

  • Yıldırım, Y., Mirzazadeh, M.: Optical pulses with Kundu-Mukherjee-Naskar model in fiber communication systems. Chin. J. Phys. 64, 183–193 (2020)

    Article  MathSciNet  Google Scholar 

  • Yu, W., Zhou, Q., Mirzazadeh, M., Liu, W., Biswas, A.: Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics. J. Adv. Res. 15, 69–76 (2019)

    Article  Google Scholar 

  • Yusuf, A., Sulaiman, T.A., Inc, M., Bayram, M.: Breather wave, lump-periodic solutions and some other interaction phenomena to the Caudrey-Dodd-Gibbon equation. Eur. Phys. J. Plus 135, 1-8 (2020)

    Article  Google Scholar 

  • Zhang, Z.Y., Li, Y.X., Liu, Z.H., Miao, X.J.: New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity via modified trigonometric function series method. Commun. Nonlinear Sci. Numer. Simul. 16(8), 3097–3106 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Zhao, Juan, and Wei Li.: Exact solitary wave solution in the ZK-BBM equation. J. Nonlinear Dyn. 2014,  (2014) 

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

    Md. Abdul Kayum & M. Ali Akbar

  2. Department of Mathematics, Bangamata Sheikh Fojilatunnesa Mujib Science & Technology University, Jamalpur, Bangladesh

    Ripan Roy

  3. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    M. S. Osman

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  1. Md. Abdul Kayum
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  2. Ripan Roy
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  3. M. Ali Akbar
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  4. M. S. Osman
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Kayum, M.A., Roy, R., Akbar, M.A. et al. Study of W-shaped, V-shaped, and other type of surfaces of the ZK-BBM and GZD-BBM equations. Opt Quant Electron 53, 387 (2021). https://doi.org/10.1007/s11082-021-03031-6

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