Skip to main content
SpringerLink
Log in

Lie vector fields, conservation laws, bifurcation analysis, and Jacobi elliptic solutions to the Zakharov–Kuznetsov modified equal-width equation

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

The present paper intends to thoroughly study an evolutionary model called the Zakharov–Kuznetsov modified equal-width (ZK–MEW) equation. More precisely, Lie symmetries as well as invariant solutions to the ZK–MEW equation describing shallow and stratified waves in nonlinear LC circuits are first derived, and then a general theorem established by Ibragimov is adopted to retrieve its conservation laws. Additionally, by applying the qualitative theory of dynamical systems, the bifurcation analysis of the dynamical system is carried out and several Jacobi elliptic solutions to the ZK–MEW equation are formally constructed. In some case studies, the impact of the nonlinear coefficient on the physical features of bright and kink solitary waves as well as periodic continuous waves is examined in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data availability

Due to the nature of the research, supporting data is not available.

References

  • Abdelrahman, M.A.E., Sohaly, M.A., Alharbi, A.: The new exact solutions for the deterministic and stochastic (2+1)-dimensional equations in natural sciences. J. Taibah Univ. Sci. 13, 834–843 (2019)

    Google Scholar 

  • Adeyemo, O.D.: Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+1)-D quantum Zakharov–Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.04.012

    Article  Google Scholar 

  • Adeyemo, O.D., Khalique, C.M.: Dynamical soliton wave structures of one-dimensional Lie subalgebras via group-invariant solutions of a higher-dimensional soliton equation with various applications in ocean physics and mechatronics engineering. Commun. Appl. Math. Comput. 4, 1531–1582 (2022)

    MathSciNet  Google Scholar 

  • Adeyemo, O.D., Khalique, C.M.: Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electromagnetism and complex media. Alex. Eng. J. 73, 751–769 (2023)

    Google Scholar 

  • Adeyemo, O.D., Zhang, L., Khalique, C.M.: Optimal solutions of Lie subalgebra, dynamical system, travelling wave solutions and conserved currents of (3+1)-dimensional generalized Zakharov–Kuznetsov equation type I. Eur. Phys. J. plus 137, 954 (2022a)

    Google Scholar 

  • Adeyemo, O.D., Zhang, L., Khalique, C.M.: Bifurcation theory, Lie group-invariant solutions of subalgebras and conservation laws of a generalized (2+1)-dimensional BK equation type II in plasma physics and fluid mechanics. Math. 10, 2391 (2022b)

    Google Scholar 

  • Ali, K.K., Abd El Salam, M.A., Mohamed, E.M., Samet, B., Kumar, S., Osman, M.S.: Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series. Adv. Differ. Equ. 2020(1), 494 (2020)

    MathSciNet  Google Scholar 

  • Ali, K.K., Tarla, S., Yusuf, A.: Quantum-mechanical properties of long-lived optical pulses in the fourth-order KdV-type hierarchy nonlinear model. Opt. Quant. Electron. 55, 590 (2023a)

    Google Scholar 

  • Ali, K.K., Yusuf, A., Yokus, A., Ali, M.R.: Optical waves solutions for the perturbed Fokas–Lenells equation through two different methods. Results Phys. 53, 106869 (2023b)

    Google Scholar 

  • Almusawa, H.: Lie symmetry analysis and conservation laws of two-mode Cahn–Allen equation. J. Geom. Phys. 179, 104579 (2022)

    MathSciNet  Google Scholar 

  • Alquran, M., Sulaiman, T.A., Yusuf, A., Alshomrani, A.S., Baleanu, D.: Nonautonomous lump-periodic and analytical solutions to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 111, 11429–11436 (2023)

    Google Scholar 

  • Alrebdi, T.A., Raza, N., Salman, F., Alshahrani, B., Abdel-Aty, A.H., Eleuch, H.: Exploration of new solitons and phase characterization for the extended Gerdjikov–Ivanov equation. Results Phys. 41, 105961 (2022)

    Google Scholar 

  • Atas, S.S., Ismael, H.F., Sulaiman, T.A., Bulut, H.: Investigation of some nonlinear physical models: exact and approximate solutions. Opt. Quant. Electron. 55, 293 (2023)

    Google Scholar 

  • Baskonus, H.M., Osman, M.S., Rehman, H.U., Ramzan, M., Tahir, M., Ashraf, S.: On pulse propagation of soliton wave solutions related to the perturbed Chen–Lee–Liu equation in an optical fiber. Opt. Quant. Electron. 53, 556 (2021)

    Google Scholar 

  • Bilal, M., Ren, J., Inc, M., Alqahtani, R.T.: Dynamics of solitons and weakly ion-acoustic wave structures to the nonlinear dynamical model via analytical techniques. Opt. Quant. Electron. 55, 656 (2023)

    Google Scholar 

  • Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2010)

    Google Scholar 

  • Cao, J., Song, M., Biswas, A.: Topological solitons and bifurcation analysis of the PHI-four equation. Bull. Malays. Math. Sci. Soc. 37, 1209–1219 (2014)

    MathSciNet  Google Scholar 

  • Faridi, W.A., Asjad, M.I., Jhangeer, A., Yusuf, A., Sulaiman, T.A.: The weakly non-linear waves propagation for Kelvin–Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory. Opt. Quant. Electron. 55, 172 (2023)

  • Hashemi, M.S., Haji-Badali, A., Alizadeh, F.: Non-classical Lie symmetry and conservation laws of the nonlinear time-fractional Kundu–Eckhaus (KE) equation. Pramana J. Phys. 95, 107 (2021)

    ADS  Google Scholar 

  • Hashemi, M.S., Haji-Badali, A., Alizadeh, F., Yang, X.J.: Non-classical Lie symmetries for nonlinear time-fractional Heisenberg equations. Math. Methods Appl. Sci. 45, 10010–10026 (2022)

    ADS  MathSciNet  Google Scholar 

  • Hosseini, K., Salahshour, S., Mirzazadeh, M., Ahmadian, A., Baleanu, D., Khoshrang, A.: The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation: Its solitons and Jacobi elliptic function solutions. Eur. Phys. J. plus 136, 206 (2021a)

    Google Scholar 

  • Hosseini, K., Matinfar, M., Mirzazadeh, M.: Soliton solutions of high-order nonlinear Schrödinger equations with different laws of nonlinearities. Regul. Chaotic Dyn. 26, 105–112 (2021b)

    ADS  MathSciNet  Google Scholar 

  • Hosseini, K., Hincal, E., Salahshour, S., Mirzazadeh, M., Dehingia, K., Nath, B.J.: On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation. Optik 272, 170215 (2023a)

    ADS  CAS  Google Scholar 

  • Hosseini, K., Alizadeh, F., Hinçal, E., Baleanu, D., Akgül, A., Hassan, A.M.: Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to the nonlinear Kodama equation. Results Phys. 54, 107129 (2023b)

    Google Scholar 

  • Hosseini, K., Hinçal, E., Ilie, M.: Bifurcation analysis, chaotic behaviors, sensitivity analysis, and soliton solutions of a generalized Schrödinger equation. Nonlinear Dyn. 111, 17455–17462 (2023c)

    Google Scholar 

  • Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)

  • Ibragimov, N.H.: Nonlinear self-adjointness and conservation laws. J. Phys. A: Math. Theor. 44, 432002 (2011)

  • Ibrahim, S., Baleanu, D.: Classes of solitary solution for nonlinear Schrödinger equation arising in optical fibers and their stability analysis. Opt. Quant. Electron. 55, 1158 (2023)

    Google Scholar 

  • Inc, M.: New exact solutions for the ZK–MEW equation by using symbolic computation. Appl. Math. Comput. 189, 508–513 (2007)

    MathSciNet  Google Scholar 

  • Jaradat, I., Sulaiman, T.A., Alshomrani, A.S., Yusuf, A., Alquran, M., Baleanu, D.: Optical wave propagation to a nonlinear phenomenon with pulses in optical fiber. Opt. Quant. Electron. 55, 361 (2023)

    Google Scholar 

  • Khalil, T.A., Badra, N., Ahmed, H.M., Rabie, W.B.: Optical solitons and other solutions for coupled system of nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index by Jacobi elliptic function expansion method. Optik 253, 168540 (2022)

    ADS  Google Scholar 

  • Khalique, C.M., Adem, K.R.: Exact solutions of the (2+1)-dimensional Zakharov–Kuznetsov modified equal width equation using Lie group analysis. Math. Comput. Model. 54, 184–189 (2011)

    MathSciNet  Google Scholar 

  • Kudryashov, N.A.: Method for finding highly dispersive optical solitons of nonlinear differential equation. Optik 206, 163550 (2020)

    ADS  Google Scholar 

  • Kumar, D., Park, C., Tamanna, N., Paul, G.C., Osman, M.S.: Dynamics of two-mode Sawada–Kotera equation: mathematical and graphical analysis of its dual-wave solutions. Results Phys. 19, 103581 (2020)

    Google Scholar 

  • Kumar, S., Kumar, A., Inc, M., Alotaibi, M., Abdou, M.A., Akgül, A.: An investigation of (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov system: Lie symmetry reductions, invariant solutions, dynamical behaviors and conservation laws. Results Phys. 43, 106034 (2022)

    Google Scholar 

  • Liu, X., Jiao, Y., Wang, Y., Zhou, Q., Wang, W.: Kink soliton behavior study for systems with power-law nonlinearity. Results Phys. 33, 105162 (2022)

    Google Scholar 

  • Ma, H.C., Zhang, Z.P., Deng, A.P.: A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation. Acta Math. Appl. Sin. 28, 409–415 (2012)

    MathSciNet  Google Scholar 

  • Narain, R., Kara, A.H.: On the redefinition of the variational and partial variational conservation laws in a class of nonlinear PDEs with mixed derivatives. Math. Comput. Appl. 15, 732–741 (2010)

    MathSciNet  Google Scholar 

  • Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer-Verlag, Heidelberg (1986)

    Google Scholar 

  • Sarker, S., Said, G.S., Tharwat, M.M., Karim, R., Akbar, M.A., Elazab, N.S., Osman, M.S., Dey, P.: Soliton solutions to a wave equation using the (ϕ'/ϕ)-expansion method. Partial Differ. Equ. Appl. Math. 8, 100587 (2023)

  • Song, M., Wang, B., Li, S.: Bifurcation of traveling wave solutions for (1+1)-dimensional resonant nonlinear Schrödinger equation. J. Math. Anal. Appl. 509, 125965 (2022)

    Google Scholar 

  • Tarla, S., Ali, K.K., Yusuf, A.: Exploring new optical solutions for nonlinear Hamiltonian amplitude equation via two integration schemes. Phys. Scr. 98, 095218 (2023)

    ADS  Google Scholar 

  • Wazwaz, A.M.: Exact solutions for the ZK–MEW equation by using the tanh and sine-cosine methods. Int. J. Comput. Math. 82, 699–708 (2005)

    MathSciNet  Google Scholar 

  • Xu, F., Yan, W., Chen, Y.L., Li, C.Q., Zhang, Y.N.: Evaluation of two-dimensional ZK–MEW equation using the Exp-function method. Comput. Math. Appl. 58, 2307–2312 (2009)

    MathSciNet  Google Scholar 

  • Yang, Q., Zhang, H.: On the exact soliton solutions of fifth-order Korteweg-de Vries equation for surface gravity waves. Results Phys. 26, 104424 (2021)

    Google Scholar 

  • Zhu, H.M., Zhang, Z.Y., Zheng, J.: The time-fractional (2+1)-dimensional Hirota–Satsuma–Ito equations: Lie symmetries, power series solutions and conservation laws. Commun. Nonlinear Sci. Numer. Simul. 115, 106724 (2022)

    MathSciNet  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, Near East University TRNC, Mersin 10, 99138, Nicosia, Turkey

    K. Hosseini, F. Alizadeh, K. Sadri & E. Hinçal

  2. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    K. Hosseini

  3. Department of Mathematics, Art-Science Faculty, Uludag University, 16059, Bursa, Turkey

    A. Akbulut

  4. Mathematics Department, Faculty of Science, King Abdulaziz University, 21521, Jeddah, Saudi Arabia

    H. M. Alshehri

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

    M. S. Osman

Authors
  1. K. Hosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Alizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Sadri
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Hinçal
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Akbulut
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. H. M. Alshehri
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. M. S. Osman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

KH and FA wrote the original draft, KS, EH, and AA revised the original draft, and HMA and MSO reviewed the revised draft. All authors reviewed the manuscript.

Corresponding authors

Correspondence to K. Hosseini or M. S. Osman.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Ethical approval

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hosseini, K., Alizadeh, F., Sadri, K. et al. Lie vector fields, conservation laws, bifurcation analysis, and Jacobi elliptic solutions to the Zakharov–Kuznetsov modified equal-width equation. Opt Quant Electron 56, 506 (2024). https://doi.org/10.1007/s11082-023-06086-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-06086-9

Keywords

Search

Navigation