Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Nonlinear Dispersion Dynamics of Optical Solitons of Zoomeron Equation with New $varphi ^{6}$-Model Expansion Approach

Journal of Vibration Testing and System Dynamics 8(3) (2024) 285--307 | DOI:10.5890/JVTSD.2024.09.002

Muhammad Abubakar Isah$^1$, Asif Yokus$^{1,2}$

$^{1}$ Firat University, Faculty of Science, Department of Mathematics, Elazig, Turkey

$^{2}$ Application and Research Center Advisory Board Member, Istanbul Commerce University, Istanbul, Turkey

Download Full Text PDF



One of the equations describing incognito evolution, the nonlinear Zoomeron equation, is studied in this work. In a variety of physical circumstances, including laser physics, fluid dynamics and nonlinear optics, solitons with particular properties arise and the Zoomeron equation is a single example of one such situation. The method of $\varphi^6$-model expansion allows for the explicit retrieval of a wide range of solution types, including kink-type solitons, these solitons are also called topological solitons in the context of water waves, their velocities do not depend on the wave amplitude, others are bright, singular, periodic and combined singular soliton solutions. The outcomes of this research may improve the Zoomeron equation's nonlinear dynamical features. The method proposes a practical and effective approach for solving a large class of nonlinear partial differential equations. The nonlinear dispersion behavior is analyzed for different values of the magnitude, which physically represents the wave velocity, from the parameters of the generated traveling wave solutions. Interesting graphs are employed to explain and highlight the dynamical aspects of the results, and all of the obtained results are put into the Zoomeron equation to show the accuracy of the results.


  1. [1]  Isah, M.A. and Kulahci, M.A. (2019), Involute Curves in 4-dimensional Galilean space G4, In Conference Proceedings of Science and Technology, 2(2), 134-141.
  2. [2]  Isah, M.A., Isah, I., Hassan, T.L., and Usman, M. (2021), Some characterization of osculating curves according to darboux frame in three dimensional euclidean space, International Journal of Advanced Academic Research, 7(12), 47-56.
  3. [3]  Isah, M.A. and Külahçı, M.A. (2020), Special curves according to bishop frame in minkowski 3-space, Applied Mathematics and Nonlinear Sciences, 5(1), 237-248.
  4. [4]  Isah, I., Isah, M.A., Baba, M.U., Hassan, T.L., and Kabir, K.D. (2021), On integrability of silver riemannian structure, International Journal of Advanced Academic Research, 7(12), 2488-9849.
  5. [5]  Myint-U, T. and Debnath, L. (2007), Linear Partial Differential Equations for Scientists and Engineers, Springer Science \& Business Media.
  6. [6]  Liu, N., Xuan, Z., and Sun, J. (2022), Triple-pole soliton solutions of the derivative nonlinear Schrödinger equation via inverse scattering transform, Applied Mathematics Letters, 125, p.107741.
  7. [7] Yokus, A. and Isah, M.A. (2022), Stability analysis and solutions of (2+1)-Kadomtsev–Petviashvili equation by homoclinic technique based on Hirota bilinear form, Nonlinear Dynamics, 109(4), 3029-3040,
  8. [8]  Yokus, A. and Isah, M.A. (2023), Dynamical behaviors of different wave structures to the Korteweg–de Vries equation with the Hirota bilinear technique, Physica A: Statistical Mechanics and its Applications, 622, p.128819.
  9. [9]  Yokus, A. and Isah, M.A. (2023), March. Stability Analysis and Soliton Solutions of the Nonlinear Evolution Equation by Homoclinic Technique Based on Hirota Bilinear Form. In , 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA) (pp. 1-6). IEEE.
  10. [10]  Durur, H., Yokus, A., and Abro, K.A. (2023), A non-linear analysis and fractionalized dynamics of Langmuir waves and ion sound as an application to acoustic waves, International Journal of Modelling and Simulation, 43(3), 235-241.
  11. [11]  Durur, H. (2021), Energy-carrying wave simulation of the Lonngren-wave equation in semiconductor materials, International Journal of Modern Physics B, 35(21), p.2150213.
  12. [12]  Tarla, S., Ali, K.K., Yilmazer, R., and Osman, M.S. (2022), Propagation of solitons for the Hamiltonian amplitude equation via an analytical technique, Modern Physics Letters B, 36(23), p.2250120.
  13. [13]  Yokuş, A. (2021), Simulation of bright–dark soliton solutions of the Lonngren wave equation arising the model of transmission lines, Modern Physics Letters B, 35(32), p.2150484.
  14. [14]  Yokuş, A. (2021), Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation, Mathematical Modelling and Numerical Simulation with Applications, 1(1), 24-31.
  15. [15]  Isah, M.A. and Yokus, A. (2023), Rogue waves and Stability Analysis of the new (2+1)-KdV Equation Based on Symbolic Computation Method via Hirota Bilinear Form. In 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA) (pp. 1-6). IEEE.
  16. [16]  Baskonus, H.M., Mahmud, A.A., Muhamad, K.A., and Tanriverdi, T. (2022), A study on Caudrey–Dodd–Gibbon–Sawada–Kotera partial differential equation, Mathematical Methods in the Applied Sciences, 45(14), 8737-8753.
  17. [17]  Ali, K.K., Yilmazer, R., Bulut, H., and Yokus, A. (2022), New wave behaviours of the generalized Kadomtsev-Petviashvili modified equal Width-Burgers equation, Applied Mathematics, 16(2), 249-258.
  18. [18]  Duran, S. and Karabulut, B. (2022), Nematicons in liquid crystals with Kerr Law by sub-equation method, Alexandria Engineering Journal, 61(2), 1695-1700.
  19. [19]  Duran, S., Yokuş, A., Durur, H., and Kaya, D. (2021), Refraction simulation of internal solitary waves for the fractional Benjamin–Ono equation in fluid dynamics, Modern Physics Letters B, 35(26), p.2150363.
  20. [20] Tarla, S. and Yilmazer, R. (2022), Investigation of time-dependent Paraxial equation with an analytical method, Optik, 261, p.169111.
  21. [21]  Kaya, D., Yokuş, A., and Demiroğlu, U. (2020), Comparison of exact and numerical solutions for the Sharma–Tasso–Olver equation, Numerical Solutions of Realistic Nonlinear Phenomena, 53-65.
  22. [22]  Isah, M.A. and Yokus, A. (2023), The novel optical solitons with complex Ginzburg--Landau equation for parabolic nonlinear form using the Q6-model expansion approach, Journal MESA, 14(1), 205-225.
  23. [23]  Ali, K.K., Tarla, S., Yusuf, A., and Yilmazer, R. (2023), Closed form wave profiles of the coupled-Higgs equation via the $\varphi^ 6$-model expansion method, International Journal of Modern Physics B, 37(07), p.2350070.
  24. [24]  Isah, M.A. and Yokus, A. (2022), Application of the newly $\varphi^6-$model expansion approach to the nonlinear reaction-diffusion equation, Open Journal of Mathematical Sciences, 6, 269-28, doi:10.30538/oms2022.0192.
  25. [25]  Isah, M.A. (2023), A novel technique to construct exact solutions for the Complex Ginzburg-Landau equation using quadratic-cubic nonlinearity law, Mathematics in Engineering, Science $\&$ Aerospace (MESA), 14(1), 239-260.
  26. [26]  Yao, S.W., Akram, G., Sadaf, M., Zainab, I., Rezazadeh, H., and Inc, M. (2022), Bright, dark, periodic and kink solitary wave solutions of evolutionary Zoomeron equation, Results in Physics, 43, p.106117.
  27. [27] Zayed, E.M.E. (2014), The $(G'/G, 1/G)$-expansion method and its applications for constructing the exact solutions of the nonlinear Zoomeron equation, Transaction on Applied Mathematics and Informatics in Engineering, 1(1).
  28. [28]  Motsepa, T., Khalique, C.M., and Gandarias, M.L. (2017), Symmetry analysis and conservation laws of the Zoomeron equation, Symmetry, 9(2), p.27.
  29. [29]  Manafian, J., Lakestani, M., and Bekir, A. (2016), Study of the analytical treatment of the (2+1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach, International Journal of Applied and Computational Mathematics, 2, 243-268.
  30. [30]  Gao, H. (2014), Symbolic computation and new exact travelling solutions for the (2+1)-dimensional Zoomeron equation, International Journal of Modern Nonlinear Theory and Application, 3(2), 23-28.
  31. [31]  Gao, W., Rezazadeh, H., Pinar, Z., Baskonus, H.M., Sarwar, S., and Yel, G. (2020), Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Optical and Quantum Electronics, 52, 1-13.
  32. [32]  Zhang, Q., Xiong, M., and Chen, L. (2020), Exact solutions of two nonlinear partial differential equations by the first integral method, Advances in Pure Mathematics, 10(01), p.12.
  33. [33] Bekir, A., Taşcan, F., and Ünsal, Ö. (2015), Exact solutions of the Zoomeron and Klein–Gordon–Zakharov equations, Journal of the Association of Arab Universities for Basic and Applied Sciences, 17, 1-5.
  34. [34]  Abazari, R. (2011), The solitary wave solutions of Zoomeron equation, Applied Mathematics and Information Sciences, 5(59), 2943-2949.
  35. [35]  Jadaun, V., Kumar, S., and Garg, Y. (2017), Symmetry analysis and soliton solution of (2+1)-dimensional Zoomeron equation. arXiv preprint arXiv:1701.05499.
  36. [36]  Qawasmeh, A. (2013), Soliton solutions of (2+1)-Zoomeron equation and Duffing equation and SRLW equation, Journal of Mathematical and Computational Science, 3(6), 1475-1480.
  37. [37]  Higazy, M., Muhammad, S., Al-Ghamdi, A., and Khater, M.M. (2022), Computational wave solutions of some nonlinear evolution equations, Journal of Ocean Engineering and Science, j.joes.2022.01.007.
  38. [38]  Khan, K. and Akbar, M.A. (2014), Traveling wave solutions of the (2+1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method, Ain Shams Engineering Journal, 5(1), 247-256.
  39. [39]  Zhou, Q., Yao, D., and Chen, F. (2013), Analytical study of optical solitons in media with Kerr and parabolic-law nonlinearities, Journal of Modern Optics, 60(19), 1652-1657.
  40. [40] Zhou, Q., Xiong, X., Zhu, Q., Liu, Y., Yu, H., Yao, P., Biswas, A., and Belicd, M. (2015), Optical solitons with nonlinear dispersion in polynomial law medium, Journal of Optoelectronics and Advanced Materials, 17, 82-86.
  41. [41]  Yokus, A. and Isah, M.A. (2022), Investigation of internal dynamics of soliton with the help of traveling wave soliton solution of Hamilton amplitude equation, Optical and Quantum Electronics, 54(8), p.528.
  42. [42]  Sajid, N. and Akram, G. (2020), Novel solutions of Biswas-Arshed equation by newly $\phi^6$-model expansion method, Optik, 211, p.164564.
  43. [43]  Zayed, E.M. and Al-Nowehy, A.G. (2017), Many new exact solutions to the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms using three different techniques, Optik, 143, 84-103.
  44. [44]  Zayed, E.M., Al-Nowehy, A.G., and Elshater, M.E. (2018), New $\varphi ^{6}$-model expansion method and its applications to the resonant nonlinear Schr\"{o}dinger equation with parabolic law nonlinearity, The European Physical Journal Plus, 133(10), 417.
  45. [45]  Isah, M.A. and Yokus, A. (2022), The investigation of several soliton solutions to the complex Ginzburg-Landau model with Kerr law nonlinearity, Mathematical Modelling and Numerical Simulation with Applications, 2(3), 147-163.
  46. [46]  Taskesen, H. and Polat, N. (2021), Existence of weak solutions with different initial energy levels to an equation modeling shallow-water waves, International Journal of Open Problems in Computer Science $\&$ Mathematics, 14(4), 29-47.
  47. [47]  Yokuş, A., Durur, H., and Abro, K.A. (2021), Symbolic computation of Caudrey–Dodd–Gibbon equation subject to periodic trigonometric and hyperbolic symmetries, The European Physical Journal Plus, 136(4), 1-16.
  48. [48]  Jadaun, V., Kumar, S., and Garg, Y. (2017), Symmetry analysis and soliton solution of (2+1)-dimensional Zoomeron equation, arXiv preprint arXiv:1701.05499.
  49. [49]  Morris, R.M. and Leach, P.G.L. (2014), Symmetry reductions and solutions to the Zoomeron equation, Physica Scripta, 90(1), p.015202.