On the Wave Solutions of (2+1)-Dimensional Time-Fractional Zoomeron Equation

Abstract

In this manuscript, we have applied the sine-Gordon expansion method and the Bernoulli sub-equation method to seek the traveling wave solutions of the (2+1)-dimensional time-fractional partial Zoomeron equation. The exact solutions of the Zoomeron equation that are obtained by the sine-Gordon method are plotted in 3D figures, as well as the effects of the fractional derivative $\alpha $ are illustrated in 2D figures, while the exact solutions of the Zoomeron equation that are obtained by the Bernoulli sub-equation method are plotted in 3D figures and contour plot. Bright solutions, kink soliton, singular soliton solution, and complex solutions to the studied equation are constructed. Also, different values of the fractional parameter $\alpha $ are tested to study the effect of the parameter. We conclude that these methods are sufficient for seeking the exact solutions.

Keywords

• time fractional
• sine-Gordon method
• Bernoulli sub-equation method
• Zoomeron equation

References

1. [1] E.M.E. Zayed, Y.A. Amer, The First Integral Method and its Application for Deriving the Exact Solutions of a Higher-Order Dispersive Cubic-Quintic Nonlinear Schrödinger Equation, Comput. Math. Model. 27 (2016), 80-94. https://doi.org/10.1007/s10598-015-9305-y.
2. [2] M. Eslami, Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations, Appl. Math. Comput. 285 (2016), 141–148.
3. [3] W.G. and H.F.I. and H.B. and H.M. Baskonus, Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media, Phys. Scr. 95 (2019), 035207. \url{http://iopscience.iop.org/10.1088/1402-4896/ab4a50.}
4. [4] M. Yavuz, T.A. Sulaiman, F. Usta, H. Bulut, Analysis and numerical computations of the fractional regularized long-wave equation with damping term, Math. Methods Appl. Sci. (2020).
5. [5] H.F. Ismael, H. Bulut, H.M. Baskonus, Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and $(m+ (G'/ G))$ -expansion method, Pramana - J. Phys. 94 (2020), 35. \url{https://doi.org/10.1007/s12043-019-1897-x.}
6. [6] W. Gao, H.F. Ismael, A.M. Husien, H. Bulut, H.M. Baskonus, Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrödinger and Resonant Nonlinear Schrödinger Equation with the Parabolic Law, Appl. Sci. 10 (2019), 219. \url{https://doi.org/10.3390/app10010219.}
7. [7] H.F. Ismael, H.M. Baskonus, H. Bulut, Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model, Discret. Contin. Dyn. Syst. - S. 0 (n.d.) 0.\url{https://doi.org/10.3934/dcdss.2020398.}
8. [8] F. Usta, Numerical analysis of fractional Volterra integral equations via Bernstein approximation method, J. Comput. Appl. Math. 384 (2020), 113198.

9. [9] W. Gao, H.F. Ismael, S.A. Mohammed, H.M. Baskonus, H. Bulut, Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in Kerr media with M-fractional, Front. Phys. 7 (2019), 197.
10. [10] F. Usta, Fractional type Poisson equations by radial basis functions Kansa approach, J. Inequalities Spec. Funct. 7 (2016), 143–149.
11. [11] F. Usta, Numerical solution of fractional elliptic PDE’s by the collocation method, Appl. Appl. Math. An Int. J. 12 (2017), 470–478.
12. [12] F. Usta, A mesh-free technique of numerical solution of newly defined conformable differential equations, Konuralp J. Math. 4 (2016), 149–157.
13. [13] F. Usta, H. Budak, M. Sarikaya, Yang-Laplace transform method Volterra and Abel’s integro-differential equations of fractional order, Int. J. Nonlinear Anal. Appl. 9 (2018), 203–214.
14. [14] P. Wan, J. Manafian, H.F. Ismael, S.A. Mohammed, Investigating One-, Two-, and Triple-Wave Solutions via Multiple Exp-Function Method Arising in Engineering Sciences, Adv. Math. Phys. 2020 (2020), 8018064. \url{https://doi.org/10.1155/2020/8018064.}
15. [15] K.K. Ali, A.R. Seadawy, A. Yokus, R. Yilmazer, H. Bulut, Propagation of dispersive wave solutions for (3+ 1)-dimensional nonlinear modified Zakharov–Kuznetsov equation in plasma physics, Int. J. Mod. Phys. B. 34 (2020), 2050227.
16. [16] K.K. Ali, H. Dutta, R. Yilmazer, S. Noeiaghdam, On the new wave behaviors of the Gilson-Pickering equation, Front. Phys. 8 (2020), 54.
17. [17] H.F. Ismael, H. Bulut, H.M. Baskonus, W. Gao, Newly modified method and its application to the coupled Boussinesq equation in ocean engineering with its linear stability analysis, Commun. Theor. Phys. 72 (2020), 115002. \url{https://doi.org/10.1088/1572-9494/aba25f.}
18. [18] H.F. Ismael, H. Bulut, C. Park, M.S. Osman, M-lump, N-soliton solutions, and the collision phenomena for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation, Results Phys. 19 (2020), 103329. \url{https://doi.org/10.1016/j.rinp.2020.103329.}
19. [19] F. Usta, A conformable calculus of radial basis functions and its applications, An Int. J. Optim. Control Theor. Appl. 8 (2018), 176–182.
20. [20] K. Khan, M. Ali Akbar, M. Abdus Salam, M. Hamidul Islam, A note on enhanced $(G'/G)$-expansion method in nonlinear physics, Ain Shams Eng. J. 5 (2014), 877-884. \url{https://doi.org/10.1016/j.asej.2013.12.013.}
21. [21] K. Hosseini, A. Korkmaz, A. Bekir, F. Samadani, A. Zabihi, M. Topsakal, New wave form solutions of nonlinear conformable time-fractional Zoomeron equation in (2 + 1)-dimensions, Waves in Random and Complex Media. (2019), 1-11. \url{https://doi.org/10.1080/17455030.2019.1579393.}
22. [22] N. Ahmed, S. Bibi, U. Khan, S.T. Mohyud-Din, A new modification in the exponential rational function method for nonlinear fractional differential equations, Eur. Phys. J. Plus. 133 (2018), 1-11. \url{https://doi.org/10.1140/epjp/i2018-11896-0.}
23. [23] E. Aksoy, A.C. Çevikel, A. Bekir, Soliton solutions of (2+1)-dimensional time-fractional Zoomeron equation, Optik, 127 (2016), 6933-6942. \url{https://doi.org/10.1016/j.ijleo.2016.04.122.}
24. [24] O. Guner, A. Bekir, Bright and dark soliton solutions for some nonlinear fractional differential equations, Chinese Phys. B. 25 (2016), 030203. \url{https://doi.org/10.1088/1674-1056/25/3/030203.}
25. [25] D. Kumar, M. Kaplan, New analytical solutions of (2+1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques, Chinese J. Phys. 56 (2018), 2173-2185. \url{https://doi.org/10.1016/j.cjph.2018.09.013.}
26. [26] T. Motsepa, C. Khalique, M. Gandarias, Symmetry Analysis and Conservation Laws of the Zoomeron Equation, Symmetry, 9 (2017), 27. \url{https://doi.org/10.3390/sym9020027.}
27. [27] Y. Zhou, S. Cai, Q. Liu, Bounded Traveling Waves of the (2+1)-Dimensional Zoomeron Equation, Math. Probl. Eng. 2015 (2015), 163597. \url{https://doi.org/10.1155/2015/163597.}
28. [28] M. Alquran, K. Al-Khaled, Mathematical methods for a reliable treatment of the (2+1)-dimensional Zoomeron equation, Math. Sci. 6 (2013), 11. \url{https://doi.org/10.1186/2251-7456-6-11.}
29. [29] K. Khan, M. Ali Akbar, 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 Eng. J. 5 (2014), 247-256. \url{https://doi.org/10.1016/j.asej.2013.07.007.}
30. [30] A. Bekir, F. Taşcan, Ö. Ünsal, Exact solutions of the Zoomeron and Klein-Gordon-Zakharov equations, J. Assoc. Arab Univ. Basic Appl. Sci. 17 (2015), 1-5. \url{https://doi.org/10.1016/j.jaubas.2013.12.002.}
31. [31] M.Z. Sarikaya, F. Usta, On comparison theorems for conformable fractional differential equations, Int. J. Anal. Appl. 12 (2016), 207–214.
32. [32] F. Usta, M.Z. Sarıkaya, The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities, Demonstr. Math. 52 (2019), 204–212.
33. [33] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
34. [34] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: Theory and applications, 1993, Gordan and Breach, Amsterdam. (1993).
35. [35] Z. Zheng, W. Zhao and H. Dai, A new definition of fractional derivative, Int. J. Non. Linear. Mech., 108 (2019), 1-6.
36. [36] H.M. Baskonus, H. Bulut, New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures, An Int. J. Optim. Control Theor. Appl. 6 (2016), 141-150. \url{https://doi.org/10.11121/ijocta.01.2016.00295.}
37. [37] K.K. Ali, R. Yilmazer, H.M. Baskonus, H. Bulut, New wave behaviors and stability analysis of the Gilson–Pickering equation in plasma physics, Indian J. Phys. 94 (2020), 1-6. \url{https://doi.org/10.1007/s12648-020-01773-9.}
38. [38] K.K. Ali, R. Yilmazer, H. Bulut, Analytical Solutions to the Coupled Boussinesq–Burgers Equations via Sine-Gordon Expansion Method, in: 4th Int. Conf. Comput. Math. Eng. Sci., Springer Nature, n.d.: p. 233.
39. [39] H.M. Baskonus, H. Bulut, On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method, Waves in Random and Complex Media. 25 (2015), 720-728. \url{https://doi.org/10.1080/17455030.2015.1080392.}
40. [40] H.H. Abdulkareem, H.F. Ismael, E.S. Panakhov, H. Bulut, Some Novel Solutions of the Coupled Whitham-Broer-Kaup Equations, in: Int. Conf. Comput. Math. Eng. Sci., Springer, 2019: pp. 200–208.
41. [41] H.F. Ismael, H. Bulut, On the Solitary Wave Solutions to the (2+ 1)-Dimensional Davey-Stewartson Equations, in: Int. Conf. Comput. Math. Eng. Sci., Springer, 2019: pp. 156–165.
42. [42] K.K. Ali, R. Yilmazer, H.M. Baskonus, H. Bulut, Modulation instability analysis and analytical solutions to the system of equations for the ion sound and Langmuir waves, Phys. Scr. 95 (2020), 065602.