New closed-form soliton solutions for the tzitzeica dodd bullough equation arising nonlinear optics via three distinct re-liable approaches

  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract

    Tzitzeica Dodd Bullough (TDB) equation appears in the field of quantum field theory and nonlinear optics. In this article, we extracted abundant new soliton solutions with free choice of arbitrary parameters to the Tzitzeica-Dodd-Bullough (TDB) equation through the three separate methods such as the enhanced -expansion method, the improved -expansion method and the -expansion method by means of the wave transformation and the Painleve property. In these schemes, we formally derived some new closed form soliton solutions of the TDB equation through with symbolic computation package Maple. Soliton solutions are expressed by hyperbolic function, trigonometric function and rational function. The attained solutions are verified by symbolic computation software Maple 17. The attained solutions can be demonstrated by two-dimensional (2D) and three-dimensional (3D) graphs. Finally, it can be concluded that the adopted methods are very effective and well-suited to find new closed-form soliton solutions to the other nonlinear evaluation equations (NLEEs) with integer or fractional order.


  • Keywords

    Tzitzeica-Dodd-Bullough Equation; Enhanced -Expansion Method; Improved -Expansion Method; - Expansion Method; Closed-Form Soliton Solution.

  • References

      [1] Debnath, L., Nonlinear partial differential equations for scientists and engineers, Springer Science & Business Media, 2011.

      [2] Whitham, G. B., Linear and nonlinear waves. Vol. 42. John Wiley & Sons, 2011.

      [3] Wazwaz, A. M., Partial differential equations and solitary waves theory. Springer Science & Business Media, 2010.

      [4] Wang, M., Zhou, Y. & Li, Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letters A, 216 (1-5), 1996, 67-75.

      [5] Wazwaz, A. M., Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method. Applied Mathematics and Computation, 190 (1) (2007) 633-640.

      [6] Fu, X., Cen, S., Li, C. F. & Chen, X., Analytical trial function method for development of new 8-node plane element based on the variational principle containing Airy stress function. Engineering Computations, 27 (4), 2010, 442-463.

      [7] Malfliet, W., Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60 (7) (1992) 650-654.

      [8] Chow, K. W., A class of exact, periodic solutions of nonlinear envelope equations. Journal of Mathematical Physics, 36 (8) (1995) 4125-4137.

      [9] Fan, E., Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method. Journal of Physics A: Mathematical and General, 35(32) (2002) 6853.

      [10] Fan, E., Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4) (2000) 212-218.

      [11] El-Wakil, S. A., & Abdou, M. A., New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons & Fractals, 31(4) ( 2007) 840-852.

      [12] Bai, C., Exact solutions for nonlinear partial differential equation: a new approach. Physics Letters A, 288 (3) (2001) 191-195.

      [13] Yusufoğlu, E., & Bekir, A., Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine–cosine method. International Journal of Computer Mathematics, 83 (12) ( 2006) 915-924.

      [14] Wu, J., & Geng, X., Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation. Communications in Nonlinear Science and Numerical Simulation, 53 (2017)83-93.

      [15] Alquran, M. & Jarrah, A., Jacobi elliptic function solutions for a two-mode KdV equation. Journal of King Saud University-Science, .31(4) (2017) 485-489.

      [16] Ebadian, A., Najafzadeh, S., Farahrooz, F. et al., On the convergence and numerical computation of two-dimensional fuzzy Volterra–Fredholm integral equation by the homotopy perturbation method. SeMA Journal, 75 (2018) 17-34.

      [17] Zhang, S., A generalized auxiliary equation method and its application to the (2+1)-dimensional KdV equations. Applied mathematics and computation,188(1) (2007)1-6.

      [18] Hosseini, K., Ansari, R. & Gholamin, P., Exact solutions of some nonlinear systems of partial differential equations by using the first integral method, Journal of Mathematical Analysis and Applications, 387(2) (2012) 807–814.

      [19] Hosseini, K. Mayeli,P. & Kumar, D., New exact solutions of the coupled sine-Gordon equation in nonlinear optics using the modified Kudryashov method. Journal of Modern Optics, 65(3) (2017) 361-364.

      [20] Khater, M. M. A. & Kumar, D.,New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water. Journal of Ocean Engineering and Science, 2(3) (2017) 223-228.

      [21] Wang, M., Li, X., & Zhang, J., The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372 (4) (2008) 417-423.

      [22] Hawlader, F. & Kumar D., A variety of exact analytical solutions of extended shallow water wave equations via the improved expansion method. International Journal of Physical Research, 5 (1) (2017) 21-27.

      [23] Khan, K. &Akbar M. A., Traveling wave solutions of nonlinear evolution equations via the enhanced -expansion method. Journal of the Egyptian Mathematical Society, 22 (2) (2014) 220-226.

      [24] Yokus A., Solutions of some nonlinear partial differential equations and comparison of their solutions, Ph.D. Thesis, Firat University, Turkey, 2011.

      [25] Khater, M.M.A. & Kumar, D., Implementation of three reliable methods for finding exact solution of (2+1) dimensional generalized fractional evolution equations, Optical and quantum electronics. 50( 427) (2018).

      [26] Li, Lx., Li, Eq. & Wang, Ml., The -expansion method and its application to travelling wave solutions of the Zakharov equations. Applied Mathematics-A Journal of Chinese Universities, 25 (4) (2010) 454-462.

      [27] Mamun Miah, M., Shahadat Ali, H.M., Ali Akbar, M. et al., Some applications of the -expansion method to find new exact solutions of NLEEs. The European Physical Journal Plus, 132 (6) (2017) 252.

      [28] Wazwaz, A. M., The tanh method for travelling wave solutions to the Zhiber–Shabat equation and other related equations. Communications in Nonlinear Science and Numerical Simulation, 13(3) (2008) 584-592.

      [29] Wazwaz, A. M., The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations. Chaos, Solitons & Fractals, 25 (1) ( 2005) 55-63.

      [30] Khan, K., & Akbar, M. A., Exact and solitary wave solutions for the Tzitzeica–Dodd–Bullough and the modified KdV–Zakharov–Kuznetsov equations using the modified simple equation method. Ain Shams Engineering Journal, 4(4), (2013) 903-909.

      [31] Tang, Y., Xu,W., Shen, J. & Gao, L. Bifurcations of traveling wave solutions for Zhiber–Shabat equation. Nonlinear Analysis: Theory, Methods & Applications, 67(2) (2007) 648-656.

      [32] Wazzan, L., Solutions of Zhiber-Shabat and Related Equations Using a Modified tanh-coth Function Method. Journal of Applied Mathematics and Physics, 4(6) (2016) 1068-1079.

      [33] Abazari, R., The -expansion method for Tzitzéica type nonlinear evolution equations. Mathematical and Computer Modelling, 52(9) (2010) 1834-1845.

      [34] Borhanifar, A., & Moghanlu, A. Z., Application of the -expansion method for the Zhiber–Shabat equation and other related equations. Mathematical and Computer Modelling, 54(9) ( 2011) 2109-2116.

      [35] Manafian, J. & Lakestani, M. Dispersive dark optical soliton with Tzitzéica type nonlinear evolution equations arising in nonlinear optics. Opt Quant Electron 48(116 ) (2016).

      [36] Islam, M. R., Application of the -expansion method for Tzitzeica type nonlinear evolution equations. Journal for Foundations and Applications of Physics, 4(1) (2017) 8-18.

      [37] Hosseini, K., Bekir,A. & Kaplan, M., New exact traveling wave solutions of the Tzitzéica-type evolution equations arising in non-linear optics. Journal of Modern Optics, 64(16) (2017) 1688-1692.

      [38] Inc, M., New type soliton solutions for the Zhiber–Shabat and related equations. Optik, 138 (2017) 1-7.

      [39] Tala-Tebue, E., Djoufack, Z. I., Tsobgni-Fozap D. C., Kenfack-Jiotsa, A., Kapche-Tagne, F., & Kofané,T. C., Traveling wave solutions along microtubules and in the Zhiber–Shabat equation. Chinese Journal of Physics, 55( 3) (2017) 939-946.

      [40] Kaplan, M., & Hosseini K., Investigation of exact solutions for the Tzitzéica type equations in nonlinear optics. Optik, 154 (2018) 393-397.

      [41] He, J. H., & Abdou M. A., New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons & Fractals, 34 (5) ( 2007) 1421-1429.

      [42] Esen, A., Yagmurlu, N.M. & Tasbozan, O., Double exp-function method for multisoliton solutions of the Tzitzeica-Dodd-Bullough equation. Acta Math. Appl. Sin. Engl. Ser, 32(2) (2016) 461–468.

      [43] Neirameh, A., New exact solitary wave solutions to the TDB and (2+1)-DZ equations. Palestine Journal of Mathematics, 4(2) (2015) 386-390.

      [44] Hosseini, K., Ayati, Z. & Ansari,R., New exact traveling wave solutions of the Tzitzéica type equations using a novel exponential rational function method. Optik, 148 (2017) 85-89.

      [45] Kumar D., Hosseini K. & Samadani F., The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzeica type equations in nonlinear optics, Optik, 149 (2017) 439-446.

      [46] Hosseini, K., Ayati, Z. & Ansari, R., New exact solutions of the Tzitzéica type equations arising in nonlinear optics using a modified version of the improved tan(Φ(ξ)/2)tan⁡(Φ(ξ)/2)-expansion method. Opt Quant Electron 49(2730 (2017).

      [47] Huber, A., A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction. Physica D: Nonlinear Phenomena, 237 (8) ( 2008) 1079-1087.

      [48] He, B., Long Y., & Rui W., New exact bounded travelling wave solutions for the Zhiber–Shabat equation. Nonlinear Analysis: Theory, Methods & Applications, 71 (5) (2009) 1636-1648.




Article ID: 31271
DOI: 10.14419/ijpr.v9i1.31271

Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.