Soliton solution of the coupled nonlinear Klein Gordon equations with two integration schemes

  • Authors

    • Anwar Ja'afar Mohamad Jawad Al-Rafidain University College
    2019-07-14
    https://doi.org/10.14419/ijet.v7i4.11794
  • Nonlinear PDEs, Klein-Gordon Equations, Exact Solutions, Sech –Csch Function Method.
  • This paper established a traveling wave solution by using Sech - Csch function algorithms as well as the Modified Simple Equation Method (MSEM) for nonlinear partial differential equations. These methods will be applied to solving the conservation laws of the coupled Klein-Gordon equations that arise in the quantum field theory. There are two types of nonlinearity that will be considered, namely the cubic and the power law. The conservation laws will be finally determined from these conserved densities from the corresponding soliton solution.

     

  • References

    1. [1] Wazwaz, A.M. (2005). The tanh-function method: Solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos Solitons and Fractals, Vol. 25, No. 1, pp. 55-63. https://doi.org/10.1016/j.chaos.2004.09.122.

      [2] Anjan Biswas, Abdul H. Kara, Luminiţa Moraru, Ashfaque H. Bokhari, F. D. Zaman, (2014),“Conservation Laws of Coupled Klein-Gordon equations with cubic and power law nonlinearities†Proceeding of the Romanian Academy, Series A, Volume 15, Number 2 , pp. 123–129.

      [3] Anwar Ja’afar Mohamed Jawad, Soliton Solutions for the Boussinesq Equations, (2013) J. Math. Comput. Sci. 3 , No. 1, 254-265

      [4] El-Wakil, S.A, Abdou, M.A. (2007). New exact travelling wave solutions using modified extended tanh-function method, Chaos Solitons Fractals, Vol. 31, No. 4, pp. 840-852. https://doi.org/10.1016/j.chaos.2005.10.032.

      [5] Jawad A J M, (2012), Soliton Solutions for Nonlinear Systems (2+ 1)-Dimensional Equations, IOSR Journal of Mathematics (IOSRJM) ISSN: 2278-5728 vol.1 no. 6. https://doi.org/10.9790/5728-0162734.

      [6] Biswas A, Ebadi G, Fessak M, Johnpillai AG, Johnson S, Krishnan EV, Yildirim A. (2012), Solutions of the perturbed Klein-Gordon equations. Iran J Sci Tech no l Trans A; 36:431–52.

      [7] Fan, E. (2000). Extended tanh-function method and its applications to nonlinear equations. Phys Lett A, Vol. 277, No.4, pp. 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8.

      [8] Feng, Z.S. (2002). The first integer method to study the Burgers-Korteweg-de Vries equation, J Phys. A. Math. Gen, Vol. 35, No. 2, pp. 343-349. https://doi.org/10.1088/0305-4470/35/2/312.

      [9] Inc, M., Ergut, M. (2005). Periodic wave solutions for the generalized shallow water wave equation by the improved Jacobi elliptic function method, Appl. Math. E-Notes, Vol. 5, pp. 89-96.

      [10] Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations, Am. J. Phys, Vol. 60, No. 7, pp. 650-654. https://doi.org/10.1119/1.17120.

      [11] Khater, A.H., Malfliet, W., Callebaut, D.K. and Kamel, E.S. (2002). The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction–diffusion equations, Chaos Solitons Fractals, Vol. 14, No. 3, PP. 513-522. https://doi.org/10.1016/S0960-0779(01)00247-8.

      [12] Zhang, Sheng. (2006), the periodic wave solutions for the (2+1) -dimensional Konopelchenko Dubrovsky equations, Chaos Solitons Fractals, Vol. 30, pp. 1213-1220. https://doi.org/10.1016/j.chaos.2005.08.201.

      [13] Wazwaz, A.M. (2006). Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos Solitons Fractals, Vol. 28, No. 2, pp. 454-462. https://doi.org/10.1016/j.chaos.2005.06.004.

      [14] Xia, T.C., Li, B. and Zhang, H.Q. (2001). New explicit and exact solutions for the Nizhnik- Novikov-Vesselov equation. Appl. Math. E-Notes, Vol. 1, pp. 139-142.

      [15] Yusufoglu, E., Bekir A. (2006), Solitons and periodic solutions of coupled nonlinear evolution equations by using Sine-Cosine method, Internat. J. Comput. Math, Vol. 83, No. 12, pp. 915-924. https://doi.org/10.1080/00207160601138756.

      [16] Anwar Ja’afar Mohamad Jawad and Mahmood Jawad Abu-AlShaeer, (2018), Soliton Solutions of the Coupled Schrödinger-Boussinesq Equations for Kerr Law Nonlinearity, Abstract and Applied Analysis. https://doi.org/10.1155/2018/8325919.

      [17] Anwar Ja’afar Mohamad Jawad, Marko D. Petkovic´, Anjan Biswas, (2010), Modified Simple Equation Method for Evolution Equations, Applied Mathematics and Computation journal,Vol. 217, No. 1. https://doi.org/10.1016/j.amc.2010.06.030.

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  • How to Cite

    Ja'afar Mohamad Jawad, A. (2019). Soliton solution of the coupled nonlinear Klein Gordon equations with two integration schemes. International Journal of Engineering & Technology, 7(4), 6781-6785. https://doi.org/10.14419/ijet.v7i4.11794