Solution of the (2+1) Dimensional Breaking Soliton Equation by Using Two Different Methods

Abstract

The non-linear partial differential (2+1) dimensional Breaking Soliton equation is studied

by using the direct integration and homotopy perturbation method. In this study, we use direct

integration to obtain the known solution in the literature in practical and shortest way by assigning

some special values to the constants in the solutions of the (2+1) dimensional Breaking Soliton

equation. We also obtain same type solution for (2+1) dimensional Breaking Soliton equation by

using the homotopy perturbation method with one iteration. Similarly, same type solutions can be

done different methods such as (G'/G)-expansion method.

Keywords

• Breaking Soliton equation,Homotopy Perturbation Method,direct integration

References

1. [1] S. Arbabi, M. Najafi, Soliton solutions of nonlinear evolution equations in mathematical Physics, Optik, 2016, 127 4270–4274.
2. [2] F. Calogero, A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform, Nuovo Cimento B, 1976, 32, 201-242.B, 1976, 32, 201.
3. [3] F. Calogero, A. Degasperis, Nonlinear evolution equations solvable by the inverse spectral transform, Nuovo Cimento B, 1977, 39, 1-54.
4. [4] B. Tian, K. Zhao, Y. Gao, Symbolic Computation in Engineering: Application to a Breaking Soliton Equation, Letters in Applied and Engineering Sciences, 1997, 35, 1081-1083.
5. [5] Z. Yan, H. Zhang, Constructing Families of Soliton-Like Solutions to a (2+l)-Dimensional Breaking Soliton Equation Using Symbolic Computation, Computers and Mathematics with Applications, 2002, 44, 1439-1444.
6. [6] X. Geng, C. Cao, Explicit solutions of the 2 + 1-dimensional breaking soliton equation, Chaos, Solitons and Fractals, 2004, 22 683–691.
7. [7] J. Mei , H. Zhang, New types of exact solutions for a breaking soliton equation, Chaos, Solitons and Fractals, 2004, 20, 771–777.
8. [8] S. Zhang, New exact non-traveling wave and coefficient function solutions of the (2 + 1)-dimensional breaking soliton equations, Physics Letters A, 2007, 368, 470–475.

9. [9] S. Zhang, A generalized new auxiliary equation method and its application to the (2 + 1)-dimensional breaking soliton equations, Applied Mathematics and Computation, 2007, 190, 510–516.
10. [10] S.-H. Ma, J.-P. Fang, C.-L. Zheng, New exact solutions of the (2 + 1)-dimensional breaking soliton system via an extended mapping method, Chaos, Solitons and Fractals, 2009, 40, 210–214.
11. [11] Z-L. Tao, Solving the breaking soliton equation by He's variational method, Computers and Mathematics with Applications, 2009, 58, 2395-2397.
12. [12] X. Da-Quan, Symmetry reduction and new non-traveling wave solutions of (2 + 1)-dimensional breaking soliton equation, Commun Nonlinear Sci Numer Simulat, 2009, 15, 2061–2065.
13. [13] Z. Zhao, Z. Dai, G. Mu, The breather-type and periodic-type soliton solutions for the (2 + 1)-dimensional breaking soliton equation, Computers and Mathematics with Applications, 2011, 61, 2048–2052.
14. [14] H. Li, X. Wan, Z. Fu, and S. Liu, New special structures to the (2 + 1)-dimensional breaking soliton equations, Phys. Scr., 2011, 84, 035005 (5pp).
15. [15] E. Zayed, M. Abdelaziz, M. Elmalky, Enhanced (G'/G) -Expansion Method and Applications to the (2 + 1) D Typical Breaking Soliton and Burgers Equations, J. Adv. Math. Stud., 2011, 4, 109-122.
16. [16] M.T. Darvishi, M. Najafi, Some exact solutions of the (2 + 1)-dimensional break-ing soliton equation using the three-wave method, World Acad. Sci. Eng.Technol., 2011, 55, 919–922.
17. [17] M. T. Darvishi, M. Najafi, Some exact solutions of the (2+1)-dimensional breaking soliton equation using the three-wave method, International Journal of Computational and Mathematical Sciences, 2012, 6, 13-16.
18. [18] G. Xu, Integrability of a (2+1)-dimensional generalized breaking soliton equation, Applied Mathematics Letters, 2015, 50, 16,22.
19. [19] He, JH.: Homotopy perturbation technique, Comp. Meth. Appl. Mech. Eng., 1999, 178, 257-262.
20. [20] He, JH.: A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J Nonlinear Mech, 2000, 35, 37-43.
21. [21] He, JH.: Homotopy perturbation method for bifurcation of nonlinear problems, Int J Nonlinear Sci Numer Simul., 2005, 6 (2), 207-208.
22. [22] El-Shahed, M.:Application of He’s homotopy perturbation method to Volterra’s integro differential equation, Int J Nonlinear Sci Numer Simul, 2005, 6(2), 163-168.