Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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TRAVELING WAVE SOLUTIONS FOR HIGHER DIMENSIONAL NONLINEAR EVOLUTION EQUATIONS USING THE $(\frac{G'}{G})$- EXPANSION METHOD

  • Zayed, E.M.E. (Mathematics Department, Faculty of Science, Zagazig University)
  • Published : 2010.01.30
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Abstract

In the present paper, we construct the traveling wave solutions involving parameters of nonlinear evolution equations in the mathematical physics via the (3+1)- dimensional potential- YTSF equation, the (3+1)- dimensional generalized shallow water equation, the (3+1)- dimensional Kadomtsev- Petviashvili equation, the (3+1)- dimensional modified KdV-Zakharov- Kuznetsev equation and the (3+1)- dimensional Jimbo-Miwa equation by using a simple method which is called the ($\frac{G'}{G}$)- expansion method, where $G\;=\;G(\xi)$ satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the travelling waves. The travelling wave solutions are expressed by hyperbolic, trigonometric and rational functions.

Keywords

References

  1. M.A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models, Appl.Math.Comput, 190 (2007) 988-996. https://doi.org/10.1016/j.amc.2007.01.070
  2. M.A. Abdou, The extended F-expansion method and its applications for a class of nonlinear evolution equation, Chaos, Solitons and Fractals 31 (2007) 95 -104. https://doi.org/10.1016/j.chaos.2005.09.030
  3. M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear Equations and Inverse Scattering Transform, Cambridge Univ. Press, Cambridge, 1991.
  4. C.L.Bai and H.Zhao, Generalized method to construct the solitonic solitonic to (3+1)dimensional nonlinear equation, Phys. Letters A, 354 (2006) 428-436. https://doi.org/10.1016/j.physleta.2006.01.084
  5. A.Bekir, Application of the $(\frac{G}{G})$-expansion method for nonlinear evolution equations, Phys.Letters A, 372 (2008) 3400-3406. https://doi.org/10.1016/j.physleta.2008.01.057
  6. A. Bekir and A.Boz, Exact solutions for nonlinear evolution equations using Exp-function method, Phys. Letters A, 372 (2008) 1619-1625. https://doi.org/10.1016/j.physleta.2007.10.018
  7. Y.Chen and Q.Wang, Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solitons to (1+1) dimensional dispersive long wave equation, Chaos, Solitons and Fractals, 24 (2005) 745-757. https://doi.org/10.1016/j.chaos.2004.09.014
  8. E.G. Fan, Extended tanh- function method and its applications to nonlinear equations, Phys.Letters A, 277 (2000) 212-218. https://doi.org/10.1016/S0375-9601(00)00725-8
  9. J.H.He and X.H.Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006) 700-708. https://doi.org/10.1016/j.chaos.2006.03.020
  10. M. Jimbo and T. Miwa, solitons and infinite dimensional Lie algebra, Publ.Res.Inst. Math.Sci., 19 (1983) 943-948. https://doi.org/10.2977/prims/1195182017
  11. H.Junqi, An algebraic method exactly solving two high dimensional nonlinear evolution equations, Chaos, Solitons and Fractals, 23 (2005) 391-398. https://doi.org/10.1016/j.chaos.2004.02.044
  12. S. Liu, Z.Fu, S.D. Liu and Q.Zhao, Jacobi elliptic function expansion method and periodic wave soltions of nonlinear wave equations, Phys. Letters A, 289 (2001) 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1
  13. X.Z. Li and M.L.Wang, A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with higher order nonlinear terms, Phys.Letters A, 361 (2007) 115-118. https://doi.org/10.1016/j.physleta.2006.09.022
  14. M.R.Miura, Backlund Transformation, Springer-Verlag, Berlin,1978.
  15. C.Rogers and W.F.Shadwick, Backlund Transformations, Academic Press, New York,1982.
  16. B.Tian and Y.T. Gao, Beyond traveling waves: a new algorithm for solving nonlinear evolution equations,Comput. Phys.Commu., 95 (1996) 139-142. https://doi.org/10.1016/0010-4655(96)00014-8
  17. Z.Wang and H.Q.Zhang, A new generalized Riccati equation rational expansion methodto a class of nonlinear evolution equation with nonlinear terms of any order, Appl.Math.Comput, 186 (2007) 693-704. https://doi.org/10.1016/j.amc.2006.08.015
  18. M.Wang and Y.Zhou, The periodic wave equations for the Klein-Gordon-Schordinger equations, Phys. Letters A, 318 (2003) 84-92. https://doi.org/10.1016/j.physleta.2003.07.026
  19. M.Wang and X. Li, Extended F-expansion and periodic wave solutions for the generalized Zakharov equations, Phys. Letters A, 343 (2005) 48-54 . https://doi.org/10.1016/j.physleta.2005.05.085
  20. M.Wang and X.Li, Applications of F-expansion to periodic wave solutions for a newHamiltonian amplitude equation, Chaos, Solitons and Fractals 24 (2005) 1257- 1268. https://doi.org/10.1016/j.chaos.2004.09.044
  21. M.L.Wang, X.Z.Li and J.L.Zhang, Sub-ODE method and solitary wave solutions for higher order nonlinear Schrodinger equation, Phys. Letters A, 363 (2007) 96-101. https://doi.org/10.1016/j.physleta.2006.10.077
  22. D.S.Wang, Y.J.Ren and H.Q.Zhang, Further extended sinh-cosh and sin-cos methods and new non traveling wave solutions of the (2+1)-dimensional dispersive long wave equations, Appl. Math.E-Notes, 5 (2005) 157-163.
  23. D.S.Wang, W. Sun, C.Kong and H. Zhang, New extended rational expansion method and exact solutions of Boussinesq and Jimbo- Miwa equation, Appl. Math. Comput., 189(2007) 878-886. https://doi.org/10.1016/j.amc.2006.11.142
  24. M.Wang, X.Li and J.Zhang, The $(\frac{G}{G})$ expansion method and traveling wave solutions of nonlinear evoltion equations in mathematical physics, Phys.Letters A, 372 (2008) 417-423. https://doi.org/10.1016/j.physleta.2007.07.051
  25. A.M. Wazwaz, New solutions of distinct physical structures to high - dimensional nonlinear evolution equations, Appl. Math. Comput., 196 (2008) 363-368. https://doi.org/10.1016/j.amc.2007.06.002
  26. A.M. Wazwaz, Multiple soliton solutions for the Calogero- Bogoyavlenskii- Schiff Jimbo Miwa and YTSF equations, Appl. Math. Comput., 203 (2008) 592-597. https://doi.org/10.1016/j.amc.2008.05.004
  27. G. Xu, An elliptic equation method and its applications in nonlinear evolution equations,Chaos, Solitons and Fractals, 29 (2006) 942-947. https://doi.org/10.1016/j.chaos.2005.08.058
  28. Z.Yan and H.Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham- Broer-Kaup equation in shallow water, Phys. Letters A, 285 (2001) 355-362. https://doi.org/10.1016/S0375-9601(01)00376-0
  29. Z.Yan, Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stawartson-type equation via a new method, Chaos,Solitons and Fractals,18 (2003) 299-309. https://doi.org/10.1016/S0960-0779(02)00653-7
  30. E.Yusufoglu, New solitary solutions for the MBBM equations using Exp-function method, Phys. Letters A, 372 (2008) 442-446. https://doi.org/10.1016/j.physleta.2007.07.062
  31. E. Yusufoglu and A.Bekir, Exact solution of coupled nonlinear evolution equation, Chaos, Solitons and Fractals, 37 (2008) 842-848. https://doi.org/10.1016/j.chaos.2006.09.074
  32. S.J. Yu, K.Toda, N. Sasa and T. Fukuyama, N- soliton solutions to Bogoyavlenskii- Schiff equation and a guest for the soliton solutions in (3+1) dimensions, J. Phys.A. Math. Gen., 31 (1998) 3337-3347. https://doi.org/10.1088/0305-4470/31/14/018
  33. E.M.E.Zayed, H.A.Zedan and K.A.Gepreel, On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations, Chaos,Solitons and Fractals, 22 (2004) 285-303. https://doi.org/10.1016/j.chaos.2003.12.045
  34. E.M.E.Zayed, H.A.Zedan and K.A.Gepreel, On the solitary wave solutions for nonlinear Euler equations, Appl. Anal., 83 (2004) 1101-1132. https://doi.org/10.1080/00036810410001689274
  35. E.M.E.Zayed, H.A.Zedan and K.A.Gepreel, Group analysis and modified tanh-function to find the invariant solutions and soliton solution for nonlinear Euler equations, Int.J.nonlinear Sci. and Numel.Simul.5 (2004) 221-234. https://doi.org/10.1515/IJNSNS.2004.5.3.221
  36. E.M.E. Zayed, A.M. Abourabia, K.A.Gepreel and M.M. Horbaty, Traveling solitary wave solutions for the nonlinear coupled KdV system, Chaos, Solitons and Fractals, 34(2007) 292-306. https://doi.org/10.1016/j.chaos.2006.03.065
  37. E.M.E.Zayed, K.A.Gepreel and M.M.Horbaty, Exact solutions for some nonlinear differential equations using complex hyperbolic function, Appl. Anal., 87 (2008) 509-522. https://doi.org/10.1080/00036810801912098
  38. E.M.E.Zayed and K. A.Gepreel, The $(\frac{G}{G})$-expansion method for finding traveling wave solutions of nonlinear PDEs in mathematical physics, J. Math. Phys., 50 (2009) 013502-013513. https://doi.org/10.1063/1.3033750
  39. E.M.E.Zayed, The C expansion method and its applications to some nonlinear evolution equations in the mathematical physics, J. Appl. Math. Computing, 31 (2009) 89-103.
  40. S. L.Zhang, B. Wu and S.Y. Lou, Painleve analvsis and special solutions of generalized Broer-Kaup equations, Phys. Lett. A, 300 (2002) 40-48. https://doi.org/10.1016/S0375-9601(02)00688-6
  41. S. Zhang and T.C. Xia, A generalized F-expansion method and new exact solutions of Konopelchenko-Dubeovsky equations, Appl. Math. Comput., 183 (2006) 1190-1200. https://doi.org/10.1016/j.amc.2006.06.043
  42. S. Zhang, Application of Exp-function method to higher dimensional nonlinear evolution equation, Chaos, Solitons and Fractals 38 (2008) 270-276. https://doi.org/10.1016/j.chaos.2006.11.014
  43. S. Zhang, Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer- Kaup- Kupershmidt equations, Phys. Letters A, 372 (2008)1873-1880. https://doi.org/10.1016/j.physleta.2007.10.086
  44. S. Zhang and T.C. Xia, A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations, AppI.Math.E-Notes, 8 (2008) 58-66.
  45. S. Zhang and T.C. Xia, Symbolic computation and new families of exact non-traveling wave solutions to (3+1)- dimensional Kadomtsev- Petviashvili equation, Appl.Math. Comput.,181 (2006) 319-331. https://doi.org/10.1016/j.amc.2006.01.033
  46. S.Zhang, J.Tong and W.Wang, A generalized $(\frac{G}{G})$- expansion method for the mKdV equation with variable coefficients, Phys.Letters A, 372 (2008) 2254-2257. https://doi.org/10.1016/j.physleta.2007.11.026
  47. J. Zhang, X.Wei and Y.Lu, A generalized $(\frac{G}{G})$-expansion method and its applications, Phys.Letters A, 372 (2008) 3653-3658. https://doi.org/10.1016/j.physleta.2008.02.027