Exact solutions of the ZK-MEWequation and the Davey-Stewartson equation

 
 
 
  • Abstract
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  • Abstract


    In this paper we introduce a new version of the trial equation method for solving non-integrable partial differential equations in mathematical physics. Some exact solutions including soliton solutions, rational and elliptic function solutions to the generalized (2+1)-dimensional ZK-MEW equation and the generalized Davey-Stewartson equation with the complex coefficients are obtained by this method.

    Keywords: Extended trial equation method, generalized (2+1)-dimensional ZK-MEW equation, Davey-Stewartson equation, soliton solution, elliptic solutions.


  • References


      M. J. Ablowitz, P.A. Clarkson Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge: Cambridge University press, 1991.
    1. A. M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Applied Mathematics and Computation 154 (2004) 713-723.
    2. P. Rosenau, J. M. Hyman, Compactons: solitons with finite wavelengths, Physical Review Letters 70 (1993) 564-567.
    3. A. M. Wazwaz, An analytic study of compactons structures in a class of nonlinear dispersive equations, Mathematics and Computers in Simulation 63 (2003) 35-44.
    4. R. Hirota, Exact solutions of the Korteweg-de-Vries equation for multiple collisions of solitons, Phys. Lett. A 27 (1971) 1192-1194.
    5. W. Malfliet, W. Hereman, The tanh method: exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54 (1996) 563-568.
    6. M. A. Abdou, The extended tanh method and its applications for solving nonlinear physical models, Appl. Math. Comput. 190 (2007) 988-996.
    7. H.X. Wu, J. H. He, Exp-function method and its application to nonlinear equations, Chaos Solitons Fractals 30 (2006) 700-708.
    8. M. Wang, X. Li, J. Zhang, The $(frac{G'}{G})$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 (2008) 417-423.
    9. G. Ebadi, A. Biswas, The $(frac{G'}{G})$ method and topological soliton solution of the K(m,n) equation, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 2377-2382.
    10. E. M. E. Zayed, K. A. Gepreel, The $(G'/G)$-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics, Journal of Mathematical Physics 50 (2009) 013502-12.
    11. E. M. E. Zayed, M. A. S. EL-Malky, The Extended $(G'/G)$-expansion method and its applications for solving the (3+1)-dimensional nonlinear evolution equations in mathematical physcis, Global Journal of Science Frontier Research 11 (2011) 13 pages.
    12. M. Ekici, D. Duran, A. Sonmezoglu, Constructing of exact solutions to the (2+1)-dimensional breaking soliton equations by the multiple $(frac{G'}{G})$-expansion method, J. Adv. Math. Stud. 7 (2014) 27-44.
    13. M. Wang, Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 (1995) 169-172.
    14. H. T. Chen, H.Q. Zhang, New double periodic and multiple soliton solutions of the generalized (2 + 1)-dimensional Boussinesq equation, Chaos Soliton. Fract. 20 (2004) 765-769.
    15. D. Zhang, Doubly periodic solutions of the modified Kawahara equation, Chaos Soliton. Fract. 25 (2005) 1155-1160.
    16. Q. Liu, J. M. Zhu, Exact Jacobian elliptic function solutions and hyperbolic function solutions for Sawada-Kotere equation with variable coefficient, Phys. Lett. A 352 (2006) 233-238.
    17. X. Zhao, H. Zhi, H. Zhang, Improved Jacobi-function method with symbolic computation to construct new double-periodic solutions for the generalized Ito system, Chaos Soliton. Fract. 28 (2006) 112-126.
    18. A. Ebaid, E. H. Aly, Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions, Wave Motion 49 (2012) 296-308.
    19. A. Filiz, M. Ekici, A. Sonmezoglu, F-expansion method and new exact solutions of the Schr"{o}dinger-KdV equation, The Scientific World Journal, 2014 (2014). Article ID 534063, 14 pages.
    20. C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta. Phys. Sin. 54 (2005) 2505-2509.
    21. C. S. Liu, A new trial equation method and its applications, Commun. Theor. Phys. 45 (2006) 395-397.
    22. C. S. Liu, Trial equation method for nonlinear evolution equations with rank inhomogeneous: mathematical discussions and applications, Commun. Theor. Phys. 45 (2006) 219-223.
    23. C. S. Liu, Using trial equation method to solve the exact solutions for two kinds of KdV equations with variable coefficients, Acta. Phys. Sin. 54 (2005) 4506-4510.
    24. C. S. Liu, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun. 181 (2010) 317-324.
    25. Y. Gurefe, A. Sonmezoglu, E Misirli, Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics, Pramana-J. Phys. 77 (2011) 1023-1029.
    26. Y. Pandir, Y. Gurefe, U. Kadak, E. Misirli, Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal. 2012 (2012). Art. ID 478531, 16 pages.
    27. Y. Gurefe, E. Misirli, A. Sonmezoglu, M. Ekici, Extended trial equation method to generalized nonlinear partial differential equations, Appl. Math. Comput. 219 (2013) 5253-5260.
    28. Y. Gurefe, E. Misirli, Y. Pandir, A. Sonmezoglu, M. Ekici, New Exact Solutions of the Davey-Stewartson Equation with Power-Law Nonlinearity, Bull. Malays. Math. Sci. Soc. (2013) in press.
    29. M. Ekici, D. Duran, A. Sonmezoglu, Soliton Solutions of the Klein-Gordon-Zakharov Equation with Power Law Nonlinearity, ISRN Computational Mathematics 2013 (2013). Article ID 716279, 7 pages.
    30. A. Filiz, A. Sonmezoglu, M. Ekici, D. Duran, A New Approach for Soliton Solutions of RLW Equation and (1+2)-Dimensional Nonlinear Schr"{o}dinger's Equation, Mathematical Reports (2014) in press.
    31. N. J. Zabusky, M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Physical Review Letters 15 (1965) 240-243.
    32. M. Wadati, The modified Kortweg-de Vries equation, Journal of Physical Society of Japan 34 (1973) 1289-1296.
    33. A. M. Wazwaz, Exact solutions for the ZK-MEW equation by using the tanh and sine-cosine methods, Int. J. Comput. Math. 82 (2005) 699-708.
    34. E. M. E. Zayed, A. H. Arnous, Exact solutions of the nonlinear ZK-MEW and the Potential YTSF equations using the modified simple equation method, AIP Conference Proceedings of ICNAAM 2012 1479 (2012) 2044-2048. American Institute of Physics, USA.
    35. G. Ebadi, A. Biswas, The $(frac{G'}{G})$ method and 1-soliton solution of the Davey-Stewartson equation, Math. Comput. Model. 53 (2011) 694-698.
    36. A. Bekir, A. C. Cevikel, New solitons and periodic solutions for nonlinear physical models in mathematical physics, Nonlinear Anal. Real World Appl. 11 (2010) 3275-3285.
    37. X. Zhao, Self-similar solutions to a generalized Davey-Stewartson system, Math. Comput. Model. 50 (2009) 1394-1399.

 

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Article ID: 2116
 
DOI: 10.14419/ijamr.v3i2.2116




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