The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations

Article

Abstract

The (G′/G, 1/G)-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the (G′/G)-expansion method proposed recently, is presented. By using this method abundant travelling wave solutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.

Keywords

The (G′/G 1/G)-expansion method travelling wave solutions homogeneous balance solitary wave solutions Zakharov equations 

MR Subject Classification

35Q20 

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References

  1. [1]
    Z T Fu, S K Liu, S D Liu, Q Zhao. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys Lett A, 2001, 290: 72–76.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J H He, X H Wu. Exp-function method for nonlinear wave equations, Chaos Soliton Fract, 2006, 30: 700–708.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    D J Huang, H Q Zhang. Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations, Acta Phys Sinica, 2004, 53: 2434–2438.MathSciNetGoogle Scholar
  4. [4]
    L X Li, M L Wang. The (G′/G)-expansion method and travelling wave solutions for a higher-order nonlinear Schrödinger equation, Appl Math Comput, 2009, 208: 440–445.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    S K Liu, Z T Fu, S D Liu, Q Zhao. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys Lett A, 2001, 289: 69–74.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    S D Liu, ZT Fu, S K Liu, Q Zhang. The envelope periodic solutions to nonlinear wave equations with Jacobi elliptic function, Acta Phys Sinica, 2002, 51: 718–722.MathSciNetGoogle Scholar
  7. [7]
    E J Parkes, B R Duffy. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput Phys Comm, 1996, 98: 288–300.MATHCrossRefGoogle Scholar
  8. [8]
    Y D Shang, Y Huang, W J Yuan. The extended hyperbolic functions method and new exact solutions to the Zakharov equations, Appl Math Comput, 2008, 200: 110–122.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A M Wazwaz. Compact and noncompact physical structures for the ZK-BBM equation, Appl Math Comput, 2005, 169: 713–725.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M L Wang. Solitary wave solutions for variant Boussinesq equations, Phys Lett A, 1995, 199: 169–172.CrossRefMathSciNetGoogle Scholar
  11. [11]
    M L Wang, Y B Zhou. The periodic wave solutions for the Klein-Gordon-Schrödinger equations, Phys Lett A, 2003, 318: 84–92.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    M L Wang, X Z Li, J L Zhang. Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation, Phys Lett A, 2007, 363: 96–101.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M L Wang, X Z Li, J L Zhang. The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys Lett A, 2008, 372: 417–423.CrossRefMathSciNetGoogle Scholar
  14. [14]
    M L Wang, Y M Wang, J L Zhang. The periodic wave solutions for two systems of nonlinear wave equations, Chinese Phys, 2003, 12: 1341–1348.CrossRefGoogle Scholar
  15. [15]
    M L Wang, X Z Li. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation, Chaos Soliton Fract, 2005, 24: 1257–1268.MATHCrossRefGoogle Scholar
  16. [16]
    C H Zhao, Z M Sheng. Explicit travelling wave solutions for Zakharov equations, Acta Phys Sinica, 2004, 53: 1629–1634.MathSciNetGoogle Scholar
  17. [17]
    J L Zhang, M L Wang. Complex tanh-function expansion method and exact solutions to two systems of nonlinear wave equations, Comm Theor Phys, 2004, 42: 491–493.MATHGoogle Scholar
  18. [18]
    J L Zhang, Y M Wang, M L Wang. Exact solutions to two nonlinear equations, Acta Phys Sinica, 2003, 52: 1574–1578.MathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina
  2. 2.Department of MathematicsLanzhou UniversityLanzhouChina

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