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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
J H He, X H Wu. Exp-function method for nonlinear wave equations, Chaos Soliton Fract, 2006, 30: 700–708.
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.
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.
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.
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.
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.
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.
A M Wazwaz. Compact and noncompact physical structures for the ZK-BBM equation, Appl Math Comput, 2005, 169: 713–725.
M L Wang. Solitary wave solutions for variant Boussinesq equations, Phys Lett A, 1995, 199: 169–172.
M L Wang, Y B Zhou. The periodic wave solutions for the Klein-Gordon-Schrödinger equations, Phys Lett A, 2003, 318: 84–92.
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.
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.
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.
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.
C H Zhao, Z M Sheng. Explicit travelling wave solutions for Zakharov equations, Acta Phys Sinica, 2004, 53: 1629–1634.
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.
J L Zhang, Y M Wang, M L Wang. Exact solutions to two nonlinear equations, Acta Phys Sinica, 2003, 52: 1574–1578.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the International Cooperation and Exchanges Foundation of Henan Province (084300510060) and the Youth Science Foundation of Henan University of Science and Technology of China (2008QN026).
Rights and permissions
About this article
Cite this article
Li, Lx., Li, Eq. & Wang, Ml. The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations. Appl. Math. J. Chin. Univ. 25, 454–462 (2010). https://doi.org/10.1007/s11766-010-2128-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-010-2128-x