American Journal of Mathematical Analysis
ISSN (Print): 2333-8490 ISSN (Online): 2333-8431 Website: http://www.sciepub.com/journal/ajma Editor-in-chief: Grigori Rozenblum
Open Access
Journal Browser
Go
American Journal of Mathematical Analysis. 2013, 1(3), 28-32
DOI: 10.12691/ajma-1-3-1
Open AccessArticle

The (G’/G)-Expansion Method and Traveling Wave Solutions for Some Nonlinear PDEs

J.F. Alzaidy1,

1Mathematics Department, Faculty of Science, Taif University, Kingdom of Saudi Arabia

Pub. Date: May 19, 2013

Cite this paper:
J.F. Alzaidy. The (G’/G)-Expansion Method and Traveling Wave Solutions for Some Nonlinear PDEs. American Journal of Mathematical Analysis. 2013; 1(3):28-32. doi: 10.12691/ajma-1-3-1

Abstract

In the present paper, we construct the traveling wave solutions involving parameters of some nonlinear PDEs in mathematical physics via the nonlinear SchrÖdinger (NLS−) equation and the regularized long-wave (RLW) equation by using a simple method which is called the (G’/G) -expansion method, where G=G(ζ) satisfies the second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by hyperbolic, trigonometric and rational functions. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear PDEs arising in mathematical physics.

Keywords:
the (G’/G) -expansion method traveling wave solutions nonlinear SchrÖdinger (NLS−) equation regularized long-wave (RLW) equation solitary wave solutions

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  M. J. Ablowitz and P.A. Clarkson, Soliton, Nonlinear Evolution Equationuations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.
 
[2]  C. H. Gu et al., Soliton Theory and Its Application, Zhejiang Science and Technology Press, Zhejiang, 1990.
 
[3]  V. B. Matveev and M.A. Salle, Darboux Transformation and Soliton, Springer, Berlin, 1991.
 
[4]  R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004.
 
[5]  S.Y. Lou, J.Z. Lu, Special solutions from variable separation approach: Davey-Stewartson equation, J. Phys. A: Math. Gen. 29 (1996) 4209.
 
[6]  E. J. Parkes and B.R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers equation, Phys. Lett. A 229 (1997) 217.
 
[7]  E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212.
 
[8]  Z. Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A 292 (2001) 100.
 
[9]  Y. Chen, Y. Zheng, Generalized extended tanh-function method to construct new explicit exact solutions for the approximate equations for long water waves, Int. J. Mod. Phys. C 14 (4) (2003) 601.
 
[10]  M. L. Wang, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67.
 
[11]  G. W. Bluman and S. Kumei, Symmetries and Differential Equationuations, Springer-Verlag, New York, 1989.
 
[12]  P. J. Olver, Applications of Lie Groups to Differential Equationuations, Springer-Verlag, New York, 1986.
 
[13]  Z. Y. Yan, A reduction mKdV method with symbolic computation to constract new doubly- periodic solutions for nonlinear wave equations, Int. J. Mod. Phys. C, 14 (2003) 661.
 
[14]  Z. Y. Yan, The new tri-function method to multiple exact solutions of nonlinear wave equations, Physica Scripta, 78 (2008) 035001.
 
[15]  Z. Y. Yan, Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov– Kuznetsov equation in dense quantum plasmas, Physics Letters A, 373 (2009) 2432.
 
[16]  D. C. Lu and B. J. Hong, New exact solutions for the (2+1)-dimensional Generalized Broer-Kaup system, Appl. Math. Comput, 199(2008)572.
 
[17]  A. V. Porubov, Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid, Phys. Lett. A, 221 (1996) 391.
 
[18]  M. Wazwaz, The tanh and sine- cosine method for compact and noncompact solutions of nonlinear Klein Gordon equation, Appl. Math. Comput, 167 (2005)1179.
 
[19]  Z. Y. Yan and H. Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Phys. Lett. A, 285 (2001) 355.
 
[20]  D. C. Lu, Jacobi elliptic functions solutions for two variant Boussinesq equations, Chaos, Solitons and Fractals, 24 (2005) 1373.
 
[21]  Z. Y. 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.
 
[22]  C. L. Bai and H. Zhao, Generalized method to construct the solitonic solutions to (3+1)- dimensional nonlinear equation, Phys. Lett. A, 354 (2006) 428.
 
[23]  F. Cariello and M. Tabor, Similarity reductions from extended Painleve’ expansions for nonintegrable evolution equations, Physica D, 53 (1991) 59.
 
[24]  M. Wang and X. Li, Extended F-expansion and periodic wave solutions for the generalized Zakharov equations, Phys. Lett. A, 343 (2005) 48.
 
[25]  X. Feng, Exploratory approach to explicit solution of nonlinear evolution equations, Int. J. Theor. Phys, 39 (2000) 222.
 
[26]  E. M. E. Zayed and K.A.Gepreel, The (G’/G) expansion method for finding traveling wave solutions of nonlinear PDEs in mathematical physics, J. Math. Phys., 50 (2009) 013502.
 
[27]  A. Bekir, Application of the (G’/G)− expansion method for nonlinear evolution equations, Phys. Lett A, 372 (2008) 3400.
 
[28]  M. Wang, X. Li and J.Zhang, The(G’/G)−expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett A, 372 (2008) 417.
 
[29]  Q. Ding, The NLS-equation and its SL(2,R) structure, J. Phys. A: Math, 33 (2000) L325.
 
[30]  A. Bekir, New Exact TravellingWave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equationuations, International Journal of Nonlinear Science,6(2008)46.
 
[31]  D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech, 25 (1966) 321.