Application of the (G'/G)-expansion method for the Zhiber-Shabat equation and other related equations

Abstract In this paper, the exact traveling wave solutions of the Zhiber–Shabat equation and the related equations: Liouville equation, Dodd–Bullough–Mikhailov (DBM) equation, sinh–Gordon equation, and Tzitzeica–Dodd–Bullough (TDB) equation are studied by using the ( G ′ G )-expansion method. Solitons and periodic solutions for these equations are formally derived.

[1]  Abdul-Majid Wazwaz,et al.  The tan h method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations , 2005 .

[2]  Hossein Jafari,et al.  New solitary wave solutions for generalized regularized long-wave equation , 2010, Int. J. Comput. Math..

[3]  Abdul-Majid Wazwaz,et al.  The tanh method for travelling wave solutions to the Zhiber–Shabat equation and other related equations , 2008 .

[4]  T. Fan,et al.  New applications of developed Jacobi elliptic function expansion methods , 2005 .

[5]  Guo-Wei Wei,et al.  Discrete singular convolution for the sine-Gordon equation , 2000 .

[6]  W. Hereman,et al.  The tanh method: I. Exact solutions of nonlinear evolution and wave equations , 1996 .

[7]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[8]  A. Borhanifar,et al.  New periodic and soliton wave solutions for the generalized Zakharov system and (2 + 1)-dimensional Nizhnik–Novikov–Veselov system , 2009 .

[9]  M. A. Abdou.  The extended tanh method and its applications for solving nonlinear physical models , 2007, Appl. Math. Comput..

[10]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[11]  Ismail Aslan,et al.  Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G'/G)-expansion method , 2009, Appl. Math. Comput..

[12]  Ahmet Bekir.  Application of the (G′G)-expansion method for nonlinear evolution equations , 2008 .

[13]  A. Borhanifar,et al.  New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations , 2009 .

[14]  Zuntao Fu,et al.  Exact Solutions to Double and Triple Sinh-Gordon Equations , 2004 .

[15]  Mingliang Wang.  Exact solutions for a compound KdV-Burgers equation , 1996 .

[16]  Ahmet Bekir,et al.  Applications of the extended tanh method for coupled nonlinear evolution equations , 2008 .

[17]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[18]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[19]  Sun Jiong,et al.  A direct method for solving sine-Gordon type equations , 2002 .

[20]  Abdul-Majid Wazwaz,et al.  Exact solutions to the double sinh-gordon equation by the tanh method and a variable separated ODE method , 2005 .

[21]  Abdul-Majid Wazwaz,et al.  Distinct variants of the KdV equation with compact and noncompact structures , 2004, Appl. Math. Comput..

[22]  Hossein Jafari,et al.  New solitary wave solutions for the bad Boussinesq and good Boussinesq equations , 2009 .

[23]  Chuntao Yan.  A simple transformation for nonlinear waves , 1996 .

[24]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[25]  G. Rowlands,et al.  Nonlinear Waves, Solitons and Chaos , 1990 .

[26]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[27]  Reza Abazari,et al.  Exact solutions for non-linear Schrödinger equations by differential transformation method , 2011 .

[28]  Turgut Özis,et al.  On the validity and reliability of the (G'/G)-expansion method by using higher-order nonlinear equations , 2009, Appl. Math. Comput..

[29]  Jiao-Ling Zhang,et al.  A generalized (G′G)-expansion method and its applications , 2008 .

[30]  Mark J. Ablowitz,et al.  On the Numerical Solution of the Sine-Gordon Equation , 1996 .

[31]  Hossein Jafari,et al.  NEW SOLITONS AND PERIODIC SOLUTIONS FOR THE KADOMTSEV-PETVIASHVILI EQUATION , 2008 .

[32]  J. Perring,et al.  A Model unified field equation , 1962 .

[33]  Xin-Wei Zhou,et al.  Exp-function method to solve the nonlinear dispersive K(m,n) equations , 2008 .

[34]  Mark J. Ablowitz,et al.  Regular ArticleOn the Numerical Solution of the Sine–Gordon Equation: I. Integrable Discretizations and Homoclinic Manifolds , 1996 .

[35]  Mingliang Wang,et al.  Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics , 1996 .

[36]  C. Dai,et al.  Exact travelling wave solutions of the discrete nonlinear Schrödinger equation and the hybrid lattice equation obtained via the exp-function method , 2008 .

[37]  E. Fan,et al.  Extended tanh-function method and its applications to nonlinear equations , 2000 .

[38]  Robert Conte,et al.  Link between solitary waves and projective Riccati equations , 1992 .

[39]  Wentao Huang,et al.  Qualitative behavior and exact travelling wave solutions of the Zhiber-Shabat equation , 2009 .

[40]  Ji-Huan He,et al.  Exp-function method for nonlinear wave equations , 2006 .

[41]  Xu-Hong Wu,et al.  EXP-function method and its application to nonlinear equations , 2008 .