The extended (G'/G)-expansion method and its applications to the Whitham-Broer-Kaup-Like equations and coupled Hirota-Satsuma KdV equations

In this paper, based on a new general ansatze, the extended G^'G-expansion method is proposed to seek exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to construct travelling wave solutions of Whitham-Broer-Kaup-Like equations and coupled Hirota-Satsuma KdV equations. By using this method, new exact solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.

[1]  Qi Wang,et al.  Multiple Riccati equations rational expansion method and complexiton solutions of the Whitham-Broer-Kaup equation [rapid communication] , 2005 .

[2]  Xiangzheng Li,et al.  Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms , 2007 .

[3]  B. Duffy,et al.  An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations , 1996 .

[4]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[5]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[6]  Huiqun Zhang,et al.  New exact travelling wave solutions for some nonlinear evolution equations, Part II , 2008 .

[7]  Mingliang Wang,et al.  The periodic wave solutions for the Klein–Gordon–Schrödinger equations , 2003 .

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

[9]  Abdelhalim Ebaid,et al.  Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method , 2007 .

[10]  Xuedong Zheng,et al.  Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation , 2003 .

[11]  Abdul-Majid Wazwaz,et al.  The sine-cosine method for obtaining solutions with compact and noncompact structures , 2004, Appl. Math. Comput..

[12]  Mingliang Wang,et al.  The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations , 2004 .

[13]  Shimin Guo,et al.  Auxiliary equation method for the mKdV equation with variable coefficients , 2010, Appl. Math. Comput..

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

[15]  Xiangzheng Li,et al.  Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation , 2007 .

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

[17]  Yong Chen,et al.  A new general algebraic method with symbolic computation to construct new travelling wave solution for the (1 + 1)-dimensional dispersive long wave equation , 2005, Appl. Math. Comput..

[18]  Hong-qing Zhang,et al.  New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics , 1999 .

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

[20]  Huiqun Zhang,et al.  New exact travelling wave solutions for some nonlinear evolution equations , 2005 .

[21]  Xiangzheng Li,et al.  A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms , 2007 .

[22]  Abdul-Majid Wazwaz,et al.  The sine-cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations , 2006, Appl. Math. Comput..

[23]  Hong-qing Zhang,et al.  New exact solutions to the generalized coupled Hirota–Satsuma Kdv system , 2005 .

[24]  C. Dai,et al.  Semifoldons with fusion and fission properties of (2 + 1)-dimensional nonlinear system , 2008 .

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

[26]  C. Dai,et al.  Periodic structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation , 2009 .

[27]  Michael Tabor,et al.  Similarity reductions from extended Painleve´ expansions for nonintegrable evolution equations , 1991 .

[28]  Xiaoling Zhang,et al.  A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equations with nonlinear terms of any order , 2007, Appl. Math. Comput..

[29]  Emmanuel Yomba,et al.  The extended Fan's sub-equation method and its application to KdV¿MKdV, BKK and variant Boussinesq equations , 2005 .

[30]  Deng-Shan Wang,et al.  Symbolic computation and non-travelling wave solutions of (2 + 1)-dimensional nonlinear evolution equations , 2008 .

[31]  Jike Wang,et al.  The novel solutions of auxiliary equation and their application to the (2 + 1)-dimensional Burgers equations , 2008 .

[32]  Yang Feng,et al.  A new auxiliary function method for solving the generalized coupled Hirota-Satsuma KdV system , 2008, Appl. Math. Comput..

[33]  Zhenya Yan,et al.  The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system , 2003 .

[34]  Zhou Yu-bin,et al.  Application of Modified G′/G-Expansion Method to Traveling Wave Solutions for Whitham–Broer–Kaup-Like Equations , 2009 .

[35]  Xianguo Geng,et al.  A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations , 1999 .

[36]  Abdul-Majid Wazwaz,et al.  Multiple kink solutions and multiple singular kink solutions for the (2 + 1)-dimensional Burgers equations , 2008, Appl. Math. Comput..

[37]  Sheng Zhang.  A further improved extended Fan sub-equation method for (2+1)-dimensional breaking soliton equations , 2008, Appl. Math. Comput..

[38]  Mingliang Wang,et al.  Periodic wave solutions to a coupled KdV equations with variable coefficients , 2003 .

[39]  A. Jamiołkowski.  Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .