Some applications of the (G'/G)-expansion method to non-linear partial differential equations

In the present paper, we construct the traveling wave solutions involving parameters of the (2+1)-dimensional higher order Broer-Kaup equations, the (2+1)-dimensional breaking soliton equations, the (2+1)- dimensional asymmetric Nizhnik-Novikov-Vesselov equations and the (2+1)-dimensional BKP equations in terms of the hyperbolic functions, trigonometric functions and the rational functions by using a new approach, namely the G^'G-expansion method, where G=G(@x) satisfies a 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 the hyperbolic functions, the trigonometric functions and the rational functions.

[1]  Mingliang Wang,et al.  Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation , 2005 .

[2]  Cheng-Lin Bai,et al.  Combined structures of single valued and multiple valued localized excitations in higher-dimensional soliton system , 2006 .

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

[4]  Elena Medina,et al.  New solutions of the 2 + 1 dimensional BKP equation through symmetry analysis: source and sink solutions, creation and diffusion of breathers… , 2004 .

[5]  Zhen Wang,et al.  Many new kinds exact solutions to (2+1)-dimensional Burgers equation and Klein-Gordon equation used a new method with symbolic computation , 2007, Appl. Math. Comput..

[6]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[7]  Zhang Hong-qing,et al.  Solving the (2 + 1)-dimensional higher order Broer–Kaup system via a transformation and tanh-function method , 2004 .

[8]  Lina Song,et al.  A new compound Riccati equations rational expansion method and its application to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system , 2008 .

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

[10]  Jianlan Hu,et al.  A new method of exact travelling wave solution for coupled nonlinear differential equations , 2004 .

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

[12]  Wei Wang,et al.  A generalized (G′G)-expansion method for the mKdV equation with variable coefficients , 2008 .

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

[14]  Jianlan Hu,et al.  A new method for finding exact traveling wave solutions to nonlinear partial differential equations , 2001 .

[15]  Cheng-Lin Bai,et al.  Complex hyperbolic-function method and its applications to nonlinear equations , 2006 .

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

[17]  Khaled A. Gepreel,et al.  On the rational solitary wave solutions for the nonlinear Hirota–Satsuma coupled KdV system , 2006 .

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

[19]  R. Hirota,et al.  Soliton solutions of a coupled Korteweg-de Vries equation , 1981 .

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

[21]  Zhang Sheng,et al.  The periodic wave solutions for the (2 + 1)-dimensional dispersive long water equations , 2007 .

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

[23]  X. Feng,et al.  Exploratory Approach to Explicit Solution ofNonlinear Evolution Equations , 2000 .

[24]  Ahmet Bekir,et al.  Exact solutions of coupled nonlinear evolution equations , 2008 .

[25]  Kongqing Yang,et al.  Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature , 2001 .

[26]  Qi Wang,et al.  Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation , 2005 .

[27]  Alexey V. Porubov,et al.  Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer , 1996 .

[28]  Khaled A. Gepreel,et al.  A modified extended method to find a series of exact solutions for a system of complex coupled KdV equations , 2005 .

[29]  Zhenya Yan.  New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations , 2001 .

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

[31]  Nikolai A. Kudryashov,et al.  Extended simplest equation method for nonlinear differential equations , 2008, Appl. Math. Comput..

[32]  Bin Wu,et al.  Painlevé analysis and special solutions of generalized Broer-Kaup equations , 2002 .

[33]  Zuntao Fu,et al.  JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS , 2001 .

[34]  M. Boiti,et al.  Spectral transform for a two spatial dimension extension of the dispersive long wave equation , 1987 .

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

[36]  Zhenya Yan,et al.  Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey–Stewartson-type equation via a new method , 2003 .

[37]  Xiangzheng Li,et al.  Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations , 2005 .

[38]  Wu Jian-Ping,et al.  N-Soliton Solution of a Generalized Hirota?Satsuma Coupled KdV Equation and Its Reduction , 2009 .

[39]  Khaled A. Gepreel,et al.  Group Analysis and Modified Extended Tanh-function to Find the Invariant Solutions and Soliton Solutions for Nonlinear Euler Equations , 2004 .

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

[41]  Sheng Zhang,et al.  A further improved tanh function method exactly solving the -dimensional dispersive long wave equations. , 2008 .

[42]  Jianlan Hu,et al.  Explicit solutions to three nonlinear physical models , 2001 .

[43]  Khaled A. Gepreel,et al.  On the solitary wave solutions for nonlinear Hirota–Satsuma coupled KdV of equations , 2004 .

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

[45]  R. Hirota.  Exact envelope‐soliton solutions of a nonlinear wave equation , 1973 .

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

[47]  Nikolai A. Kudryashov,et al.  Exact solutions of the generalized Kuramoto-Sivashinsky equation , 1990 .

[48]  Zhuosheng Lü,et al.  Soliton like and multi-soliton like solutions for the Boiti–Leon–Pempinelli equation , 2004 .

[49]  Engui Fan,et al.  Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method , 2002 .

[50]  Dazhao Lü,et al.  Jacobi elliptic function solutions for two variant Boussinesq equations , 2005 .

[51]  Ding-jiang Huang,et al.  Exact travelling wave solutions for the Boiti–Leon–Pempinelli equation , 2004 .

[52]  E. Yomba.  Construction of new soliton-like solutions of the (2 + 1) dimensional dispersive long wave equation , 2004 .

[53]  Kwok Wing Chow,et al.  A class of exact, periodic solutions of nonlinear envelope equations , 1995 .

[54]  Zhenya Yan,et al.  New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water , 2001 .

[55]  Deng-Shan Wang,et al.  Further extended sinh-cosh and sin-cos methods and new nontraveling wave solutions of the -dimensional dispersive long wave equations. , 2005 .

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

[57]  Khaled A. Gepreel,et al.  On the solitary wave solutions for nonlinear Euler equations , 2004 .

[58]  David J. Evans,et al.  On travelling wave solutions of some nonlinear evolution equations , 2004, Int. J. Comput. Math..

[59]  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..

[60]  Elsayed M. E. Zayed,et al.  Travelling solitary wave solutions for the nonlinear coupled Korteweg–de Vries system , 2007 .