A novel (G'/G)-expansion method for solving the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation in mathematical physics

A novel G′/G-expansion method is presented in this article to construct more general type and new travelling wave solutions of non-linear evolution equations. The method is illustrated in its application via the 3 + 1-dimensional modified KdV-Zakharov-Kuznetsov equation which is used to model different non-linear phenomena. Through the assistance of Maple, we obtain several new and more general travelling wave solutions in terms of the hyperbolic functions, the trigonometric functions, the rational functions and their combination with parameters. The solutions obtained in this article can successfully recover previously known solutions by setting particular values of the parameters that have been found by other sophisticated methods.

[1]  D. D. Ganji,et al.  Exact travelling solutions for the Lax's seventh-order KdV equation by sech method and rational exp-function method , 2008, Appl. Math. Comput..

[2]  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 .

[3]  Shimin Guo,et al.  The extended (G'/G)-expansion method and its applications to the Whitham-Broer-Kaup-Like equations and coupled Hirota-Satsuma KdV equations , 2010, Appl. Math. Comput..

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

[5]  Abdul-Majid Wazwaz,et al.  Partial Differential Equations , 2002 .

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

[7]  B. Rhoads.  On the Expansion Method , 1991 .

[8]  expansion Method and Its Application to the MKdV Equation , 2009 .

[9]  M. Akbar,et al.  Some New Exact Traveling Wave Solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation , 2012 .

[10]  Abdul-Majid Wazwaz,et al.  Partial differential equations : methods and applications , 2002 .

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

[12]  A. Wazwaz.  The extended tanh method for the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms , 2008 .

[13]  Abdul-Majid Wazwaz,et al.  A sine-cosine method for handlingnonlinear wave equations , 2004, Math. Comput. Model..

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

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

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

[17]  M. Ali Akbar,et al.  New Solitary and Periodic Solutions of Nonlinear Evolution Equation by Exp-function Method , 2012 .

[18]  Davood Domiri Ganji,et al.  Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation by homotopy perturbation method , 2006 .

[19]  New Applications of an Improved (G'/G)-expansion Method to Construct the Exact Solutions of Nonlinear PDEs , 2010 .

[20]  Jiao Zhang,et al.  An improved (G′/G)-expansion method for solving nonlinear evolution equations , 2010, Int. J. Comput. Math..

[21]  M. A. Abdou,et al.  New periodic wave solutions via Exp-function method , 2008 .

[22]  E. Zayed,et al.  New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized -expansion method , 2009 .

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

[24]  E. Zayed,et al.  The ( $\frac{G'}{G})$ -expansion method and its applications to some nonlinear evolution equations in the mathematical physics , 2009 .

[25]  E. Zayed,et al.  THE ($\frac{G'}{G}$)- EXPANSION METHOD COMBINED WITH THE RICCATI EQUATION FOR FINDING EXACT SOLUTIONS OF NONLINEAR PDES , 2011 .

[26]  Peter A. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: Remarks on Riemann-Hilbert problems , 1991 .

[27]  Norhashidah Hj. Mohd. Ali,et al.  A Generalized and Improved -Expansion Method for Nonlinear Evolution Equations , 2012 .

[28]  Nasir Taghizadeh,et al.  New solutions for some time fractional differential equations , 2012, Int. J. Comput. Sci. Math..

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

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

[31]  Shun-dong Zhu,et al.  The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation , 2008 .

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