Exact travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation using the new approach of generalized G′/G$\left (\boldsymbol { {G^{\prime }/G}} \right )$-expansion method

In this paper, the new generalized (G′/G)$G^{\prime } /G)$-expansion method is executed to find the travelling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVB equation. The efficiency of this method for finding exact and travelling wave solutions has been demonstrated. It is shown that the new approach of generalized (G′/G)${G}^{\prime }/G)$-expansion method is a straightforward and effective mathematical tool for solving nonlinear evolution equations in applied mathematics, mathematical physics and engineering. Moreover, this procedure reduces the large volume of calculations.

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

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

[3]  M. Kruskal,et al.  A two-parameter Miura transformation of the Benjamin-Ono equation , 1979 .

[4]  Abdul-Majid Wazwaz,et al.  The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations , 2005, Appl. Math. Comput..

[5]  Md. Nur Alam,et al.  Traveling wave solutions of the nonlinear Drinfel’d–Sokolov–Wilson equation and modified Benjamin–Bona–Mahony equations , 2013 .

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

[7]  S. Ibrahim,et al.  Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method , 2012 .

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

[9]  Engui Fan,et al.  Two new applications of the homogeneous balance method , 2000 .

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

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

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

[13]  Abdul-Majid Wazwaz,et al.  Multiple-soliton solutions for a (3 + 1)-dimensional generalized KP equation , 2012 .

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

[15]  Farah Aini Abdullah,et al.  New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation , 2013 .

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

[17]  Norhashidah Hj. Mohd. Ali,et al.  Abundant Exact Traveling Wave Solutions of Generalized Bretherton Equation via Improved (G′/G)-Expansion Method , 2012 .

[18]  Farah Aini Abdullah,et al.  New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method , 2012, J. Appl. Math..

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

[20]  Anjan Biswas,et al.  Modified simple equation method for nonlinear evolution equations , 2010, Appl. Math. Comput..

[21]  Elsayed M. E. Zayed,et al.  A note on the modified simple equation method applied to Sharma-Tasso-Olver equation , 2011, Appl. Math. Comput..

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

[23]  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 .

[24]  M. A. Abdou,et al.  New exact travelling wave solutions using modified extended tanh-function method , 2007 .

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