The Two‐Variable (G^′/G, 1/G)‐Expansion Method for Solving the Nonlinear KdV‐mKdV Equation

Abstract

We apply the two-variable (, )-expansion method to construct new exact traveling wave solutions with parameters of the nonlinear ()-dimensional KdV-mKdV equation. This method can be thought of as the generalization of the well-known ()-expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

New Exact Solutions for Boussinesq Type Equations by Using (G'/G, 1/G) and (1/G')-Expansion Methods

“…As a pioneer work, Li et al [39] have applied the two-variable (G /G, 1/G)--expansion method and found the exact solutions of the Zakharov equations. Then Zayed and Abdelaziz [40,41] determined exact solutions of some nonlinear evolution equations. (1/G )-expansion method has rst been introduced by Yoku³ [42].…”

Close Form Solutions of the Fractional Generalized Reaction Duffing Model and the Density Dependent Fractional Diffusion Reaction Equation

“…The exact solutions to NLEEs help us to provide information about the structure of complex phenomena. As a key problem, finding their exact solutions is of great importance and it is actually executed through various efficient and powerful method, such as, the Hirota method [1], the Backlund transform method [2,3], the inverse scattering transform method [4], the Jacobi elliptic function expansion method [5][6][7], the truncated Painleve expansion method [8][9][10][11], the tanh function method [12][13][14][15], the Exp-function method [16][17][18][19][20][21][22], the ( ⁄ ) -expansion method [23][24][25][26][27][28][29][30], the improved ( ⁄ ) -expansion method [31][32], the two variable ( ⁄ , 1 ⁄ ) -expansion method [33,34], the first integral method [35] etc. The main concept of the ( ⁄ ) -expansion method is the exact solution of nonlinear NLEEs are revealed by a polynomial in one variable ( ⁄ ) in which = ( ) satisfies the second order ordinary differential equation (ODE) ( ) + ( ) + ( ) = 0, where and are constants.…”

“…The degree of the polynomial can be evaluated by taking homogeneous balance between the highest-order derivatives and nonlinear terms in the given nonlinear PDEs, where the coefficient of the polynomial can be determined by solving a set of algebraic equations. Recently, Li et al [33] Density Dependent Fractional Diffusion Reaction Equation and Zayed et al [34] applied the two variable ( ⁄ , 1 ⁄ )-expansion method and determined the exact solution of nonlinear NLEEs.…”

Exact Solutions and Optical Soliton Solutions of the Nonlinear Biswas-Milovic Equation with Dual-Power Law Nonlinearity

“…Several methods for finding the exact solutions to nonlinear equations in mathematical physics have been presented, such as the inverse scattering method [1], the Hirota bilinear transform method [2], the truncated Painlevé expansion method [3][4][5][6], the Bäcklund transform method [7,8], the exp-function method [9][10][11], the tanhfunction method [12,13], the Jacobi elliptic function expansion method [14][15][16][17][18], the (G /G)-expansion method [19][20][21][22][23][24][25], the modified (G /G)-expansion method [26], the (G /G, 1/G)-expansion method [27][28][29][30], the modified simple equation method [11,[31][32][33], the multiple expfunction algorithm method [34,35], the transformed rational function method [36], the local fractional series expansion method [37], the first integral method [38][39][40], the generalized Riccati equation mapping method [17,18,41,…”