Some Remarks on Exp-Function Method and Its Applications

Abstract

Recently, many important nonlinear partial differential equations arising in the applied physical and mathematical sciences have been tackled by a popular approach, the so-called Exp-function method. In this paper, we present some shortcomings of this method by analyzing the results of recently published papers. We also discuss the possible improvement of the effectiveness of the method.

Exact Solutions of Non-Linear Lattice Equations by an Improved Exp-Function Method

“…To the best of our knowledge, they have not been reported in the literature. Solutions (38) and (39) contain arbitrary function α(t) and arbitrary constants b 1 and b −1 , which provide enough freedom for us to describe rich spatial structures of these obtained solutions. Applying the improved exp-function method to some other non-linear lattice equations with variable coefficients are worthy of study.…”

Some Remarks on Exp-Function Method and Its Applications-A Supplement

“…The solutions (18) and (19) are not general since they contain only one arbitrary constant while Eq. (1) has the second order.…”

“…In the mean time, a number of pitfalls of Exp-function method came to our attention, see Refs. [10][11][12][13][14][15][16][17][18][19][20]. Exp-function method assumes an ansatz (a rational combination of exponential functions) involving many unknown parameters to be specified at the stage of solving the problem.…”

“…The functions (18) and (19) admit singularity for C 1 < 0 and C 2 < 0. Assuming C 1 > 0 and C 2 > 0, Eqs.…”

Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method

“…In recent years, various methods have been established to obtain exact solutions for nonlinear partial differential equations (PDEs). For example, the Hirota bilinear transformation method, 1, 2 the Backlund transformation method, 3 the Jacobi elliptic function expansion method, 4 the tanh-coth method, 5,6 the homogeneous balance method, 7 the variational iteration method, 8 the F-expansion method, 9, 10 the direct algebraic method, 11 the Cole-Hopf transformation method, 12,13 the Exp-function method [14][15][16][17][18][19][20][21][22] and others. [23][24][25] Recently, Wang et al 26 presented the G /G -expansion method to obtain traveling wave solutions of the nonlinear partial differential equations.…”