IOP PUBLISHING PHYSICA SCRIPTA Phys. Scr. 79 (2009) 025006 (6pp) doi:10.1088/0031-8949/79/02/025006 A note on the Exp-function method and its application to nonlinear equations Ahmad T Ali Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt E-mail: atali71@yahoo.com Received 23 July 2008 Accepted for publication 19 November 2008 Published 4 February 2009 Online at stacks.iop.org/PhysScr/79/025006 Abstract An improved Exp-function method is described and used for finding a unified solution of a nonlinear wave equation. The combined KdV–MKdV equation is selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free constants is obtained. This method can be applied also to many other equations. PACS numbers: 02.30.Jr, 04.20.Jb, 05.45.Yv 1. Introduction The investigation of exact solutions of nonlinear evolution equations (NLEEs) plays an important role in the study of nonlinear physical phenomena and has gradually become one of the most important and significant tasks. Recently, many new approaches to nonlinear wave equations have been proposed, for example, Bäcklund transformation [8, 20], the homogeneous balance method [22], the similarity transformation method [1, 21], the homotopy perturbation method [10–12], the variational iteration method [4, 9], the asymptotic methods [13], the tanh-function method [23], the F-expansion method [26, 30], the extended Fan’s sub-equation method [27, 28] and the Jacobi elliptic function expansion method [2, 5–7, 18, 19]. Recently, He et al [15–17, 24] proposed a straightforward and concise method, called the Exp-function method, to obtain generalized solitary solutions of NLEEs. The most detailed development of the Exp-function method and its applications have been given by the originator [14, 32]. The purpose of this paper is to present an improved Exp-function method and apply it to the combined KdV–MKdV equation for obtaining new exact solutions for it as a model for other partial differential equations (PDEs). In section 2, the basic idea of the Exp-function method is given. In section 3, we introduced the improved Exp-function method. The applications of the proposed method to the combined KdV–MKdV equation are illustrated in section 4. The conclusion is then given in section 5. 2. Basic idea of the Exp-function method We consider a general nonlinear partial differential equation (NLPDE) of the form P (u , u t , u x , u tt , u tx , u xx , . . .) = 0. (2.1) Using a travelling wave transformation ξ = kx + ω t , (2.2) where k and ω are constants, we can rewrite equation (2.1) as the following nonlinear ordinary differential equation (NLODE): Q(u , u ′ , u ′′ , u ′′′ ,...) = 0, (2.3) where the prime denotes the derivation with respect to ξ . According to the Exp-function method, it is assumed that the solution can be expressed in the form [16] u (ξ) = ∑ d i =−c a i exp (iξ) ∑ q j =− p b j exp (jξ) , (2.4) where c, d , p and q are positive integers that could be freely chosen, and a i and b j are unknown constants to be determined. To determine the values of c and p, we balance the linear term of highest order in equation (2.3) with the highest order nonlinear term. Similarly, to determine the values of d and q , we balance the linear term of lowest order in equation (2.3) with the lowest order nonlinear term. We note that it takes a lot of effort to do the balancing process twice to determine c, p and d , q . Besides it should be noted that so far in all studied examples, such as the modified Benjamin–Bona–Mahony equation [29], the Broer–Kaup–Kupershmidt equations [31] and the symmetric regularized long wave equation [25] and the foam drainage equation [3], the results amount to the fact that p = c and q = d . This suggests another way to improve the Exp-function method. 0031-8949/09/025006+06$30.00 1 © 2009 The Royal Swedish Academy of Sciences Printed in the UK