New Exact Solutions of the Brusselator Reaction Diffusion Model Using the Exp-Function Method
Firstly, using a series of transformations, the Brusselator reaction diffusion model is reduced into a nonlinear reaction diffusion equation, and then through using Exp-function method, more new exact solutions are found which contain soliton solutions. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The results show the reliability and efficiency of the proposed method.
The Brusselator reaction model plays an important role both in biology and in chemistry. Since the model was put forward by Prigogine and Lefever in 1968, much attention had been paid to the model and many properties of it had been researched by marly people via using different methods [1–5]. In this paper, we mainly consider the model improved by Lefever et al. in 1977  . The model is described as follows: where is a constant, and is the diffusion coefficient; the functions and denote the concentrations. System (1.1) describes a biochemical model. Recently, many approaches have been suggested to solve the nonlinear equations, such as the variational iteration method [6–8], the homotopy perturbation method [9–11], the tanh-method , the extended tanh-method , the sinh-method , the homogeneous balance method [15, 16], the F-expansion method , and the extended Fan's subequation method . Recently, He and Wu  have proposed a straightforward method called Exp-function method to obtain the exact solutions of nonlinear evolution equations (NLEEs). It should be pointed out that the method is also valid for difference-different equations [20, 21]. The solution's procedure of this method is of utter simplicity, and this method has been successfully applied to many kinds of NLEEs [22–33]. The Exp-function method not only provides generalized solitonary solutions but also provides periodic solutions. Taking advantage of the generalized solitonary solutions, we can recover some known solutions obtained by the most existing methods such as decomposition method, tanh-function method, algebraic method, extended Jacobi elliptic function expansion method, F-expansion method, auxiliary equation method, and others [22–33].
2. Exp-Function Method and Exact Solutions
In this section we intend to find a solitary wave solution of (1.1). Therefore by using the following transformations: the system (1.1) is reduced to a nonlinear reaction diffusion equation with respect to :
After that we use the transformation where and are constants to be determined later, and is arbitrary constant. Therefore (2.2) converts to By virtue of the Exp-function method , we assume that the solution of (2.4) is of the form
where and are unknown positive integers and to be determined later; and are constants.
By balancing the highest order of linear term with the highest order of nonlinear term , the values of and can be determined easily.
Since setting leads easily to
Similarly, to determine and , we balance the lowest-order linear term of Exp-function in (2.4): This requires which leads to
We can freely choose the values of and , but we will illustrate that the final solution does not strongly depend on the choice of the values of and [19, 28]. Choosing and for simplicity causes the trial function (2.5) to become By substituting (2.10) into (2.4), we get where ; are constants. If the coefficients of are set to zero, we have By solving this system of algebraic equations, we obtain the following sets of solutions.
Case 2 E.
Substituting (2.13)–(2.17) into (2.10) gives the generalized solitonary solution where and the solutions where , and where denotes , and , and where , and where , respectively. The choice of and in our solution (2.18) gives the same bell solitary wave solution presented in  which was obtained on using the sine-cosine method Also if we set in (2.18), we obtain the new solutions Also, setting and causes (2.19) to lead to the new kink solitary wave solution This solution is similar to the solution obtained in .
By choosing and in (2.22), we can find solitary wave solution where .
An investigation on the Brusselator reaction diffusion model was established by using the Exp-function method. During this procedure some new exact solitary wave solutions, mostly solitons and kinks solutions, were obtained as well as some special cases. In particular, Yan's solution  can be considered as a special case of our result, and our result can turn into kink, soliton, and bell solutions with a suitable choice of the parameters. The study reveals the power of the method.
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