New Exact Solutions of the Brusselator Reaction Diffusion Model Using the Exp-Function Method

Abstract

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.

1. Introduction

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 [1] . The model is described as follows: 𝐾𝜓_𝑥𝑥−𝜓_𝑡+𝜓²𝜑−𝐵𝜓=0,𝐾𝜑_𝑥𝑥−𝜑_𝑡−𝜓²𝜑+𝐵𝜓=0, (1.1) where 𝐵 is a constant, and 𝐾 is the diffusion coefficient; the functions 𝜓(𝑥,𝑡) and 𝜑(𝑥,𝑡) 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 [12], the extended tanh-method [13], the sinh-method [14], the homogeneous balance method [15, 16], the F-expansion method [17], and the extended Fan's subequation method [18]. Recently, He and Wu [19] 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: 𝜓=𝑣,𝜑=−𝑣,𝑥=±𝑦,𝑡=𝑡, (2.1) the system (1.1) is reduced to a nonlinear reaction diffusion equation with respect to 𝑣(𝑦,𝑡):

𝐾𝑣_𝑦𝑦−𝑣_𝑡−𝐵𝑣−𝑣³=0. (2.2) After that we use the transformation 𝑣=𝜙(𝜉),𝜉=𝜆(𝑦−𝑤𝑡+𝛾), (2.3) where 𝜆 and 𝑤 are constants to be determined later, and 𝛾 is arbitrary constant. Therefore (2.2) converts to 𝐾𝜆²𝑑²𝜙𝑑𝜉²+𝜆𝑤𝑑𝜙𝑑𝜉−𝐵𝜙−𝜙³=0. (2.4) By virtue of the Exp-function method [19], we assume that the solution of (2.4) is of the form

𝑎𝜙(𝜉)=^𝑐exp(𝑐𝜉)+⋯+𝑎_−𝑑exp(−𝑑𝜉)𝑏_𝑓exp(𝑓𝜉)+⋯+𝑏_−𝑔,exp(−𝑔𝜉) (2.5) where 𝑐,𝑑,𝑓, and 𝑔 are unknown positive integers and to be determined later; 𝑎_𝑛 and 𝑏_𝑚 are constants.

By balancing the highest order of linear term 𝜙^′′ with the highest order of nonlinear term 𝜙³, the values of 𝑐 and 𝑓 can be determined easily.

Since 𝜙^′′=𝑐₁exp[(𝑐+3𝑓)𝜉]+⋯𝑐₂,𝜙exp[4𝑓𝜉]+⋯₃=𝑐₃exp[(3𝑐+𝑓)𝜉]+⋯𝑐₄,exp[4𝑓𝜉]+⋯ (2.6) setting 𝑐+3𝑓=3𝑐+𝑓 (2.7) leads easily to 𝑐=𝑓.

Similarly, to determine 𝑑 and 𝑔, we balance the lowest-order linear term of Exp-function in (2.4): 𝜙^′′=⋯+𝑑₁exp[−(𝑑+3𝑔)𝜉]⋯+𝑑₂,𝜙exp[−3𝑔𝜉]₃=⋯+𝑑₃exp[−(3𝑑+𝑔)𝜉]⋯+𝑑₄.exp[−3𝑔𝜉] (2.8) This requires −(𝑑+3𝑔)=−(3𝑑+𝑔), (2.9) which leads to 𝑔=𝑑.

We can freely choose the values of 𝑐 and 𝑑, but we will illustrate that the final solution does not strongly depend on the choice of the values of 𝑐 and 𝑑 [19, 28]. Choosing 𝑓=𝑐=1 and 𝑑=𝑔=1 for simplicity causes the trial function (2.5) to become 𝑎𝜙(𝜉)=¹exp(𝜉)+𝑎₀+𝑎₋₁exp(−𝜉)𝑏₁exp(𝜉)+𝑏₀+𝑏₋₁.exp(−𝜉) (2.10) By substituting (2.10) into (2.4), we get 1𝑄(𝑘₃exp(3𝜉)+𝑘₂exp(2𝜉)+𝑘₁exp(𝜉)+𝑘₀+𝑘₋₁exp(−𝜉)+𝑘₋₂exp(−2𝜉)+𝑘₋₃exp(−3𝜉))=0, (2.11) where 𝑄=(𝑏₁exp(𝜉)+𝑏₀+𝑏₋₁exp(−𝜉))³; 𝑘_𝑖(𝑖=−3,−2,…,2,3) are constants. If the coefficients of exp(𝑛𝜉) are set to zero, we have 𝑘₃=0,𝑘₂=0,𝑘₁𝑘=0,₀𝑘=0,₋₃=0,𝑘₋₂=0,𝑘₋₁=0. (2.12) By solving this system of algebraic equations, we obtain the following sets of solutions.

Case A. 𝑎₁=0,𝑎₀=𝑎₀,𝑎₋₁=0,𝜆²=𝐵𝐾,𝑏₁=𝑏₁,𝑏₀=𝑏₀,𝑏₋₁𝑎=−²⁰8𝐵𝑏₁,𝑤=0. (2.13)

Case B. 𝑎₁=0,𝑎₀=𝑎₀,𝑎₋₁=𝐵𝑏₋₁√−𝐵,𝜆²=−2𝐵𝐾,𝑏₁=0,𝑏₀=𝑎₀√−𝐵,𝑏₋₁=𝑏₋₁,𝑤=0. (2.14)

Case C. 𝑎₁=𝑎₁,𝑎₀=𝑎₀,𝑎₋₁=𝐵𝑏²₀+𝑎²₀4𝑎₁,𝜆²=−2𝐵𝐾,𝑏₁=𝑎₁√−𝐵,𝑏₀=𝑏₀,𝑏₋₁=14𝐵𝑎₀𝑏²₀+𝑎³₀+𝑏³₀(𝐵²/√−𝐵)+𝑏₀(𝐵𝑎²₀/√−𝐵)𝑎₁𝐵(𝑎₀/√−𝐵−𝑏₀),𝑤=0. (2.15)

Case D. 𝑎₀=𝑎₀,𝑎₋₁=0,𝑎₁=0𝜆²=−𝐵,𝑏2𝐾₀=𝑎₀√−𝐵,𝑏₋₁=𝑏₋₁,𝑏₁=0,𝑤=3𝐵.−2𝐵/𝐾 (2.16)

Case 2 E. 𝑎₀=𝑎₀,𝑎₋₁=0,𝑎₁=𝑎₁,𝜆²=−𝐵,𝑏2𝐾₀=−𝑎²₀+𝐵𝑎₁𝑏₋₁/√−𝐵𝐵𝑎₀/√−𝐵,𝑏₋₁=𝑏₋₁,𝑏₁=𝑎₁√−𝐵,𝑤=3𝐵.−2𝐵/𝐾 (2.17)

Substituting (2.13)–(2.17) into (2.10) gives the generalized solitonary solution 𝑎𝜙(𝜉)=⁰𝑏₁exp(𝜉)−(𝑎²₀/8𝐵𝑏₁,)exp(−𝜉) (2.18) where 𝜉=𝐵/𝐾(±𝑥+𝛾) and the solutions 𝑎𝜙(𝜉)=⁰+(𝐵𝑏₋₁/√−𝐵)exp(−𝜉)𝑎₀/√−𝐵+𝑏₋₁exp(−𝜉), (2.19) where 𝜉=−2𝐵/𝐾(±𝑥+𝛾), and 𝑎𝜙(𝜉)=¹exp(𝜉)+𝑎₀+(𝐵𝑏²₀+𝑎²₀/4𝑎₁)exp(−𝜉)(𝑎₁/√−𝐵)exp(𝜉)+𝑏₀,+𝔖exp(−𝜉) (2.20) where 𝔖 denotes 1/4((𝐵𝑎₀𝑏²₀+𝑎³₀+𝑏³₀𝐵²/√−𝐵+𝑏₀𝐵𝑎²₀/√−𝐵)/(𝑎₁𝐵(𝑎₀/√−𝐵−𝑏₀))), and 𝜉=−2𝐵/𝐾(±𝑥+𝛾), and 𝑎𝜙(𝜉)=⁰𝑎₀/√−𝐵+𝑏₋₁,exp(−𝜉) (2.21) where 𝜉=−𝐵/2𝐾(±𝑥−(3𝐵/−2𝐵/𝐾)𝑡+𝛾), and 𝑎𝜙(𝜉)=⁰+𝑎₁exp(𝜉)(𝑎₁/√−𝐵)exp(𝜉)+(−𝑎²₀+𝐵𝑎₁𝑏₋₁/√−𝐵)/(𝐵𝑎₀/√−𝐵)+𝑏₋₁,exp(−𝜉) (2.22) where 𝜉=−𝐵/2𝐾(±𝑥−(3𝐵/−2𝐵/𝐾)𝑡+𝛾), respectively. The choice of 𝑎₀=2√−2𝐵 and 𝑏₁=1 in our solution (2.18) gives the same bell solitary wave solution presented in [34] which was obtained on using the sine-cosine method 𝑣(𝑥,𝑡)=√−2𝐵sech𝐵𝐾(±𝑥+𝛾). (2.23) Also if we set 𝑏₁=±1,𝑎₀=2√𝑏2𝐵,₁=−1,𝑎₀=4√−𝐵, (2.24) in (2.18), we obtain the new solutions 𝑣(𝑥,𝑡)=±√2𝐵csch𝐵𝐾4√(±𝑥+𝛾),𝑣(𝑥,𝑡)=𝐵cosh𝐵/𝐾(±𝑥+𝛾)−3sinh.𝐵/𝐾(±𝑥+𝛾) (2.25) Also, setting 𝑎₀=√−𝐵 and 𝑏₋₁=±1 causes (2.19) to lead to the new kink solitary wave solution 𝑣(𝑥,𝑡)=√−𝐵coth−𝐵√2𝐾(±𝑥+𝛾),𝑣(𝑥,𝑡)=−𝐵tanh−𝐵2𝐾(±𝑥+𝛾). (2.26) This solution is similar to the solution obtained in [34].

By setting

𝑏₀=0,𝑎₁=√−𝐵,𝑎₀=±2𝑖√−𝐵, (2.27) in (2.20), we will have the solitary wave solution 𝑣(𝑥,𝑡)=√−𝐵(tanh−2𝐵𝐾(±𝑥+𝛾)±𝑖sech−2𝐵𝐾(±𝑥+𝛾)), (2.28) where 𝑖=√−1. Now if we set 𝑎₀=√−𝐵 and 𝑏₋₁=1 in (2.21), we will have another kink solitary wave solution √𝑣(𝑥,𝑡)=−𝐵2+√−𝐵2tanh−𝐵8𝐾(±𝑥−3𝐵−2𝐵/𝐾𝑡+𝛾), (2.29) which is similar to the solution obtained in [34].

If we also set 𝑏₋₁=1 and 𝑎₀=−√−𝐵 in (2.21), we obtain the new kink solitary wave solution √𝑣(𝑥,𝑡)=−𝐵2+√−𝐵2coth−𝐵8𝐾(±𝑥−3𝐵−2𝐵/𝐾𝑡+𝛾). (2.30) Also, setting 𝑏₋₁=𝑎₀ causes (2.21) to lead to the new soliton solution 1𝑣(𝑥,𝑡)=−(√−𝐵/𝐵)+cosh−𝐵/2𝐾(±𝑥−(3𝐵/,−2𝐵/𝐾)𝑡+𝛾)−ℌ (2.31) where ℌ denotes sinh−𝐵/2𝐾(±𝑥−(3𝐵/−2𝐵/𝐾)𝑡+𝛾).

By choosing 𝑎₁=√−𝐵,𝑎₋₁=1, and 𝑎₀=𝑖√−𝐵 in (2.22), we can find solitary wave solution √𝑣(𝑥,𝑡)=−𝐵2(1+tanh−𝐵2𝐾(±𝑥−3𝐵−2𝐵/𝐾𝑡+𝛾)+𝑖sech−𝐵2𝐾(±𝑥−3𝐵−2𝐵/𝐾𝑡+𝛾)), (2.32) where 𝑖=√−1.

3. Conclusion

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 [34] 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.