Traveling wave solutions of the time-delayed generalized Burgers-type equations

Background Recently, nonlinear time-delayed evolution equations have received considerable interest due to their numerous applications in the areas of physics, biology, chemistry and so on. Methods In this paper, we obtain traveling wave solutions by using the extended \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{G^{\prime}}{G}\right)$$\end{document}G′G-expansion method. Results Based on the method, we get many solutions of the time-delayed generalized Burgers-type equations. Conclusions The results reveal that the extended \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{G^{\prime}}{G}\right)$$\end{document}G′G-expansion method is direct, effective and can be used for many other nonlinear time-delayed evolution equations.

• The time-delayed generalized Burgers equation: where p, s are constants and τis a time-delayed constant.
• The time-delayed generalized Burgers-Fisher equation: This paper is organized as follows: in "Methods" section, the main steps of extended G ′ G -expansion method for obtaining traveling wave solutions of nonlinear time-delayed evolution equation are given. In "Results" section, we construct traveling solutions of the time-delayed generalized Burgers-type equation. Some conclusions are given in "Conclusions" section.

Considering the following nonlinear evolution equation:
where P is a polynomial in v = v(x 1 , x 2 , x 3 , . . . , t) and its various partial derivatives.
Step 1 By means of the traveling wave transformation where the coefficients k i , h are constants. Equation (1) can be transformated as follows: Step 2 We suppose that the Eq. (3) has the following solution: where a l are constants to be determined later, and G(η) satisfies the following equation: where α and β are arbitrary constants. Based on Eq. (5), we have

Results
In this section, we apply the extended G ′ G -expansion method to obtain traveling wave solutions of the time-delayed generalized Burgers-type equations.

Solutions to the time-delayed generalized Burgers equation
We consider the following time-delayed generalized Burgers equation: By using transformations v(x, t) = V (η) and η = k(x − ωt), Eq. (6) can be reduced as follows: Balancing V ′′ with V s V ′ gives n = 1 s which is not an integer as s � = 1. So we use a transformation V = W 1 s to change Eq. (7) into the form: We suppose that the solutions of (8) have the form (4) and (5), so From above two equations, we can get the degrees of W ′′ W and W ′ W 2 are 2n + 2 and 3n + 1 respectively. Balancing W ′′ W and W ′ W 2 in Eq. (8) yields 2n + 2 = 3n + 1, namely n = 1. Therefore Eq. (8) have the following solutions: Substituting Eqs. (9) and (5) into Eq. (8), we get a set of under-determined algebraic equations for a l (l = 0, ±1), k, ω, α and β.
Solving this algebraic equations by Maple, we can obtain the two results:

Case 2
where α, β and ω are arbitrary constants. Using Eqs. (9) and (10), we obtain the following solution of Eq. (6): Based on Eqs. (9) and (11), we get the solution of Eq. (6) as follows: Substituting the general solutions of Eq. (5) into Eq. (12), we have two kinds of travelling wave solutions as follows: When Tang et al. SpringerPlus (2016) 5:2094 where Substituting the general solutions of Eq. (5) into Eq. (13), we have the following two kinds of travelling wave solutions: When In Figs. 1, 2, 3 and 4, we show the effect of the time-delayed solution (14). It should be noted that when τ → 0, we can recover some traveling wave solutions of the generalized Burgers equation.
Remark 1 By using extended G ′ G -expansion method, we can obtain solutions including all the solutions given in Deng et al. (2009) as special cases. For example, if setting C 2 = 0, then solution (28) is the same as Eq. (19) in Deng et al. (2009). Similarly, solution (28) is also the same as Eq. (20) obtained in Deng et al. (2009) when we set C 1 = 0. It shows that extended G ′ G -expansion method is more powerful than the method in Deng et al. (2009) in constructing exact solutions.
Remark 2 Rosa et al. (2015) applied Lie classical method and G ′ G -expansion method to Fisher equation and derived some new traveling wave solutions. If setting a l (l = −n . . . − 1) = 0, then Eq. (4) becomes Eq. (14) in Rosa and Gandarias, (2015). So if we applied Lie classical method and extended G ′ G -expansion method to Fisher equation, then many more exact solutions can be obtained. Searching exact solutions by use of Lie classical method and extended G ′ G -expansion method is our future work. Taking C 1 = a 1 , C 2 = a 2 , we can convert Eq. (33) into the following form: