Many Exact Solutions of the Nonlinear KPP Equation Using the Bäcklund Transformation of the Riccati Equation

Citation: Zayed EME, Alurrfi KAE, Al Nowehy AG (2017) Many Exact Solutions of the Nonlinear KPP Equation Using the Bäcklund Transformation of the Riccati Equation. Int J Opt Photonic Eng 2:006 Copyright: © 2017 Zayed EME, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *Corresponding author: Elsayed ME Zayed, Department of Mathematics, Faculty of Sciences, Zagazig University, Zagazig, Egypt, E-mail: eme_zayed@yahoo.com VIBGYOR ISSN: 2631-5092

The objective of this article is to use the Bäcklund transformation of the Riccati equation to construct new exact traveling wave solutions of the following nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation [22,26,46]. It is well-known [16] that Eq. (2.6) has the following solutions: Step 3 We determine the positive integer N in (2.4) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in Eq. (2.3). More precisely we define the degree of u(ξ) as D[u(ξ)] = N which gives rise to the degree of other expressions as follows: Therefore, we can get the value of N in (2.4). In some nonlinear equations the balance number N is not a positive integer. In this case, we make the following transformations [47]: where p q is a fraction in the lowest terms, we let Step 4 We substitute (2.4) along with Eq. (2.6) into Eq. (2.3), collect all the terms with the same powers of φ I (ξ) and set them to zero, we obtain a system of algebraic equations, which can be solved by Maple to get the values of a i and k,ω. Consequently, we obtain the exact traveling wave solutions of Eq. (2.1).
Finally if B = 0, then the above method reduces to the well-known modified extended tanh-function method [16].
Where µ,γ,δ are real constants? Eq. (1.1) includes the Fisher equation, Huxley equation, Burgers-Huxley equation, Chaffee-Infanfe equation and Fitzhugh-Nagumo equation as special cases. Recently, Feng, et al. [22] have discussed Eq. (1.1) using the (G ' /G)-expansion method and found its exact solutions, while Zayed, et al. [26,46] have applied two methods via the modified simple equation method and the Riccati equation method combined with the (G ' /G)-expansion method respectively, to Eq. (1.1) and determined the exact traveling wave solutions of it. This paper is organized as follows: In Sec. 2, the description of the Bäcklund transformation of the Riccati equation is given. In Sec. 3, we use the given method described in Sec. 2, to find traveling wave solutions of the nonlinear KPP equation. In Sec. 4, physical explanations of some results are presented. In Sec. 5, some conclusions are obtained.

Description of the Bäcklund Transformation of the Riccati Equation
Suppose that we have the following nonlinear PDE: Where F is a polynomial in u(x,t) and its partial derivatives, in which the highest order derivatives and the nonlinear terms are involved. In the following, we give the main steps of this method [5,47]: Step 1 Using the wave transformation. Where P is a polynomial in u(ξ) and its total derivatives while = d dξ ′ .

Step 2
Assume that Eq. (2.3) has the formal solution.
Where a i are constants to be determined, such that a N ≠ 0 or a -N ≠ 0, while Ψ(ξ) comes from the following Bäcklund transformation

An Application
In this section, we will apply the method described in Sec. 2 to find the exact traveling wave solutions of the nonlinear KPP equation (1.1). To this end, we use the wave transformation (2.2) to reduce Eq. (1.1) to the following ODE: By balancing u " with u 3 in Eq. (3.1), we get N = 1. Consequently, we have the formal solution Where a 0 , a 1 , a -1 are constants to be determined, such that a1 ≠ 0 or a -1 ≠ 0 while Ψ(ξ) is given by (2.5). Now, substituting (3.2) along with Eqs. (2.5) and (2.6) into (3.1), collecting the coefficients of φ i (ξ), (i = 0,1,…, 6) and setting them to zero, we get a system of algebraic equations. Solving this system of algebraic equations with aid of Maple, we have the following results: Provided that δ > 0.
If b = 0, then we get Provided that δ > 0.
From (3.1), (3.2) and (3.9), we deduce the exact traveling solutions of Eq. (1.1) as follows: If b > 0 then we have the solutions

Result 3
( ) under the constraint condition   (3.18) under the constraint condition If b > 0, then we have the solutions

Physical Explanations of Our Obtained Solutions
The obtained exact traveling wave solutions for the nonlinear KPP equation (1.1) are hyperbolic, trigonometric and rational. In this section, we have presented some graphs of the exact solutions constructed by taking suitable values Figure 1, Figure 2, Figure 3 and Figure 4 as it illustrates some of our results obtained in this article. To this end, we select some special values of the obtained parameters, for example, in some of the hyperbolic solutions (3.4), (3.5) and the trigonometric solutions (3.12), (3.13) of the nonlinear KPP equation (1.1) with B = k = δ = γ =1, -10 < x,t < 1, respectively.

Conclusions
In this article, we have employed the Bäcklund transformation of the Riccati equation to obtain exact traveling wave solutions of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation (1.1). On comparing our results in this paper with the well-known results obtained in [22,26,46] we deduce that our results in this article are new and are not published elsewhere. Further, all solutions obtained in this article have been checked with the Maple by putting them back into the original equations. Finally, the proposed method in this article can be applied to many other nonlinear PDEs in mathematical physics, which will be done in forthcoming papers.