A Note about the General Meromorphic Solutions of the Fisher Equation

1 Department of Mathematics and Physics, Shanghai Dianji University, Shanghai 201306, China 2 Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, China 3Department of Information and Computing Science, Guangxi University of Technology, Liuzhou 545006, China 4 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 5 Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China


Introduction
Consider the Fisher equation which is a nonlinear diffusion equation as a model for the propagation of a mutant gene with an advantageous selection intensity .It was suggested by Fisher as a deterministic version of a stochastic model for the spatial spread of a favored gene in a population in 1936.
Finding solutions of nonlinear models is a difficult and challenging task.Recently, the complex method was introduced by Yuan et al. [1][2][3].
In this paper, we employ the complex method to obtain the general solutions of (3).The general traveling wave exact solutions of the Fisher equation can be deduced by the traveling wave transform (, ) = (),  −  = .In order to state our results, we need some concepts and notations.
A meromorphic function () means that () is a holomorphic excepting for pole in the complex plane C except for poles.℘(;  2 ,  3 ) is the Weierstrass elliptic function with invariants  2 and  3 .We say that a meromorphic function  belongs to the class  if  is an elliptic function, or a rational function of   ,  ∈ C, or a rational function of .
Our main result is the following theorem.
(II) When  = ±5/ √ 6, the general solutions of (3) (see [4]) where both  0 and  3 are arbitrary constants.In particular, the degenerate one-parameter family of solutions is given by where  0 ∈ C.
(III) When  = ±5/ √ 6, the general solutions of (3) where both  0 and  3 are arbitrary constants.In particular, the degenerate one-parameter family of solutions is given by where  0 ∈ C.
Remark 2. The Fisher equation is classic and simplest case of the nonlinear reaction-diffusion equation, but there are many applications about it and many authors researched it [5].The first explicit form of a traveling wave solution for the Fisher equation was obtained by Ablowitz and Zeppetella [4] using the Painlevé analysis.Many authors obtained only  , () by using other methods [5][6][7];  , () are new general meromorphic solutions of the Fisher equation for  = ±5/ √ 6.
This paper is organized as follows.In the next section, the preliminary lemmas and the complex method are given.The proof of Theorem 1 is given and the general meromorphic solutions of (3) are derived by complex method in Section 3. Some conclusions and discussions are given in the final section.

Preliminary Lemmas and the Complex Method
In order to give complex method and the proof of Theorem 1, we need some notations and results.
Step 5. Substituting the inverse transform  −1 into these meromorphic solutions ( −  0 ), then we get all exact solutions (, ) of the original given PDE.
For  = ±5/ √ 6, (3) is integrable by Ablowitz and Zeppetella [4] using the Painlevé analysis and the general solutions were given.That is, when  = ±5/ √ 6, the general solutions of (3) where both  0 and  3 are arbitrary constants, in particular, which degenerates the one-parameter family of solutions where  0 ∈ C. Thus we consider only for cases  = ±5/ √ 6, where  2 = −1.By the same arguments of Ablowitz and Zeppetella, we transform (3) with  = ±5/ √ 6 into the first Painlevé type equation.In this way we find the general solutions.

Conclusions
Complex method is a very important tool in finding the exact solutions of nonlinear evolution equations, and the Fisher equation is classic and simplest case of the nonlinear reactiondiffusion equation.In this paper, we employ the complex method to obtain the general meromorphic solutions of the Fisher equation, which improves the corresponding result obtained by Ablowitz and Zeppetella and other authors [4][5][6], and  , () are new general meromorphic solutions of the Fisher equation for  = ±5/ √ 6.Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.