The General Traveling Wave Solutions of the Fisher Equation with Degree Three

1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 2 Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China 3 Cisco School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510420, China 4Department of Mathematics and Physics, Shanghai Dianji University, Shanghai 201306, China 5 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China


Introduction and Main Result
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.
In 2005 and 2009, Feng et al. [1,2] proposed an analytic method to construct explicitly exact and approximate solutions for nonlinear evolution equations.By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation were obtained explicitly.These solutions included solitary wave solutions, singular traveling wave solutions, and periodical wave solutions.These results indicated that in some cases their analytic approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations.It can also be applied to some related nonlinear dynamical systems.
In 2010 and 2011, Demina et al. [3][4][5] studied the meromorphic solutions of autonomous nonlinear ordinary differential equations.An algorithm for constructing meromorphic solutions in explicit form was presented.General expressions for meromorphic solutions (including rational, periodic, elliptic) were found for a wide class of autonomous nonlinear ordinary differential equations.
Remark 2. The Fisher equation is a classic and the simplest case of the nonlinear reaction-diffusion equation, but there are many applications about it and many authors have been researching it (cf.[11]).The first explicit form of a traveling wave solution for the Fisher equation was obtained by Ablowitz and Zeppetella [10] using the Painleve analysis.
In this paper, we consider the Fisher equation with degree three where  is a constant.Our main result is the following theorem.
Remark 4. The Fisher equation is a classic and the simplest case of the nonlinear reaction-diffusion equation, but there are many applications about it and many authors have been researching it (cf.[13]).Many authors obtained only  ,1 () by using other methods (cf.[13]).Moreover, all  ,1 () are new general meromorphic solutions of the Fisher equations with degree three for  = ±3/ √ 2.
This paper is organized as follows.In Section 2, the preliminary lemmas and the complex method are given.The proof of Theorem 3 is given and the general meromorphic solutions of (10) 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 our complex method and the proof of Theorem 1, we need some notations and results.
(II) Degeneracy to rational functions of  according to In the proof of our main result, the following lemmas are very useful, which can be deduced by Theorem 1 in [12].
By the aforementioned lemmas and results, we can give a new method below, say complex method, to find exact solutions of some PDEs.
Step 4. By the addition formula of Lemma 6 we obtain the general meromorphic solutions ( −  0 ).
Step 5. Substituting the inverse transform  −1 into these meromorphic solutions  ( −  0 ), then we get all exact solutions (, ) of the original given PDE.
For  = 0, (10) is completely integrable by standard techniques and the solutions are expressible in terms of elliptic functions (cf.[12]).That is, by Lemmas 6 and 7, the elliptic general solutions of (10) where  0 and  2 are arbitrary.In particular, it degenerates the simply periodic solutions where  0 ∈ C.
For  = ±3/ √ 2, we transform (10) into the second 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 general Fisher equations are simplest case of the nonlinear reactiondiffusion equations.In this paper, we employ the complex method to research the integrality of the Fisher equations with degree three.We obtain the sufficient and necessary condition of integrable of the Fisher equations with degree three and the general meromorphic solutions of the integrable Fisher equations with degree three, which improves the corresponding results obtained by Feng and Li [11], Guo and Chen [16], and Agırseven and Özis ¸ [13].Moreover, all  ,1 () are new general meromorphic solutions of the Fisher equations with degree three for  = ±3/ √ 2. Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics.