A Note about the General Meromorphic Solutions of the Fisher Equation

Abstract

We employ the complex method to obtain the general meromorphic solutions of the Fisher equation, which improves the corresponding results obtained by Ablowitz and Zeppetella and other authors (Ablowitz and Zeppetella, 1979; Feng and Li, 2006; Guo and Chen, 1991), and are new general meromorphic solutions of the Fisher equation for Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

Dedicated to Professor Hongxun Yi 70th birthday

1. 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.

Set and and drop the primes; (1) becomes Substituting the traveling wave transform , into (2) gives a nonlinear ordinary differential equation where is a constant.

Finding solutions of nonlinear models is a difficult and challenging task. Recently, the complex method was introduced by Yuan et al. [1–3].

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 except for poles. is the Weierstrass elliptic function with invariants and . We say that a meromorphic function belongs to the class if is an elliptic function, or a rational function of , , or a rational function of .

Our main result is the following theorem.

Theorem 1. Equation (3) is integrable if and only if , , . Furthermore, the general solutions of (3) are of the forms below.
(I) When , the elliptic general solutions of (3) (see [1]) where , , and are arbitrary, in particular, which degenerates the simply periodic solutions where , .
(II) When , the general solutions of (3) (see [4]) where both and are arbitrary constants. In particular, the degenerate one-parameter family of solutions is given by where .
(III) When , the general solutions of (3) where both and are arbitrary constants. In particular, the degenerate one-parameter family of solutions is given by where .

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 [5–7]; are new general meromorphic solutions of the Fisher equation for .

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.

2. Preliminary Lemmas and the Complex Method

In order to give complex method and the proof of Theorem 1, we need some notations and results.

Set , , , and . We define a differential monomial denoted by

is called the degree of . A differential polynomial is defined as follows where are constants, and is a finite index set. The total degree of is defined by .

We will consider the following complex ordinary differential equations where , are constants, .

Let . Suppose that (12) has a meromorphic solution with at least one pole, we say that (12) satisfies weak condition if substituting Laurent series into (12) we can determine distinct Laurent singular parts below

In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [8].

Let , be two given complex numbers such that , be discrete subset , which is isomorphic to . The discriminant and

Weierstrass elliptic function is a meromorphic function with double periods , and satisfying the equation where , , and .

By changing (16) to the form we have , , .

Inversely, given two complex numbers and such that , then there exists double periods , Weierstrass elliptic function such that above results hold.

Lemma 3 (see [8, 9]). Weierstrass elliptic functions have two successive degeneracies and addition formula:
(I) degeneracy to simply periodic functions (i.e., rational functions of one exponential ) according to if one root is double ,
(II) degeneracy to rational functions of according to if one root is triple ,
(III) addition formula

By above lemma and results, we can give a new method below, say complex method, to find exact solutions of some PDEs.

Step 1. Substituting the transform , into a given PDE gives a nonlinear ordinary differential equations (12).

Step 2. Substitute (13) into (12) to determine that weak condition holds, and pass the Painlevé test for (12).

Step 3. Find the meromorphic solutions of (12) with pole at , which have integral constants.

Step 4. By Lemma 3, we obtain the general meromorphic solutions .

Step 5. Substituting the inverse transform into these meromorphic solutions , then we get all exact solutions of the original given PDE.

3. Proof of Theorem 1

Substituting (13) into (3), we have , , , , , , , , and

For the Laurent expansion (13) to be valid satisfies this equation and is an arbitrary constant. Therefore, or or , where . For other it would be necessary to add logarithmic terms to the expansion, thus giving a branch point rather than a pole. For , the solution of (3) has been given in Theorem 1 and can be also found by direct integration.

For , (3) is integrable by Ablowitz and Zeppetella [4] using the Painlevé analysis and the general solutions were given. That is, when , the general solutions of (3) where both and are arbitrary constants, in particular, which degenerates the one-parameter family of solutions where .

Thus we consider only for cases , where . By the same arguments of Ablowitz and Zeppetella, we transform (3) with into the first Painlevé type equation. In this way we find the general solutions.

Setting , , and substituting in Fisher’s equation (3), we obtain that the equation for is where The general solutions of (24) are the Weierstrass elliptic functions , where and are two arbitrary constants.

Therefore, when , the general solutions of (3) where both and are arbitrary constants. In particular, by Lemma 3 and , degenerate the one-parameter family of solutions where .

4. 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 reaction-diffusion 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 [4–6], and are new general meromorphic solutions of the Fisher equation for . Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors appreciate reviewers very much for their positive and constructive comments and suggestions on their paper. This work was supported by Tianyuan Youth Fund of the NSF of China (11326083), the NSF of China (11171184, 11271090), Shanghai University Young Teacher Training Program (ZZSDJ12020), Innovation Program of Shanghai Municipal Education Commission (14YZ164), NSF of Guangdong Province (S2012010010121), and projects 13XKJC01 from the Leading Academic Discipline Project of Shanghai Dianji University.