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

Abstract

We employ the complex method to research the integrality of the Fisher equations with degree three. We obtain the sufficient and necessary condition of the 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 (2006), Guo and Chen (1991), and Ağırseven and Öziş (2010). Moreover, all 𝑤_𝑔,1(𝑧) are new general meromorphic solutions of the Fisher equations with degree three for 𝑐=±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.

1. Introduction and Main Result

Consider the Fisher equation 𝑢_𝑡=𝑣𝑢_𝑥𝑥+𝑠𝑢(1−𝑢), (1) which is a nonlinear diffusion equation as a model for the propagation of a mutant gene with an advantageous selection intensity 𝑠. 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 𝑡^′=𝑠𝑡 and 𝑥^′=(𝑠/𝑣)^1/2𝑥 and drop the primes; (1) becomes 𝑢_𝑡=𝑢_𝑥𝑥+𝑢(1−𝑢). (2) Substituting the traveling wave transform 𝑢(𝑥,𝑡)=𝑤(𝑧), 𝑥−𝑐𝑡=𝑧 into (2), it gives a nonlinear ordinary differential equation 𝑤^′′+𝑐𝑤^′+𝑤(1−𝑤)=0, (3) where 𝑐 is a constant.

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. [3–5] 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.

Recently, the complex method was introduced by Yuan et al. [6–8].

Very recently, Yuan et al. [9] employed the complex method to obtain the general solutions of (3). In order to state our results, we need some concepts and notations.

A meromorphic function 𝑤(𝑧) means that 𝑤(𝑧) is holomorphic in the complex plane ℂ except for poles. Denote by ℘(𝑧;𝑔₂,𝑔₃) the Weierstrass elliptic function with invariants 𝑔₂ and 𝑔₃. We say that a meromorphic function 𝑓 belongs to the class 𝑊 if 𝑓 is an elliptic function, or a rational function of 𝑒^𝛼𝑧, 𝛼∈ℂ, or a rational function of 𝑧.

Theorem 1. Equation (3) is integrable if and only if 𝑐=0,±5/√6,±5𝑖/√6. Furthermore, the general solutions of (3) are of the following forms. (I) (See [6]), when 𝑐=0, the elliptic general solutions of (3) 𝑤_𝑑(𝑧)(𝑧)+1=6−℘4℘^′(𝑧)+𝐹℘(𝑧)−𝐸²1−6𝐸+2, (4) where, 𝑔₂=1/12, 𝐹²=4𝐸³−𝑔₂𝐸−𝑔₃, 𝑔₃ and 𝐸 are arbitrary. In particular, it degenerates the simply periodic solutions 𝑤_𝑠(𝑧)=6𝛼²coth²𝛼2𝑧−𝑧₀+12𝛼²+12, (5) where 𝛼⁴=1, 𝑧₀∈ℂ. (II) (See [10]), when 𝑐=±5/√6, the general solutions of (3) 𝑤_𝑔(𝑧)=exp∓2𝑧√6℘∓𝑧exp√6−𝑠₀;0,𝑔₃, (6) where both 𝑠₀ and 𝑔₃ are arbitrary constants. In particular, it degenerates the one parameter family of solutions 𝑤_𝑓(𝑧)=11−exp±𝑧−𝑧₀/√6², (7) where 𝑧₀∈ℂ. (III) (See [9]), when 𝑐=±5𝑖/√6, the general solutions of (3) 𝑤_𝑔,𝑖(𝑧)=exp∓2𝑖𝑧√6℘𝑖exp∓𝑖𝑧√6−𝑠₀;0,𝑔₃+1, (8) where both 𝑠₀ and 𝑔₃ are arbitrary constants. In particular, it degenerates the one parameter family of solutions 𝑤_𝑓,𝑖(𝑧)1=−1−exp±𝑖𝑧−𝑧₀/√6²+1, (9) where 𝑧₀∈ℂ.

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 𝑤^′′+𝑐𝑤^′+𝑤1−𝑤²=0, (10) where 𝑐 is a constant.

Our main result is the following theorem.

Theorem 3. Equation (10) is integrable if and only if 𝑐=0, ±3/√2. Furthermore, the general solutions of (10) are of the following form.(I) (See [12]), when 𝑐=0, the elliptic general solutions of (10) 𝑤_𝑑,1(𝑧)√=±22℘^′𝑧−𝑧₀∶𝑔₂,0℘𝑧−𝑧₀∶𝑔₂,,0 (11) where 𝑧₀ and 𝑔₂ are arbitrary. In particular, it degenerates the simply periodic solutions 𝑤_𝑠,1(𝑧)=±√2tanh𝑧−𝑧₀, (12) where 𝑧₀∈ℂ.(II)When 𝑐=±3/√2, the general solutions of (10) 𝑤_𝑔,1(𝑧)1=±2∓𝑧exp√2℘^′exp∓𝑧/√2−𝑠₀;𝑔₂,0℘exp∓𝑧/√2−𝑠₀;𝑔₂,,0 (13) where ℘(𝑠:𝑔₂,0) is the Weierstrass elliptic function, both 𝑠₀ and 𝑔₂ are arbitrary constants. In particular, 𝑤_𝑔,1(𝑧) degenerates the one parameter family of solutions 𝑤_𝑓,1(𝑧)1=±1−exp±𝑧−𝑧₀/√2, (14) where 𝑧₀∈ℂ.

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 𝑤_𝑓,1(𝑧) by using other methods (cf. [13]). Moreover, all 𝑤_𝑔,1(𝑧) are new general meromorphic solutions of the Fisher equations with degree three for 𝑐=±3/√2.

Remark 5. The growth order of each 𝑤_𝑔(𝑧),𝑤_𝑔,𝑖(𝑧),𝑤_𝑔,1(𝑧) is infinite.

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.

2. 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.

Set 𝑚∈ℕ∶={1,2,3,…}, 𝑟_𝑗∈ℕ₀=ℕ∪{0}, 𝑟=(𝑟₀,𝑟₁,…,𝑟_𝑚), 𝑗=0,1,…,𝑚. We define a differential monomial denoted by 𝑀_𝑟[𝑤](𝑧)∶=[𝑤(𝑧)]^𝑟0𝑤^′(𝑧)^𝑟1𝑤^′′(𝑧)^𝑟2⋯𝑤^(𝑚)(𝑧)^𝑟𝑚. (15) And 𝑝(𝑟)∶=𝑟₀+𝑟₁+⋯+𝑟_𝑚 is called the degree of 𝑀_𝑟[𝑤]. A differential polynomial is defined by 𝑃𝑤,𝑤^′,…,𝑤^(𝑚)∶=_𝑟∈𝐼𝑎_𝑟𝑀_𝑟[𝑤], (16) where 𝑎_𝑟 are constants, and 𝐼 is a finite index set. The total degree is defined by deg𝑃(𝑤,𝑤^′,…,𝑤^(𝑚))∶=max_𝑟∈𝐼{𝑝(𝑟)}.

We will consider the following complex ordinary differential equations: 𝑃𝑤,𝑤^′,…,𝑤^(𝑚)=𝑏𝑤^𝑛+𝑐, (17) where 𝑏≠0, 𝑐 are constants, 𝑛∈ℕ.

Let 𝑝,𝑞∈ℕ. Suppose that (17) has a meromorphic solution 𝑤 with at least one pole; we say that (17) satisfies weak ⟨𝑝,𝑞⟩ condition if by substituting Laurent series 𝑤(𝑧)=^∞_𝑘=−𝑞𝑐_𝑘𝑧^𝑘,𝑞>0,𝑐_−𝑞≠0 (18) into (17), we can determin 𝑝 distinct Laurent singular parts below ⁻¹_𝑘=−𝑞𝑐_𝑘𝑧^𝑘. (19)

In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [14].

Let 𝜔₁, 𝜔₂ be two given complex numbers such that Im𝜔₁/𝜔₂>0, 𝐿=𝐿[2𝜔₁,2𝜔₂] be discrete subset 𝐿[2𝜔₁,2𝜔₂]={𝜔∣𝜔=2𝑛𝜔₁+2𝑚𝜔₂,𝑛,𝑚∈ℤ}, which is isomorphic to ℤ×ℤ. The discriminant Δ=Δ(𝑐₁,𝑐₂)∶=𝑐³₁−27𝑐²₂ and 𝑠_𝑛=𝑠_𝑛(𝐿)∶=_{{0}𝜔∈𝐿⧵}1𝜔^𝑛. (20)

Weierstrass elliptic function ℘(𝑧)∶=℘(𝑧,𝑔₂,𝑔₃) is a meromorphic function with double periods 2𝜔₁, 2𝜔₂ and satisfying the equation ℘^′(𝑧)²=4℘(𝑧)³−𝑔₂℘(𝑧)−𝑔₃, (21) where 𝑔₂=60𝑠₄, 𝑔₃=140𝑠₆ and Δ(𝑔₂,𝑔₃)≠0.

Lemma 6 (see [14, 15]). Weierstrass elliptic functions ℘(𝑧)∶=℘(𝑧,𝑔₂,𝑔₃) have two successive degeneracies and addition formula.(I) Degeneracy to simply periodic functions (i.e., rational functions of one exponential 𝑒^𝑘𝑧) according to ℘𝑧,3𝑑²,−𝑑³=2𝑑−3𝑑2coth²3𝑑2𝑧, (22) if one root 𝑒_𝑗 is double (Δ(𝑔₂,𝑔₃)=0).(II) Degeneracy to rational functions of 𝑧 according to ℘()=1𝑧,0,0𝑧² (23) if one root 𝑒_𝑗 is triple (𝑔₂=𝑔₃=0).(III) Addition formula ℘𝑧−𝑧₀=−℘(𝑧)−℘𝑧₀+14℘^′(𝑧)+℘^′𝑧₀℘(𝑧)−℘𝑧₀². (24)

In the proof of our main result, the following lemmas are very useful, which can be deduced by Theorem  1 in [12].

Lemma 7 (see [12]). The differential equation 𝑤^′′+𝑤−𝑤³=0 (25) has the elliptic solutions and simply periodic solution with pole at 𝑧=0𝑤(𝑧)√=±22℘^′𝑧∶𝑔₂,0℘𝑧∶𝑔₂,𝑤(𝑧)√,0=±2tanh𝑧, (26) respectively, where ℘(𝑧:𝑔₂,0) is the Weierstrass elliptic function with 𝑔₃=0 and 𝑔₂ arbitrary.

Lemma 8 (see [12]). The differential equation 𝑤^′′=2𝑤³ (27) has the elliptic solutions with pole at 𝑧=0𝑤(𝑧)1=±2℘^′𝑧∶𝑔₂,0℘𝑧∶𝑔₂,,0 (28) where ℘(𝑧∶𝑔₂,0) is the Weierstrass elliptic function with 𝑔₃=0 and 𝑔₂ arbitrary.

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

Step 1. Substituting the transform 𝑇∶𝑢(𝑥,𝑡)→𝑤(𝑧), (𝑥,𝑡)→𝑧 into a given PDE gives a nonlinear ordinary differential equations (17).

Step 2. Substitute (18) into (17) to determine that weak ⟨𝑝,𝑞⟩ condition holds and pass the Painlevé test for (17).

Step 3. Find the meromorphic solutions 𝑤(𝑧) of (17) with pole at 𝑧=0, which have 𝑚−1 integral constants.

Step 4. By the addition formula of Lemma 6 we obtain the general meromorphic solutions 𝑤(𝑧−𝑧₀).

Step 5. Substituting the inverse transform 𝑇⁻¹ into these meromorphic solutions 𝑤(𝑧−𝑧₀), then we get all exact solutions 𝑢(𝑥,𝑡) of the original given PDE.

3. Proof of Theorem 3

Proof. Substituting (18) into (10) we have 𝑞=1, 𝑝=2, 𝑐₋₁=±√2, 𝑐₀=∓𝑐/3√2, 𝑐₁=±(6−𝑐²)/18√2, 𝑐₂=0, and 0×𝑐₃+𝑐3²𝑐−23⁴=0. (29)
For the Laurent expansion (18) to be valid 𝑐 satisfies this equation and 𝑐₃ is an arbitrary constant. Therefore, 𝑐=0,±3/√2. For other 𝑐 it would be necessary to add logarithmic terms to the expansion, thus giving a branch point rather than a pole.
For 𝑐=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) 𝑤_𝑑,1(𝑧)√=±22℘^′𝑧−𝑧₀∶𝑔₂,0℘𝑧−𝑧₀∶𝑔₂,,0 (30) where 𝑧₀ and 𝑔₂ are arbitrary. In particular, it degenerates the simply periodic solutions 𝑤_𝑠,1(𝑧)=±√2tanh𝑧−𝑧₀, (31) where 𝑧₀∈ℂ.
For 𝑐=±3/√2, we transform (10) into the second Painlevé type equation. In this way we find the general solutions.
Setting 𝑤(𝑧)=𝑓(𝑧)𝑢(𝑠), 𝑠=𝑔(𝑧) and substituting in Fisher’s equation (10), we obtain that the equation for 𝑢(𝑠) is 𝑔^′2𝑢^′′=−𝑢^′𝑔^′𝑐+2𝑓^′𝑓+𝑔^′′𝑔^′−𝑓1+𝑐^′𝑓+𝑓^′′𝑓𝑢+𝑓²𝑢³. (32) If we take 𝑓 and 𝑔 such that 𝑐+2𝑓^′𝑓+𝑔^′′𝑔^′𝑓=0,1+𝑐^′𝑓+𝑓^′′𝑓=0, (33) then (32) for 𝑢 is integrable. By (33), one takes 𝑓(𝑧)=exp𝛼𝑧 and 𝑔(𝑧)=𝛽exp{−(𝑐+2𝛼)𝑧}, (34) where 𝛼=∓1/√2, 𝛽²=1.
Thus (32) reduces to 𝑢^′′=2𝑢³. (35)
Both Lemmas 6 and 8 give that the general solutions of (35) are of the form 𝑢(𝑠)1=±2℘^′𝑠−𝑠₀;𝑔₂,0℘𝑠−𝑠₀;𝑔₂,,0 (36) where ℘(𝑠) is the Weierstrass elliptic function 𝑠₀ and 𝑔₂ are two arbitrary constants.
Therefore, when 𝑐=±3/√2, by Lemma 6, the general solutions of (10) 𝑤_𝑔,1(𝑧)1=±2∓𝑧exp√2℘^′exp∓𝑧/√2−𝑠₀;𝑔₂,0℘exp∓𝑧/√2−𝑠₀;𝑔₂,,0 (37) where both 𝑠₀ and 𝑔₂ are arbitrary constants. In particular, by Lemma 6 and 𝑔₂=0, 𝑤_𝑔,1(𝑧) degenerates the one parameter family of solutions 𝑤_𝑓,1(𝑧)1=±1−exp±𝑧−𝑧₀/√2, (38) where 𝑧₀∈ℂ.

4. 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 reaction-diffusion 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 Ağırseven and Öziş [13]. Moreover, all 𝑤_𝑔,1(𝑧) are new general meromorphic solutions of the Fisher equations with degree three for 𝑐=±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.

Acknowledgments

This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as Visiting Scholars. This work was completed with the support of the NSF of China (11271090, 11101048, and 11171184) and NSF of Guangdong Province (S2012010010121) and Shanghai University Young Teacher Training Program (ZZSDJ12020), projects 10XKJ01, 12C401, and 12C104, from the Leading Academic Discipline Project of Shanghai Dianji University.