The general traveling wave solutions of the Fisher type equations and some related problems
 Wenjun Yuan^{1, 2},
 Bing Xiao^{3}Email author,
 Yonghong Wu^{4} and
 Jianming Qi^{5}Email author
https://doi.org/10.1186/1029242X2014500
© Yuan et al.; licensee Springer. 2014
Received: 29 September 2014
Accepted: 28 November 2014
Published: 18 December 2014
Abstract
In this article, we introduce two recent results with respect to the integrality and exact solutions of the Fisher type equations and their applications. We obtain the sufficient and necessary conditions of integrable and general meromorphic solutions of these equations by the complex method. Our results are of the corresponding improvements obtained by many authors. All traveling wave exact solutions of many nonlinear partial differential equations are obtained by making use of our results. Our results show that the complex method provides a powerful mathematical tool for solving a great number of nonlinear partial differential equations in mathematical physics. We will propose four analogue problems and expect that the answer is positive, at last.
MSC:30D35, 34A05.
Keywords
differential equation general solution meromorphic function elliptic function1 Introduction
Nonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of science, particularly in fluid mechanics, solid state physics, plasma physics and nonlinear optics. Exact solutions of NLPDEs of mathematical physics have attracted significant interest in the literature. Over the last years, much work has been done on the construction of exact solitary wave solutions and periodic wave solutions of nonlinear physical equations. Many methods have been developed by mathematicians and physicists to find special solutions of NLPDEs, such as the inverse scattering method [1], the Darboux transformation method [2], the Hirota bilinear method [3], the Lie group method [4], the bifurcation method of dynamic systems [5–7], the sinecosine method [8], the tanhfunction method [9, 10], the Fanexpansion method [11], and the homogenous balance method [12]. Practically, there is no unified technique that can be employed to handle all types of nonlinear differential equations. Recently, Kudryashov et al. [13–16] found exact meromorphic solutions for some nonlinear ordinary differential equations by using Laurent series and gave some basic results. Following their work, the complex method was introduced by Yuan et al. [17–19]. In this article, we survey two recent results with respect to the integrality and exact solutions of the Fisher type equations and their applications. We obtain the sufficient and necessary conditions of integrable and general meromorphic solutions of these equations by the complex method. Our results are of the corresponding improvements obtained by many authors. All traveling wave exact solutions of many nonlinear partial differential equations are obtained by making use of our results. Our results show that the complex method provides a powerful mathematical tool for solving a great number of nonlinear partial differential equations in mathematical physics. We will propose four analogue problems and expect that the answer is positive, at last.
2 Fisher type equations with degree two
where A, B, C and D are arbitrary constants.
In order to state these 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. α, b, c, ${c}_{i}$ and ${c}_{ij}$ are constants which may be different from each other in different place. We say that a meromorphic function f belongs to the class W if f is an elliptic function, or a rational function of ${e}^{\alpha z}$, $\alpha \in \mathbb{C}$, or a rational function of z.
 (I)The elliptic general solutions${w}_{1d}(z)=6\frac{A}{C}\{\mathrm{\wp}(z)+\frac{1}{4}{\left[\frac{{\mathrm{\wp}}^{\prime}(z)+F}{\mathrm{\wp}(z)E}\right]}^{2}\}+6\frac{AE}{C}\frac{B}{2C}.$
Here, $4DC=12{A}^{2}{g}_{2}+{B}^{2}$, ${F}^{2}=4{E}^{3}{g}_{2}E{g}_{3}$, ${g}_{3}$ and E are arbitrary.
 (II)The simply periodic solutions${w}_{1s}(z)=6\frac{A}{C}{\alpha}^{2}{coth}^{2}\frac{\alpha}{2}(z{z}_{0})\frac{A}{2C}{\alpha}^{2}\frac{B}{2C},$
where $4DC={A}^{2}{\alpha}^{4}+{B}^{2}$, ${z}_{0}\in \mathbb{C}$.
 (III)The rational function solutions${w}_{1r}(z)=\frac{6\frac{A}{C}}{{(z{z}_{0})}^{2}}\frac{B}{2C},$
where $4CD={B}^{2}$, ${z}_{0}\in \mathbb{C}$.
Equation (1) is an important auxiliary equation, because many nonlinear evolution equations can be converted to Eq. (1) using the traveling wave reduction. For instance, the classical KdV equation, the Boussinesq equation, the $(3+1)$dimensional JimboMiwa equation and the BenjaminBonaMahony equation can be converted to Eq. (1) [17].
where A, B, C, D, E are arbitrary constants.
 (i)If $B=0$, then we have the elliptic general solutions of Eq. (2)${w}_{2d}(z)=6\frac{A}{D}\{\mathrm{\wp}(z)+\frac{1}{4}{\left[\frac{{\mathrm{\wp}}^{\prime}(z)+M}{\mathrm{\wp}(z)N}\right]}^{2}\}+6\frac{AN}{D}\frac{C}{2D}.$
Here, $12{A}^{2}{g}_{2}={C}^{2}$, ${M}^{2}=4{N}^{3}{g}_{2}N{g}_{3}$, ${g}_{3}$ and N are arbitrary.
In particular, which degenerates to the simply periodic solutions${w}_{2s}(z)=6\frac{A}{D}{\alpha}^{2}{coth}^{2}\frac{\alpha}{2}(z{z}_{0})\frac{A}{2D}{\alpha}^{2}\frac{C}{2D},$where ${A}^{2}{\alpha}^{4}={C}^{2}$, ${z}_{0}\in \mathbb{C}$.
And the rational function solutions${w}_{2r}(z)=\frac{6\frac{A}{D}}{{(z{z}_{0})}^{2}}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{C}{2D},$where ${C}^{2}=4DE$, ${z}_{0}\in \mathbb{C}$.
 (ii)If $B=\pm \frac{5}{\sqrt{6}}\sqrt{2AD\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}$, then the general solutions of Eq. (2)$\begin{array}{rcl}{w}_{g2}(z)& =& exp\{\mp \frac{2}{\sqrt{6}}\sqrt{\frac{2D}{A}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}z\}\\ \times \mathrm{\wp}(\sqrt{\frac{D}{A}}exp\{\mp \frac{1}{\sqrt{6}}\sqrt{\frac{D}{A}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}z\}{s}_{0};0,{g}_{3})\\ \sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{C}{2D},\end{array}$
where $\sqrt{\frac{{C}^{2}}{4{D}^{2}}}=\frac{C}{2D}$, both ${s}_{0}$ and ${g}_{3}$ are arbitrary constants.
In particular, which degenerates to the one parameter family of solutions${w}_{f2}(z)=2\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{1}{{\{1exp\{\pm \frac{(z{z}_{0})}{\sqrt{6}}\sqrt{\frac{2D}{A}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}\}\}}^{2}}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{C}{2D},$where $\sqrt{\frac{{C}^{2}}{4{D}^{2}}}=\frac{C}{2D}$, ${z}_{0}\in \mathbb{C}$.
 (iii)If $B=\pm \frac{5i}{\sqrt{6}}\sqrt{2AD\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}$, then the general solutions of Eq. (2)$\begin{array}{rcl}{w}_{g2,i}(z)& =& exp\{\mp \frac{2i}{\sqrt{6}}\sqrt{\frac{2D}{A}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}z\}\\ \times \mathrm{\wp}(\sqrt{\frac{D}{A}}exp\{\mp \frac{i}{\sqrt{6}}\sqrt{\frac{D}{A}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}z\}{s}_{0};0,{g}_{3})\\ \sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{3C}{2D},\end{array}$
where $\sqrt{\frac{{C}^{2}}{4{D}^{2}}}=\frac{C}{2D}$, and both ${s}_{0}$ and ${g}_{3}$ are arbitrary constants.
In particular, which degenerates to the one parameter family of solutions$\begin{array}{rcl}{w}_{f2,i}(z)& =& 2\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{1}{{\{1exp\{\pm \frac{i(z{z}_{0})}{\sqrt{6}}\sqrt{\frac{2D}{A}\sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}}\}\}}^{2}}\\ \sqrt{\frac{{C}^{2}}{4{D}^{2}}\frac{E}{D}}\frac{3C}{2D},\end{array}$where $\sqrt{\frac{{C}^{2}}{4{D}^{2}}}=\frac{C}{2D}$, ${z}_{0}\in \mathbb{C}$.
The Fisher equation with degree two
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 the stochastic model for the spatial spread of a favored gene in a population in 1936.
Three nonlinear pseudoparabolic physical models
The onedimensional Oskolkov equation, the BenjaminBonaMahonyPeregrineBurgers equation and the OskolkovBenjaminBonaMahonyBurgers equation are the special cases of our Eq. (2).
where $\lambda \ne 0$, $\alpha \in \mathbb{R}$.
where α is a positive constant, θ and β are nonzero real numbers.
where α is a positive constant, θ is a nonzero real number.
The KdVBurgers equation
where α is a constant.
3 Fisher type equations with degree three
where A, B, C and D are arbitrary constants.
Theorem 3.1 [21]
 (I)The elliptic function solutions$\begin{array}{rcl}{w}_{3d}(z)& =& \pm \frac{1}{2}\sqrt{\frac{2A}{C}}\\ \times \frac{(\mathrm{\wp}+c)(4\mathrm{\wp}{c}^{2}+4{\mathrm{\wp}}^{2}c+2{\mathrm{\wp}}^{\prime}d\mathrm{\wp}{g}_{2}c{g}_{2})}{((12{c}^{2}{g}_{2})\mathrm{\wp}+4{c}^{3}3c{g}_{2}){\mathrm{\wp}}^{\prime}+(4{\mathrm{\wp}}^{3}+12c{\mathrm{\wp}}^{2}3{g}_{2}\mathrm{\wp}c{g}_{2})d}.\end{array}$
Here, ${g}_{3}=0$, ${d}^{2}=4{c}^{3}{g}_{2}c$, ${g}_{2}$ and c are arbitrary.
 (II)The simply periodic solutions${w}_{3s,1}(z)=\alpha \sqrt{\frac{A}{2C}}(coth\frac{\alpha}{2}(z{z}_{0})coth\frac{\alpha}{2}(z{z}_{0}{z}_{1})coth\frac{\alpha}{2}{z}_{1})$and${w}_{3s,2}(z)=\alpha \sqrt{\frac{A}{2C}}tanh\frac{\alpha}{2}(z{z}_{0}),$
where ${z}_{0}\in \mathbb{C}$, $B=A{\alpha}^{2}(\frac{1}{2}+\frac{3}{2{sinh}^{2}\frac{\alpha}{2}{z}_{1}})$, $D=\sqrt{\frac{A}{2C}}\frac{tanh\frac{\alpha}{2}{z}_{1}}{{sinh}^{2}\frac{\alpha}{2}{z}_{1}}$, ${z}_{1}\ne 0$ in the former formula, or $B=\frac{A{\alpha}^{2}}{2}$, $D=0$.
 (III)The rational function solutions${w}_{3r,1}(z)=\pm \sqrt{\frac{2A}{C}}\frac{1}{z{z}_{0}}$and${w}_{3r,2}(z)=\pm \sqrt{\frac{2A}{C{z}_{1}^{2}}}(\frac{{z}_{1}}{z{z}_{0}}\frac{{z}_{1}}{z{z}_{0}{z}_{1}}1),$
where ${z}_{0}\in \mathbb{C}$. $B=0$, $D=0$ in the former case, or given ${z}_{1}\ne 0$, $B=\frac{6A}{{z}_{1}^{2}}$, $D=\mp 2C{(\frac{2A}{C{z}_{1}^{2}})}^{3/2}$.
where A, B, C and D are arbitrary constants. They obtained the following result and gave its two applications.
 (I)All elliptic function solutions$\begin{array}{rcl}{w}_{4d}(z)& =& \frac{C}{3}\pm \sqrt{\frac{A}{2}}\\ \times \frac{(\mathrm{\wp}+E)(4\mathrm{\wp}{E}^{2}+4{\mathrm{\wp}}^{2}E+2{\mathrm{\wp}}^{\prime}F\mathrm{\wp}{g}_{2}E{g}_{2})}{((12{E}^{2}{g}_{2})\mathrm{\wp}+4{E}^{3}3E{g}_{2}){\mathrm{\wp}}^{\prime}+4F{\mathrm{\wp}}^{3}+12FE{\mathrm{\wp}}^{2}3F{g}_{2}\mathrm{\wp}FE{g}_{2}},\end{array}$
where $A({C}^{2}9B)=12C\sqrt{\frac{A}{2}}$, $27D={C}^{3}$, ${g}_{3}=0$, ${F}^{2}=4{E}^{3}{g}_{2}E$, ${g}_{2}$ and E are arbitrary constants.
 (II)All simply periodic solutions${w}_{4s,1}(z)=\pm \sqrt{\frac{A}{2}}\alpha coth\frac{\alpha}{2}(z{z}_{0})\frac{C}{3}$and$\begin{array}{rcl}{w}_{4s,2}(z)& =& \pm \sqrt{\frac{A}{2}}\alpha (coth\frac{\alpha}{2}(z{z}_{0})coth\frac{\alpha}{2}(z{z}_{0}{z}_{1}))\\ \frac{C}{3}\mp \sqrt{\frac{A}{2}}\alpha coth\frac{\alpha}{2}{z}_{1},\end{array}$where ${z}_{0}\in \mathbb{C}$. $A(2{C}^{2}+9A{\alpha}^{2}18B)=24C\sqrt{\frac{A}{2}}$, $27D{C}^{3}=27{\alpha}^{2}\sqrt{\frac{A}{2}}$ in the former case, or ${z}_{1}\ne 0$, $8C\sqrt{\frac{A}{2}}+6AB=3{A}^{2}{\alpha}^{2}(\frac{3}{{sinh}^{2}\frac{\alpha}{2}{z}_{1}}+1)$,$\begin{array}{rcl}162D\sqrt{\frac{A}{2}}& =& (2C\sqrt{\frac{A}{2}}\mp 3A\alpha coth\frac{\alpha}{2}{z}_{1})\\ \times (\frac{108A{\alpha}^{2}}{{sinh}^{2}\frac{\alpha}{2}{z}_{1}}+3{C}^{2}\mp 9C\alpha \sqrt{\frac{A}{2}}coth\frac{\alpha}{2}{z}_{1}).\end{array}$
 (III)All rational function solutions${w}_{4r,1}(z)=\pm \frac{2\sqrt{\frac{A}{2}}}{z{z}_{0}}\frac{C}{3}$and${w}_{4r,2}(z)=\pm \frac{2\sqrt{\frac{A}{2}}}{z{z}_{0}}\mp \frac{2\sqrt{\frac{A}{2}}}{z{z}_{0}{z}_{1}}\mp \frac{2\sqrt{\frac{A}{2}}}{{z}_{1}}\frac{C}{3},$
where ${z}_{0}\in \mathbb{C}$. $A({C}^{2}9B)=12C\sqrt{\frac{A}{2}}$, $27D={C}^{3}$ in the former case, or $A(\frac{54A}{{z}_{1}^{2}}+{C}^{2}9B)=12C\sqrt{\frac{A}{2}}$, $\frac{4{A}^{2}}{{z}_{1}^{3}}=(\frac{{C}^{3}}{27}+\frac{2C}{{z}_{1}^{2}}D)\sqrt{\frac{A}{2}}$, ${z}_{1}\ne 0$.
where A, B, C, D are arbitrary constants.
They got the following theorem.
 (I)[21]When $B=0$, the elliptic general solutions of Eq. (5)${w}_{5d,1}(z)=\pm \sqrt{\frac{2A}{D}}\frac{{\mathrm{\wp}}^{\prime}(z{z}_{0};{g}_{2},0)}{\mathrm{\wp}(z{z}_{0};{g}_{2},0)},$where ${z}_{0}$ and ${g}_{2}$ are arbitrary. In particular, it degenerates to the simply periodic solutions and rational solutions${w}_{5s,1}(z)=\alpha \sqrt{\frac{A}{2D}}tanh\frac{\alpha}{2}(z{z}_{0})$and${w}_{5r}(z)=\pm \sqrt{\frac{2A}{D}}\frac{1}{z{z}_{0}},$
where $C=\frac{A{\alpha}^{2}}{2}$ and ${z}_{0}\in \mathbb{C}$.
 (II)When $B=\pm \frac{3}{\sqrt{2}}\sqrt{AC}$, the general solutions of Eq. (5)${w}_{5g,1}(z)=\pm \frac{1}{2}exp\{\mp \frac{1}{\sqrt{2}}\sqrt{\frac{C}{A}}z\}\frac{{\mathrm{\wp}}^{\prime}(\sqrt{\frac{D}{C}}exp\{\mp \frac{1}{\sqrt{2}}\sqrt{\frac{C}{A}}z\}{s}_{0};{g}_{2},0)}{\mathrm{\wp}(\sqrt{\frac{D}{C}}exp\{\mp \frac{1}{\sqrt{2}}\sqrt{\frac{C}{A}}z\}{s}_{0};{g}_{2},0)},$where $\mathrm{\wp}(s:{g}_{2},0)$ is the Weierstrass elliptic function, both ${s}_{0}$ and ${g}_{2}$ are arbitrary constants. In particular, ${w}_{5g,1}(z)$ degenerates to the one parameter family of solutions${w}_{5f,1}(z)=\pm \sqrt{\frac{C}{D}}\frac{1}{1exp\{\mp \frac{1}{\sqrt{2}}\sqrt{\frac{C}{A}}(z{z}_{0})\}},$
where ${z}_{0}\in \mathbb{C}$.
All exact solutions of Eq. (NewellWhitehead), the nonlinear Schrödinger Eq. (NLS) and Eq. (Fisher 3) can be converted to Eq. (5) making use of the traveling wave reduction.
The NewellWhitehead equation
where r, s are constants.
The NLS equation
where α, β are nonzero constants.
The Fisher equation with degree three
4 The complex method and some problems
In order to state our complex method, we need some notations and results.
where ${a}_{r}$ are constants, and I is a finite index set. The total degree is defined by $degP(w,{w}^{\prime},\dots ,{w}^{(m)}):={max}_{r\in I}\{p(r)\}$.
where $b\ne 0$, c are constants, $n\in \mathbb{N}$.
In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic functions [24].
where ${g}_{2}=60{s}_{4}$, ${g}_{3}=140{s}_{6}$, and $\mathrm{\Delta}({g}_{2},{g}_{3})\ne 0$.
 (i)Degeneracy to simply periodic functions (i.e., rational functions of one exponential ${e}^{kz}$) according to$\mathrm{\wp}(z,3{d}^{2},{d}^{3})=2d\frac{3d}{2}{coth}^{2}\sqrt{\frac{3d}{2}}z$(9)
if one root ${e}_{j}$ is double ($\mathrm{\Delta}({g}_{2},{g}_{3})=0$).
 (ii)Degeneracy to rational functions of z according to$\mathrm{\wp}(z,0,0)=\frac{1}{{z}^{2}}$
if one root ${e}_{j}$ is triple (${g}_{2}={g}_{3}=0$).
 (iii)We have the addition formula$\mathrm{\wp}(z{z}_{0})=\mathrm{\wp}(z)\mathrm{\wp}({z}_{0})+\frac{1}{4}{\left[\frac{{\mathrm{\wp}}^{\prime}(z)+{\mathrm{\wp}}^{\prime}({z}_{0})}{\mathrm{\wp}(z)\mathrm{\wp}({z}_{0})}\right]}^{2}.$(10)

Step 1. Substituting the transform $T:u(x,t)?w(z)$, $(x,t)?z$ into a given PDE gives nonlinear ordinary differential equations (6).

Step 2. Substitute Eq. (7) into Eq. (6) to determine that the weak $p,q>$ condition holds, and pass the Painlevé test for Eq. (6).

Step 3. Find the meromorphic solutions $w(z)$ of Eq. (6) with a pole at $z=0$, which have $m1$ integral constants.

Step 4. By the addition formula of Theorem 4.1 we obtain all meromorphic solutions $w(z{z}_{0})$.

Step 5. Substituting the inverse transform ${T}^{1}$ into these meromorphic solutions $w(z{z}_{0})$, we get all exact solutions $u(x,t)$ of the original given PDE.
For the Laurent expansion (7) to be valid, B satisfies this equation and ${c}_{4}$ is an arbitrary constant. Therefore, $B=0$ or $B=\pm \frac{5\sqrt{AC}}{\sqrt{6}}$ or $B=\pm \frac{5i\sqrt{AC}}{\sqrt{6}}$, where ${i}^{2}=1$. For other B it would be necessary to add logarithmic terms to the expansion, thus giving a branch point rather than a pole.
Here, $12{A}^{2}{g}_{2}={C}^{2}$, ${M}^{2}=4{N}^{3}{g}_{2}N{g}_{3}$, ${g}_{3}$ and N are arbitrary.
where ${A}^{2}{\alpha}^{4}={C}^{2}$, ${z}_{0}\in \mathbb{C}$.
where $C=0$, ${z}_{0}\in \mathbb{C}$.
For $B=\pm \frac{5\sqrt{AC}}{6},\pm \frac{5i\sqrt{AC}}{6}$, we transform Eq. (2) into the autonomous part of the first Painlevé equation. In this way we find the general solutions.
where $\alpha =\mp \frac{2}{\sqrt{6}}\sqrt{\frac{C}{A}}$, ${\beta}^{2}=\frac{D}{C}$.
The general solutions of Eq. (13) are the Weierstrass elliptic functions $u(s)=\mathrm{\wp}(s{s}_{0};0,{g}_{3})$, where ${s}_{0}$ and ${g}_{3}$ are two arbitrary constants.
where ${z}_{0}\in \mathbb{C}$.
where $\alpha =\mp \frac{2i}{\sqrt{6}}\sqrt{\frac{C}{A}}$, ${\beta}^{2}=\frac{D}{C}$. The general solutions of Eq. (14) are the Weierstrass elliptic functions $u(s)=\mathrm{\wp}(s{s}_{0};0,{g}_{3})$, where ${s}_{0}$ and ${g}_{3}$ are two arbitrary constants.
where ${z}_{0}\in \mathbb{C}$. □
Obviously, Eqs. (14) and (15) are also special cases of Eqs. (1) and (3), respectively. We also know that there are six classes of Painlevé equations. Therefore, we ask naturally whether or not there exist other four classes autonomous parts of Painlevé equations could be transformed by $w(z)=f(z)u(s)$, $s=g(z)$ from the related equations; i.e, we propose the following open questions.
Question 4.1 Find all meromorphic solutions of the other four classes autonomous parts of Painlevé equations:
(AP_{3}) ${u}^{\u2033}=\frac{{({u}^{\prime})}^{2}}{u}+\gamma {u}^{3}+\frac{\delta}{u}$;
(AP_{4}) ${u}^{\u2033}=\frac{{({u}^{\prime})}^{2}}{2u}+\frac{3}{2}{u}^{3}2\alpha u+\frac{\beta}{u}$;
(AP_{5}) ${u}^{\u2033}=(\frac{1}{2u}+\frac{1}{u1}){({u}^{\prime})}^{2}+\frac{\delta u(u+1)}{u1}$;
(AP_{6}) ${u}^{\u2033}=\frac{1}{2}(\frac{1}{u}+\frac{1}{u1}){({u}^{\prime})}^{2}$;
where α, β, γ and δ are arbitrary constants.
Question 4.2 Determine the related equations and find their meromorphic general solutions for each of the above equations (AP_{ i }), $i=3,4,5,6$.
Declarations
Acknowledgements
This work was supported by the NANUM 2014 Grant to the SEOUL ICM 2014 and the Visiting Scholar Program of the Department of Mathematics and Statistics at Curtin University of Technology when the first author worked as a visiting scholar (200001807894). The first author would like to thank his School, University and Guangzhou Education Bureau for supplying him financial supports such that he has organized the International Workshop of Complex Analysis and its Applications at Guangzhou University successfully. The first author would also like to thank Professor Junesang Choi for inviting him to visit Dongguk University in Republic of Korea and for supplying him some useful information and partial financial aid. This work was completed with the support with the NSF of China (No. 11271090), Tianyuan Youth Fund of the NSF of China (No. 11326083) and NSF of Guangdong Province (S2012010010121), Shanghai University Young Teacher Training Program (ZZSDJ12020), Innovation Program of Shanghai Municipal Education Commission (14YZ164) and project (13XKJC01) from the Leading Academic Discipline Project of Shanghai Dianji University. The authors wish to thank the editor and referees for their very helpful comments and useful suggestions.
Authors’ Affiliations
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