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The general solutions of an auxiliary ordinary differential equation using complex method and its applications
Advances in Difference Equations volume 2014, Article number: 147 (2014)
In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first, and then find all meromorphic general solutions of in combination the Newell-Whitehead equation, the NLS equation, and the Fisher equation with degree three. Our result shows that all rational and simply periodic exact solutions of the combined the Newell-Whitehead equation, NLS equation, and Fisher equation with degree three are solitary wave solutions, and the method is simpler than other methods.
1 Introduction and main results
Nonlinear partial differential equations (NLPDEs) are widely used as models to describe many important dynamical systems in various fields of sciences, 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 , the Darboux transformation method , the Hirota bilinear method , the Lie group method , the bifurcation method of dynamic systems [5–7], the sine-cosine method , the tanh-function method [9, 10], the Fan-expansion method , and the homogeneous balance method . Practically, there is no unified technique that can be employed to handle all types of nonlinear differential equations. Recently, the complex method was introduced by Yuan et al. [13, 14]. It is shown that the complex method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.
Recently, Yuan et al.  derived all traveling wave exact solutions by using the complex method for a type of ordinary differential equations (ODEs):
where A, B, C, and D are arbitrary constants.
In order to state this result, we need some concepts and notations.
A meromorphic function means that is holomorphic in the complex plane ℂ except for poles. α, b, c, and are constants, which may be different from each other in different places. We say that a meromorphic function f belongs to the class W if f is an elliptic function, or a rational function of , , or a rational function of z.
Theorem 1.1 
Suppose that , then all meromorphic solutions w of an Eq. (1) belong to the class W. Furthermore, Eq. (1) has the following three forms of solutions:
The elliptic function solutions
Here , , , , , and c are arbitrary.
The simply periodic solutions
where , , , in the former formula, or , .
The rational function solutions
where , , in the former case, or given by , , .
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 modified ZK equation, the modified KdV equation, the nonlinear Klein-Gordon equation, and the modified BBM equation can be converted to Eq. (1) .
In this paper, we employ the complex method to obtain first all meromorphic solutions of Eq. (2) below,
where A, B, C, D are arbitrary constants.
Our main result is the following theorem.
Theorem 1.2 Suppose that , then Eq. (2) is integrable if and only if . Furthermore, the general solutions of Eq. (2) are of the following form.
When , we have the elliptic general solutions of Eq. (2),
where and are arbitrary. In particular, it degenerates to the simply periodic solutions and rational solutions,
where and .
When , we have the general solutions of Eq. (2),
where is the Weierstrass elliptic function, and both and are arbitrary constants. In particular, degenerates to the one parameter family of solutions,
This paper is organized as follows: In the next section, the preliminary lemmas and the complex method are given. The proof of Theorem 1.2 will be given in Section 3. All exact solutions of the auxiliary Eq. (2) are derived by complex method. In Section 4, we obtain all exact solutions of the Newell-Whitehead equation, the nonlinear Scrödinger equation (NLS), and the Fisher equation, which can be converted to Eq. (2) making use of the traveling wave reduction. 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 1.1, we need some notations and results.
Set , , , . We define a differential monomial denoted by
is called the degree of . A differential polynomial is defined by
where are constants, and I is a finite index set. The total degree is defined by .
We will consider the following complex ordinary differential equations:
where , c are constants, .
into Eq. (3), we can determine p distinct Laurent singular parts as below,
In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic functions .
Let , be two given complex numbers such that , be discrete subset , which is isomorphic to . The discriminant is and we have
The Weierstrass elliptic function is a meromorphic function with double periods , , satisfying the equation
where , , and .
The Weierstrass elliptic functions have two successive degeneracies and we have the addition formula:
Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) according to(6)
if one root is double ().
Degeneracy to rational functions of z according to
if one root is triple ().
We have the addition formula(7)
In the proof of our main result, the following lemmas are very useful, which can be deduced by Theorem 1 in .
Lemma 2.2 
The differential equation
has elliptic solutions, a simply periodic solution, and a rational solution with pole at ,
respectively, where is the Weierstrass elliptic function with and arbitrary and .
By the above lemmas and results, we can give a new method below, let us call it the complex method, to find exact solutions of some PDEs.
Step 1. Substituting the transform , into a given PDE gives a nonlinear ordinary differential equation (3).
Step 3. Find the meromorphic solutions of Eq. (3) with a pole at , which have integral constants.
Step 4. By the addition formula of Lemma 2.1 we obtain the general meromorphic solutions .
Step 5. Substituting the inverse transform into these meromorphic solutions , we get all exact solutions of the original given PDE.
3 Proof of Theorem 1.2
Proof Substituting (4) into Eq. (2) we have , , , , , , and
For the Laurent expansion (4) to be valid B satisfies this equation and is an arbitrary constant. Therefore, . For other B it would be necessary to add logarithmic terms to the expansion, thus giving a branch point rather than a pole.
For , Eq. (2) is completely integrable by standard techniques and the solutions are expressible in terms of elliptic functions (cf. ), i.e., by Lemmas 2.1 and 2.2, the elliptic general solutions of Eq. (2)
where and are arbitrary. In particular, it degenerates to the simply periodic solutions and rational solutions,
where and .
For , we transform Eq. (2) into the second Painlevé type equation. In this way we find the general solutions.
Setting , , and substituting in Eq. (2), we find that the equation for is
If we take f and g such that
then Eq. (8) for u is integrable. By (9), one takes and
where , .
Thus Eq. (8) reduces to
Both Lemmas 2.1 and 2.2 show that the general solutions of Eq. (10) are of the form
where is the Weierstrass elliptic function, and are two arbitrary constants.
Therefore, when , by Lemma 2.1, we have the general solutions of Eq. (2),
where both and are arbitrary constants. In particular, by Lemma 2.1 and , degenerates to the one parameter family of solutions,
This completes the proof of Theorem 1.2. □
4 Some applications of Theorem 1.2
Equation (2) include many well-known nonlinear equations that are with applied background as special examples, such as Newell-Whitehead equation, NLS equation, Fisher equation with degree three. In this section, the Newell-Whitehead equation, NLS equation, and Fisher equation with degree three are considered again and the exact solutions are derived with the aid of Eq. (2).
4.1 Newell-Whitehead equation
where r, s are constants.
into Eq. (A) gives
4.2 NLS equation
where α, β are nonzero constants.
into Eq. (B) gives
4.3 Fisher equation with degree three
The Fisher equation with degree three  has the form
into Eq. (C) gives
Apparently, if we set appropriate coefficients in Eq. (2), certain well-known equations will be converted to it.
The complex method is a very important tool in finding the exact solutions of nonlinear evolution equation, and Eq. (3) is one of the most important auxiliary equations, because many nonlinear evolution equations can be converted to it. In this article, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first, and then find all meromorphic exact solutions of the combined Newell-Whitehead equation, nonlinear Scrödinger equation, and Fisher equation with degree three. Our result shows that all rational and simply periodic exact solutions of the combined the Newell-Whitehead equation, nonlinear Scrödinger equation, and Fisher equation with degree three are solitary wave solutions, and the method is simpler than other methods.
Ablowitz MJ, Clarkson PA London Mathematical Society Lecture Note Series 149. In Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge; 1991.
Matveev VB, Salle MA Springer Series in Nonlinear Dynamics. In Darboux Transformations and Solitons. Springer, Berlin; 1991.
Hirota R, Satsuma J: Soliton solutions of a coupled KdV equation. Phys. Lett. A 1981, 85(8–9):407–408. 10.1016/0375-9601(81)90423-0
Olver PJ Graduate Texts in Mathematics 107. In Applications of Lie Groups to Differential Equations. 2nd edition. Springer, New York; 1993.
Li JB, Liu Z: Travelling wave solutions for a class of nonlinear dispersive equations. Chin. Ann. Math., Ser. B 2002, 3(3):397–418.
Tang S, Huang W: Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation. Appl. Math. Comput. 2007, 189(2):1774–1781. 10.1016/j.amc.2006.12.082
Feng D, He T, Lü J:Bifurcations of travelling wave solutions for -dimensional Boussinesq type equation. Appl. Math. Comput. 2007, 185(1):402–414. 10.1016/j.amc.2006.07.039
Tang S, Xiao Y, Wang Z: Travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa-Holm equation. Appl. Math. Comput. 2009, 210(1):39–47. 10.1016/j.amc.2008.10.041
Tang S, Zheng J, Huang W: Travelling wave solutions for a class of generalized KdV equation. Appl. Math. Comput. 2009, 215(7):2768–2774. 10.1016/j.amc.2009.09.019
Malfliet W, Hereman W: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 1996, 54(6):563–568. 10.1088/0031-8949/54/6/003
Fan E: Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos Solitons Fractals 2003, 16(5):819–839. 10.1016/S0960-0779(02)00472-1
Wang ML: Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A 1995, 199: 169–172. 10.1016/0375-9601(95)00092-H
Yuan WJ, Huang Y, Shang YD: All travelling wave exact solutions of two nonlinear physical models. Appl. Math. Comput. 2013, 219(11):6212–6223. 10.1016/j.amc.2012.12.023
Yuan WJ, Shang YD, Huang Y, Wang H: The representation of meromorphic solutions of certain ordinary differential equations and its applications. Sci. Sin., Math. 2013, 43(6):563–575. 10.1360/012012-159
Yuan, WJ, Xiong, WL, Lin, JM, Wu, YH: All meromorphic solutions of an auxiliary ordinary differential equation using complex method. Acta Math. Sci. (2012, to appear)
Lang S: Elliptic Functions. 2nd edition. Springer, New York; 1987.
Conte R, Musette M: Elliptic general analytic solutions. Stud. Appl. Math. 2009, 123(1):63–81. 10.1111/j.1467-9590.2009.00447.x
Vitanov NK: On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: the role of the simplest equation. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 4215–4231. 10.1016/j.cnsns.2011.03.035
Liu CS: Canonical-like transformation method and exact solutions to a class of diffusion equations. Chaos Solitons Fractals 2009, 42: 441–446. 10.1016/j.chaos.2009.01.006
Newell AC, Whitehead JA: Stability of stationary periodic structures for weakly supercritical convection and related problems. J. Fluid Mech. 1969, 38: 279. 10.1017/S0022112069000176
Wazwaz AM: The tanh-coth method for solitons and kink solutions for nonlinear parabolic equation. Appl. Math. Comput. 2007, 188: 1467–1475. 10.1016/j.amc.2006.11.013
Chen HT, Yin HC: A note on the elliptic equation method. Commun. Nonlinear Sci. Numer. Simul. 2008, 13: 547–553. 10.1016/j.cnsns.2006.06.007
Gong LX: Some new exact solutions of the Jacobi elliptic functions of NLS equation. Chin. J. Phys. 2006, 55(9):4414–4419.
Aslan Í: Travelling wave solutions to nonlinear physical models by means of the first integral method. Pramana 2011, 76(4):533–542. 10.1007/s12043-011-0062-y
This work was supported by the Visiting Scholar Program of Department of Mathematics and Statistics at Curtin University of Technology when the first author worked as a visiting scholar (200001807894) also this work was completed with the support with the NSF of China (11271090) and NSF of Guangdong Province (S2012010010121). The authors wish also specially to thank the managing editor and referees for their very helpful comments and useful suggestions. The authors finally wish to thank Professor Robert Conte for supplying his useful reprints and suggestions.
The authors declare that they have no competing interests.
The main idea of this paper was proposed by WY and ZH. MF and JL prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Yuan, W., Huang, Z., Fu, M. et al. The general solutions of an auxiliary ordinary differential equation using complex method and its applications. Adv Differ Equ 2014, 147 (2014). https://doi.org/10.1186/1687-1847-2014-147
- differential equation
- general solution
- meromorphic function
- elliptic function