ALL MEROMORPHIC SOLUTIONS OF AN AUXILIARY ORDINARY DIFFERENTIAL EQUATION AND ITS APPLICATIONS∗

Wenjun YUAN（袁文俊）

School of Mathematics and Information Science，Guangzhou University，Guangzhou 510006，China Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes，Guangzhou University，Guangzhou 510006，China

Weiling XIONG（熊维玲）

Department of Information and Computing Science，Guangxi University of Technology，Liuzhou 545006，China

Jianming LIN（林剑鸣）†

School of Economic and Management，Guangzhou University of Chinese Medicine，Guangzhou 510006，China

Yonghong WU（吴永洪）

Department of Mathematics and Statistics，Curtin University of Technology，GPO Box U 1987，Perth WA 6845，Australia

ALL MEROMORPHIC SOLUTIONS OF AN AUXILIARY ORDINARY DIFFERENTIAL EQUATION AND ITS APPLICATIONS∗

Wenjun YUAN（袁文俊）

School of Mathematics and Information Science，Guangzhou University，Guangzhou 510006，China Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes，Guangzhou University，Guangzhou 510006，China

E-mail：gzywj@tom.com

Weiling XIONG（熊维玲）

Department of Information and Computing Science，Guangxi University of Technology，Liuzhou 545006，China

E-mail：xiongwl@163.com

Jianming LIN（林剑鸣）†

School of Economic and Management，Guangzhou University of Chinese Medicine，Guangzhou 510006，China

E-mail：ljmguanli@21cn.com

Yonghong WU（吴永洪）

Department of Mathematics and Statistics，Curtin University of Technology，GPO Box U 1987，Perth WA 6845，Australia

E-mail：y.wu@curtin.edu.au

In this paper，we first employ the complex method to deritive all meromorphic solutions of an auxiliary ordinary differential equation，and then find all meromorphic exact solutions of the modified ZK equation，modified KdV equation，nonlinear Klein-Gordon equation and modified BBM equation.Our work shows that there exist some classes of rational solutions wr，2（z）and simple periodic solutions ws，1（z）which are new and are not degenerated successively to by the elliptic function solutions.

differential equation；exact solution；meromorphic function；elliptic function

2010 MR Subject Classification 30D35；34A05

1 Introduction and Main Result

In this paper，we employ the complex method to obtain first all meromorphic solutions of the auxiliary ordinary differential equations［AODEq.（1）］below

where A，B，C and D are arbitrary constants.Then，we discuss the applications of the solutions for finding meromorphic exact solutions of varies classes of partial differential equations including the modified ZK equation，modified KdV equation，nonlinear Klein-Gordon equation and modified BBM equation.

In order to state our main result，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，ciand cijare 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 eαz，α∈C，or a rational function of z.

Our main result is summarized by the following theorem.

Theorem 1 Suppose that AC/=0，then all meromorphic solutions w of an AODEq.（1）belong to the class W.Furthermore，AODEq.（1）has the following three forms of solutions:（I）The elliptic function solutions

here g3=0，d2=4c3-g2c，g2and c are arbitrary.

（II）The simply periodic solutions

and

（III）The rational function solutions

and

This paper is organized as follows:In the next section，the preliminary lemmas and the Complex Method are given.The proof of Theorem 1 will be given in Section 3.All exact solutions of the auxiliary AODEq.（1）are derived by complex method.In Section 4，we obtain all exact solutions of modified ZK equation，modified KdV equation，nonlinear Klein-Gordon equation and modified BBM equation，which can be converted to the AODEq.（1）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，we need some lemmas and results.

Lemma 1［2，3］Let k∈N，then any meromorphic solution w with at least one pole of a k-th order Briot-Bouquet equation

belongs to W，where Pi（w）are polynomials in w with constant coefficients.

Set m∈N:=｛1，2，3，···｝，j=0，1，···，m，rj∈N0=N∪｛0｝，r=（r0，r1，···，rj，···，rm）. A differential monomial is defined by

p（r）:=r0+r1+···+rmis called the degree of Mr［w］.A differential polynomial is defined by

where arare constants，and I is a finite index set.

The total degree of P（w，w′，···，w（m））is defined by degP（w，w′，···，w（m））:=max

r∈I｛p（r）｝.

We will consider the following complex ordinary differential equations

where b/=0，c are constants，n∈N.

Let p，q∈N.Suppose that equation（5）has a meromorphic solution w with at least one pole，we say that equation（5）satisfies the weakcondition if substituting Laurent series

into equation（5）we can determine p distinct Laurent principle part

with pole of multiplicity q at z=0.

Lemma 2［6，7，16］Let p，l，m，n∈N，degP（w，w（m））＜n，and an m-th order Briot-Bouquet

satisfy the weakcondition.Then all meromorphic solutions w belong to the class W.If for some values of parameters such solution w exists，then other meromorphic solutions form a one-parametric family w（z-z0），z0∈C.Furthermore，each elliptic solution with pole at z=0 can be written as

Each rational function solution w:=R（z）has the form of

with l（≤p）distinct poles of multiplicity q.

Each simply periodic solution is a rational function R（ξ）of ξ=eαz（α∈C）.R（ξ）has l（≤p）distinct poles of multiplicity q，and has the form of

To give the representations of elliptic solutions，we need some notations and results concerning elliptic function［7］.

Let ω1，ω2be two given complex numbers such that Imω1

ω2＞0，L=L［2ω1，2ω2］be discrete set L［2ω1，2ω2］=｛ω|ω=2nω1+2mω2，n，m∈Z｝，which is isomorphic to Z×Z. The discriminant∆=∆（c1，c2）:=c31-27c22and

The Weierstrass elliptic function℘（z）:=℘（z，g2，g3）is a meromorphic function with periods 2ω1，2ω2and satisfying the equation

where g2=60s4，g3=140s6and∆（g2，g3）/=0.

we have e1=℘（ω1），e2=℘（ω2），e3=℘（ω1+ω2）.

Inversely，given two complex numbers g2and g3such that∆（g2，g3）/=0，then there exists a Weierstrass elliptic function℘（z）with double periods 2ω1，2ω2such that above relations hold.

It is easy to see that the set of poles of the Weierstrass elliptic function℘（z）is L，℘（z）has 4 distinct complete multiple values e1，e2，e3and infinite，and thus any other value must be simple.

Lemma 3［1，7］The Weierstrass elliptic functions℘（z）:=℘（z，g2，g3）have two successive degeneracies and addition formula:

（I）To simply periodic functions（i.e.，rational functions of one exponential ekz）according to

if one root ejis double（∆（g2，g3）=0）.

（II）To rational functions of z according to

When ABC crank rotate at speed ω1=1 rad/s, the motion equation of point C on the upper horizon line track is in the following

if one root ejis triple（g2=g3=0）.

（III）Addition formula

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

Step 1 Substitute the transform T:u（x，t）→w（z）， （x，t）→z into a given PDE gives a nonlinear ordinary differential equation（5）or（7）.

Step 2 Substitute（6）into equation（5）or（7）to determine that weakcondition holds.

Step 3 By indeterminant relation（8），（9）and（10）we find the elliptic，rational and simply periodic solutions w（z）of equation（5）or（7）with pole at z=0，respectively.

Step 4 By Lemma 1 and Lemma 2 we obtain all meromorphic solutions w（z-z0）.

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

3 Proof of the Main Result

Hence，AODEq.（1）satisfies weak<2，1>condition and is an 2 order Briot-Bouquet differential equation.Obviously，AODEq.（1）satisfies the dominant condition.So，by Lemma 2，we know that all meromorphic solutions of AODEq.（1）belong to W.Now we will give the forms of all meromorphic solutions of AODEq.（1）.

By（9），we infer the indeterminant rational solutions of AODEq.（1）with pole at z=0 that

Substituting R21（z）into AODEq.（1），we get two classes，one is following

here B=0，D=0.The other is below

Thus all rational solutions of AODEq.（1）

and

where z0∈C，B=0，D=0 in the former formula，or given z1/=0，B=∓2C

In order to have simply periodic solutions，set ξ=exp（αz），put w=R（ξ）into AODEq.（1），then

Substituting

into eq.（15），we obtain that

and

Substituting ξ=eαzinto above relation，and then we get simply periodic solutions of AODEq.（1）with pole at z=0

and

So all simply periodic solutions of AODEq.（1）are obtained by

and

From（8）of Lemma 2，we have indeterminant relations of elliptic solutions of AODEq.（1）with pole at z=0

where F2=4E3-g2E-g3.Applying the conclusion II of Lemma 2 to wd0（z），and noting that the results of rational solutions obtained above，we deduce that c30=0，E=F=0，g3=0. Then we get that

here g3=0.Therefore，all elliptic solutions of AODEq.（1）

where z0∈C，g3=0.Making use of addition formula of Lemma 3，we rewrite it to the form

here g3=0，d2=4c3-g2c，g2and c are arbitrary.

The proof of Theorem 1 is completed.

4 Some Applications of Theorem 1

In this section，the modified ZK equation，modified KdV equation，nonlinear Klein-Gordon equation and modified BBM equation are considered again and the exact solutions are derived with the aid of AODEq.（1）.

4.1 Modified ZK Equation

Modified ZK equation（Zakharov and Kuznetsov［17］，Li et al.［9］，Hassan［5］，Wazwaz［13］，Zhao et al.［18］，Peng［12］and Yomba［15］）is expressed as

where β is a constant.

Substituting

into eq.（mZK），and integrating it yields

Eq.（17）is converted to AODEq.（1），where

By Theorem 1，therefore，all meromorphic solutions w of eq.（mZK）belong to the class W. Furthermore，eq.（mZK）has the following three forms of solutions:

（I）The elliptic general solutions

here g3=0，B2=4A3-g2A，g2and A are arbitrary.

（II）The simply periodic solutions

and

（III）The rational function solutions

and

4.2 Modified KdV Equation

Modified KdV equation（Wu et al.［14］，Li and Zhang［10］，Mei and Zhang［11］）has the form

where τ，β are constant.

Substituting

into eq.（mKdV），and integrating it yields

Eq.（19）is converted to AODEq.（1），where

By Theorem 1，therefore，all meromorphic solutions w of eq.（mKdV）belong to the class W.

4.3 Nonlinear Klein-Gordon Equation

Nonlinear Klein-Gordon equation（Wu et al.［14］，Han［4］）is of the form

where c，τ，β are constants.

Substituting

into eq.（B）gives

Eq.（21）is converted to AODEq.（1），where

4.4 Modified BBM Equation

Modified BBM equation（L¨u［8］）is considered as

where β is constant.Substituting

into eq.（mBBM），and integrating it deduces

Eq.（23）is converted to AODEq.（1），where

Remark 1 By Theorem 1，therefore，all meromorphic solutions w of eq.（mZK），eq.（mKdV），eq.（KG）and eq.（mBBM）belong to the class W.Similar to Section 4.1，by making using of（2），（3）and（4）of Theorem 1，we can obtain all exact solutions of eqs.（19），（21）and（23）.Here we omit the detail with them for simplicity.

Remark 2 There are some other partial differential equations which can be converted to AODEq.（1）with the aid of the traveling wave reduction.Here we omit the detail with them for simplicity.

5 Conclusions

Complex method is a very important tool in finding the exact solutions of nonlinear evolution equations，and AODEq.（1）is one of most important auxiliary equations because many nonlinear evolution equations can be converted to it.In this article，we employ complex method to derive all exact solutions of the auxiliary AODEq.（1）.All exact solutions of the modified ZK equation，modified KdV equation，nonlinear Klein-Gordon equation and modified BBM equation are derived with the aid of the auxiliary AODEq.（1）.The idea introduced in this paper can be applied to other nonlinear evolution equations.Our work shows that there exist some classes of rational solutions wr，2（z）and simple periodic solutions ws，1（z）which are new and are not degenerated successively to by the elliptic function solutions.

Acknowledgements This work was supported by the Visiting Scholar Program of Department of Mathematics and Statistics at Curtin University of Technology（200001807894）when the first author worked as visiting scholars.The authors finally wish to thank Professor Robert Conte for supplying his useful reprints and suggestions.

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∗Received December 2，2012；revised March 2，2015.The first author is supported by the NSFC（11271090）and NSF of Guangdong（S2012010010121）.

†Corresponding author:Jianming LIN.