JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2017/6521357 6521357 Research Article Exact Solutions of the Vakhnenko-Parkes Equation with Complex Method http://orcid.org/0000-0002-6651-1714 Gu Yongyi 1 http://orcid.org/0000-0003-4660-9228 Yuan Wenjun 1 Aminakbari Najva 1 Jiang Qinghua 1 Su Hua School of Mathematics and Information Science Guangzhou University Guangzhou 510006 China gzhu.edu.cn 2017 9112017 2017 11 05 2017 26 09 2017 9112017 2017 Copyright © 2017 Yongyi Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We derive exact solutions to the Vakhnenko-Parkes equation by means of the complex method, and then we illustrate our main results by some computer simulations. We can apply the idea of this study to related nonlinear differential equation.

National Natural Science Foundation of China 11271090 Natural Science Foundation of Guangdong Province 2016A030310257 Young Talents Innovation Project of Guangdong Province 2015KQNCX116 Joint PHD Program of Guangzhou University and Curtin University
1. Introduction and Main Results

Nonlinear differential equations are widely used as models to describe many important dynamical systems in various fields of science, especially in nonlinear optics, plasma physics, solid state physics, and fluid mechanics. It has aroused extensive interest in the study of nonlinear differential equations .

In 1992, Vakhnenko  first presented the nonlinear differential equation (1)xux+t+u=0and obtained solitary wave solutions to (1). The equation above gives a description of high-frequency waves in the relaxation medium .

In 1998, Vakhnenko and Parkes  found the soliton solution to the transformed form of (1) as follows: (2)uuxxt-uxuxt+u2ut=0.Hereafter (2) is called the Vakhnenko-Parkes equation (VPE). In recent years, many powerful methods for constructing the solutions of VPE have been used, for instance, the Hirota-Backlund transformations method , the inverse scattering method [20, 21], the exp-function method , and the (G/G)-expansion method . In this article, we would like to use the complex method  to obtain traveling wave solutions of VPE.

Substituting traveling wave transform (3)ux,t=wz,z=νx-δt, into Vakhnenko-Parkes (2), we get (4)ν2ww-ν2ww+w2w=0,Integrating (4) yields (5)3ν2ww-3ν2w2+w3-λ=0,where ν and λ are constants.

If a meromorphic function g is a rational function of z, or a rational function of eμz,μC, or an elliptic function, then we say that g belongs to the class W .

Theorem 1.

If ν0, then the meromorphic solutions w of (5) belong to the class W. In addition, (5) has the following classes of solutions.

(I) The Rational Function Solutions (6) W r z = - 6 ν 2 z - z 0 2 , where λ=0,z0C.

(II) The Simply Periodic Solutions (7) W s z = - 3 ν 2 μ 2 2 coth 2 μ z - z 0 2 + 3 μ 2 - 1 ν 2 2 , where λ=ν6/8, z0C.

(III) The Elliptic Function Solutions (8) W d z = - 6 ν 2 - z + 1 4 z + D z - C 2 + 6 ν 2 C , where D2=4C3-c3, c2=0, and c3=-λ/108ν6.

2. Preliminaries

At first, we give some notations and definitions, and then we introduce some lemmas.

Let mN{1,2,3,}, rjN=N{0}, r=(r0,r1,,rm),j=0,1,,m, and (9)Krwzwzr0wzr1wzr2wmzrm; then d(r)j=0mrj is the degree of Kr[w]. Let the differential polynomial be defined by (10)Fw,w,,wmrJarKrw, where J is a finite index set and ar are constants; then degF(w,w,,w(m))maxrJ{d(r)} is the degree of F(w,w,,w(m)).

Consider the following differential equation: (11)Fw,w,,wm=cwn+d,where c0,d are constants, nN.

Let p,qN, and assume that the meromorphic solutions w of (11) have at least one pole. If (11) has exactly p distinct meromorphic solutions, and their multiplicity of the pole at z=0 is q, then (11) is said to satisfy the p,q condition. It is not easy to verify that the p,q condition of (11) holds, so we need the weak p,q condition as follows.

By substituting the Laurent series (12)wz=τ=-qβτzτ,β-q0,q>0,into (11), we determine exactly p different Laurent singular parts: (13)τ=-q-1βτzτ; then (11) is said to satisfy the weak p,q condition.

Given two complex numbers l1, l2 such that Iml1/l2>0, and let L be the discrete subset L2l1,2l2{ll=2al1+2bl2,a,bZ}, and L is isomorphic to Z×Z. Let the discriminant Δ=Δ(b1,b2)b13-27b22 and (14)Hn=HnLlL\01ln.

A meromorphic function (z)(z,c2,c3) with double periods 2l1, 2l2, which satisfies the equation (15)z2=4z3-c2z-c3,where c2=60H4, c3=140H6, and Δ(c2,c3)0, is called the Weierstrass elliptic function.

In 2009, Eremenko et al.  studied the m-order Briot-Bouquet equation (BBEq) (16)Fw,wm=j=0nFjwwmj=0, where Fj(w) are constant coefficients polynomials, mN. For the m-order BBEq, we have the following lemma.

Lemma 2 (see [<xref ref-type="bibr" rid="B26">26</xref>, <xref ref-type="bibr" rid="B29">29</xref>, <xref ref-type="bibr" rid="B30">30</xref>]).

Let m,n,p,sN, degF(w,w(m))<n. If a m-order BBEq (17)Fw,wm=cwn+d satisfies the weak p,q condition, then the meromorphic solutions w belong to the class W. Suppose that for some values of parameters such solution w exists; then other meromorphic solutions form a one-parametric family (z-z0), z0C. Furthermore, each elliptic solution with pole at z=0 can be written as (18)wz=i=1s-1j=2q-1jβ-ijj-1!dj-2dzj-214z+Ciz-Di2-z+i=1s-1β-i12z+Ciz-Di+j=2q-1jβ-sjj-1!dj-2dzj-2z+β0,where β-ij are determined by (12), i=1sβ-i1=0, and Ci2=4Di3-c2Di-c3.

Each rational function solution has s(p) distinct poles of multiplicity q and is expressed as (19)Rz=i=1sj=1qβijz-zij+β0.

Each simply periodic solution has s(p) distinct poles of multiplicity q and is expressed as (20)Rη=i=1sj=1qβijη-ηij+β0,which is a rational function of η=eμz(μC).

Lemma 3 (see [<xref ref-type="bibr" rid="B27">27</xref>, <xref ref-type="bibr" rid="B30">30</xref>]).

Weierstrass elliptic functions (z)(z,c2,c3) have an addition formula as below: (21)z-z0=14z+z0z-z02-z0-z.When c2=c3=0, Weierstrass elliptic functions can be degenerated to rational functions according to (22)z,0,0=1z2.When Δ(c2,c3)=0, Weierstrass elliptic functions can be degenerated to simple periodic functions according to (23)z,3d2,-d3=2d-3d2coth23d2z.

3. Proof of Theorem <xref ref-type="statement" rid="thm1">1</xref>

Substituting (12) into (5) we obtain p=1,q=2,β-2=-6ν2,β-1=0, β0 is an arbitrary constant, β1=0,β2=-β02/10ν2, and β3=0.

Multiplying (5) by w/w3, we get (24)3ν2www2-3ν2w3w3+w-λww3=0.Integrating (24) yields (25)3ν2w2+2w3+2γw2-λ=0,where ν is an arbitrary constant and γ is the integrable constant.

Therefore, (25) is a first-order BBEq and satisfies the weak 1,2 condition. Hence, by Lemma 2, the meromorphic solutions of (25) wW. It means that the meromorphic solutions of (5) wW. The forms of the meromorphic solutions to (5) will be given in the following.

By (19), we infer that the indeterminate rational solutions of (5) are (26)R1z=β11z2+β12z+β10, with pole at z=0.

Substituting R1(z) into (5), we have (27)β1126ν2+β11z6+3β11β124ν2+β11z5+36β11β10ν2+β122ν2+β112β10+β11β122z4+β126β10ν2+6β11β10+β122z3+3β10β11β10+β122z2+3β12β102z+β103+λ=0; then we get β11=-6ν2,β12=β10=0.

Therefore, we can determine that (28)R1z=-6ν2z2, where λ=0.

So the rational solutions of (5) are (29)Wrz=-6ν2z-z02, where λ=0,z0C.

To obtain simply periodic solutions, let η=eμz, and substitute w=R(η) into (5); then (30)3ν2μ2RηR+η2R-3ν2μ2Rη2+R3+λ=0.Substituting (31)R2z=β21η-12+β22η-1+β20 into (30), we obtain that (32)R2z=-6ν2μ2η-12-6ν2μ2η-1-ν22,where λ=ν6/8.

Substituting η=eμz into (32), we can get simply periodic solutions to (5) with pole at z=0(33)Ws0z=-6ν2μ2eμz-12-6ν2μ2eμz-1-ν22=-6ν2μ2eμzeμz-12-ν22=-3ν2μ22coth2μz2+3μ2-1ν22, where λ=ν6/8.

So simply periodic solutions of (5) are (34)Wsz=-3ν2μ22coth2μz-z02+3μ2-1ν22, where λ=ν6/8,z0C.

From (18) of Lemma 2, we can express the elliptic solutions of (5) as (35)wd0z=β-2z+β30, with pole at z=0.

Putting wd0(z) into (5), we obtain that (36)wd0z=-6ν2z, where c2=0 and c3=-λ/108ν6.

Therefore, the elliptic solutions of (5) are (37)Wdz=-6ν2z-z0, where z0C.

Applying the addition formula, we can rewrite it as (38)Wdz=-6ν2-z+14z+Dz-C2+6ν2C, where D2=4C3-c3,c2=0, and c3=-λ/108ν6.

4. Computer Simulations

In this section, we illustrate our main results by some computer simulations. We carry out further analysis to the properties of simply periodic solutions Ws(z) and the rational solutions Wr(z) as in Figures 1 and 2.

For Ws(z), take ν=1,δ=1, and μ=2/2.

For Wr(z), take ν=1 and δ=1.

The solution of the Vakhnenko-Parkes equation corresponding to Wr(z), (a) z0=5, (b) z0=0, and (c) z0=-5.

The solution of the Vakhnenko-Parkes equation corresponding to Wr(z), (a) z0=0.5, (b) z0=0, and (c) z0=-0.5.

5. Conclusions

Employing the complex method, we can easily find exact solutions to some nonlinear differential equation. By this method, we get the meromorphic exact solutions of VPE, and then we obtain the traveling wave solutions to VPE. In Wr(z) of our solutions, let z=kx+wt and z0=-c1/c2; then it will be equivalent to Eq. (20) in . Simply periodic solutions Ws(z) are new and cannot be degenerated through elliptic function solutions. Our results demonstrate that the complex method is more simpler, and we can apply the idea of this study to related nonlinear evolution equation.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Yongyi Gu and Wenjun Yuan carried out the design of this paper and performed the analysis. Najva Aminakbari and Qinghua Jiang participated in the calculations and computer simulations. All authors typed, read, and approved the final manuscript.

Acknowledgments

This work was supported by the NSF of China (11271090), the NSF of Guangdong Province (2016A030310257), Young Talents Innovation Project of Guangdong Province (2015KQNCX116), and Joint PHD Program of Guangzhou University and Curtin University.

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