# Exact Traveling Wave Solutions to the (2 + 1)-Dimensional Jaulent-Miodek Equation

Authors: Yongyi Gu; Bingmao Deng; Jianming Lin
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We derive exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation by means of the complex method, and then we illustrate our main result by some computer simulations. It has presented that the applied method is very efficient and is practically well suited for the nonlinear differential equations that arise in mathematical physics.


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## Body

## 1. Introduction and Main Results

Nonlinear differential equations widely 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 widespread attention in the study of nonlinear differential equations [1–28]. Exact solutions of nonlinear differential equations play an important role in the study of mathematical physics phenomena. Hence, seeking explicit solutions of physics equations is an interesting and significant subject.In 2001, Geng et al. [29] developed some (2 + 1)-dimensional models from the Jaulent-Miodek hierarchy [30]. Over the past few years, many research results for the (2 + 1)-dimensional Jaulent-Miodek equations have been generated [31–34], such as the algebraic-geometrical solutions, the bifurcation and exact solutions, the N-soliton solution, and Multiple kink solutions for the (2 + 1)-dimensional Jaulent-Miodek equations.In 2012, Zhang et al. [35] studied the following (2 + 1)-dimensional Jaulent-Miodek equation: (1)a1uxt+a2ux2uxx-uxxxx-a3uxxuy-a4uxuxy+a5uyy=0,where ai are constants, i=1,2,…,5.Substituting traveling wave transform(2)ux,y,t=vz,z=x+ly+λt,into (1), and then integrating it we get(3)v′′′-a1λ+a5l2v′+lb2v′2-a23v′3-δ=0,where b=a3+a4, l and λ are constants, and δ is the integration constant. Setting w=v′, (3) becomes (4)w′′-a1λ+a5l2w+lb2w2-a23w3-δ=0.We say that a meromorphic functionζ belongs to the class W if ζ is an elliptic function, or a rational function of z, or a rational function of eμz, μ∈C. Only these functions can satisfy an algebraic addition theorem which was proved by Weierstrass, so the letter W was utilized [36]. In 2006, Eremenko [36] proved that all meromorphic solutions of the Kuramoto-Sivashinsky algebraic differential equation belong to the class W. Recently, Kudryashov et al. [37, 38] used Laurent series to seek meromorphic exact solutions of some nonlinear differential equations. Following their work, the complex method was proposed by Yuan et al. [39, 40]. They employed the Nevanlinna value distribution theory to investigate the existence of meromorphic solutions to some differential equations and then obtain the representations of meromorphic solutions to these differential equations [41, 42]. It shows that the complex method has a strong theoretical basis which can proof that meromorphic solutions of certain differential equations belong to the class W and obtain exact solutions by the indeterminate forms of the solutions. Besides, this method can be applied to get all traveling wave exact solutions or general solutions of related differential equations [43, 44]. In this article, we would like to use the complex method to obtain exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation.Theorem 1.
Ifa2≠0, then the meromorphic solutions w of (4) belong to the class W. In addition, (4) has the following classes of solutions.
(i) The rational function solutions(5)wr,1z=±6a21z-z0+lb2a2,wr,2z=±6a21z-z0-1z-z0-z1-1z1+lb2a2,where z0∈C, z1≠0. λ=-l2(4a2a5-b2)/4a1a2, δ=-l3b3/24a22 in the former case, or λ=-(4a2a5-b2)l2z12+24a2/4a1a2z12, δ=-l3b3z13-72lba2z1-966a23/2/24a22z13 in the latter case.
(ii) The simply periodic solutions(6)ws,1z=±32a2μcoth⁡μ2z-z0+lb2a2,ws,2z=±32a2μcoth⁡μ2z-z0-cothμ2z-z0-z1-cothμ2z1+lb2a2,where z0∈C, z1≠0. λ=-(4a2a5-b2)l2+6a2μ2/4a1a2, δ=-l3b3+126μ3a23/2-18lba2μ2/24a22 in the former case, or λ=3μ2/2coth2⁡μ/2z1-l2b2/4a2, δ=3/2a2μ2coth2⁡μ/2z1(μcoth⁡μ/2z1+3/2a2lb/2)-l3b3/24a22 in the latter case.
(iii) The elliptic function solutions(7)wdz=±32a2-℘+E4℘E2+4℘2E+2℘′F-℘g2-Eg212E2-g2℘+4E3-3Eg2℘′+4℘3+12E℘2-3g2℘-Eg2F+lb2a2,where g3=0, F2=4E3-g2E, E and g2 are arbitrary.Theorem 2.
Ifa2≠0, then traveling wave exact solutions of (3) have the following forms.
(i) The rational function solutions(8)vr,1z=±6a2ln⁡z-z0+lb2a2z-z0+c1,vr,2z=±6a2ln⁡z-z0z-z1-z0+lb2a2-6a21z1z-z0+c2,where z0∈C, z1≠0, c1 and c2 are integral constants.
(ii) The simply periodic solutions(9)vs,1z=∓32a2μln⁡coth2⁡μ2z-z0-1+lb2a2z-z0+c3,vs,2z=∓32a2μln⁡coth2⁡μ2z-z0-1±32a2μln⁡coth2⁡μ2z-z0-z1-1+lb2a2∓32a2μcoth⁡μ2z1z-z0+c4,where z0∈C, z1≠0, c3 and c4 are integral constants.
(iii) The elliptic function solutions(10)vdz=±32a2ln⁡-℘z+14℘′z+G℘z-H2-H+lb2a2z-z0+c5,where c5 is the integral constant, G2=4H3-g2H, g3=0.The rest of this paper is organized as follows. Section2 introduces some preliminary theory and the complex method. In Section 3, we will give the proof of Theorems 1 and 2. Some computer simulations will be given to illustrate our main results in Section 4. Conclusions are presented at the end of the paper.


## 2. Preliminary Theory and the Complex Method

At first, we give some notations and definitions, and then we introduce some lemmas and the complex method.Letm∈N≔1,2,3,…, rj∈Z, j=0,1,…,m, r=(r0,r1,…,rm), and(11)Krwz≔wzr0w′zr1w′′zr2⋯wmzrm,then d(r)≔∑j=0mrj is the degree of Kr[w]. Let the differential polynomial be defined by (12)Fw,w′,…,wm≔∑r∈JarKrw,where J is a finite index set, and ar are constants, then deg⁡F(w,w′,…,w(m))≔maxr∈J⁡dr is the degree of F(w,w′,…,w(m)).Consider the following differential equation:(13)Fw,w′,…,wm=cwn+d,where n∈N, c≠0,d are constants.Setp,q∈N, and meromorphic solutions w of (13) have at least one pole. If (13) has exactly p distinct meromorphic solutions, and their multiplicity of the pole at z=0 is q, then (13) is said to satisfy the p,q condition. It could be not easy to show that the p,q condition of (13) holds, so we need the weak p,q condition as follows.Inserting the Laurent series(14)wz=∑τ=-q∞βτzτ,β-q≠0,q>0,into (13), we can determine exactly p different Laurent singular parts:(15)∑τ=-q-1βτzτ;then (13) is said to satisfy the weak p,q condition.Given two complex numbersν1, ν2, Imν1/ν2>0, and let L be the discrete subset L[2ν1,2ν2]≔{ν∣ν=2c1ν1+2c2ν2,c1,c2∈Z}, and L is isomorphic to Z×Z. Let the discriminant Δ=Δ(b1,b2)≔b13-27b22 and (16)hn=hnL≔∑ν∈L∖01νn.A meromorphic function℘(z)≔℘(z,g2,g3) with double periods 2ν1, 2ν2, which satisfies the equation(17)℘′z2=4℘z3-g2℘z-g3,in which g2=60h4, g3=140h6, and Δ(g2,g3)≠0, is called the Weierstrass elliptic function.In 2009, Eremenko et al. [45] studied the m-order Briot-Bouquet equation (BBEq)(18)Fw,wm=∑j=0nFjwwmj=0,where Fj(w) are constant coefficient polynomials, m∈N. For the m-order BBEq, we have the following lemma.Lemma 3 (see [37, 40, 46]).
Letm,n,p,s∈N,deg⁡Fw,wm<n, and a m-order BBEq (19)Fw,wm=cwn+dsatisfies the weak p,q condition; then the meromorphic solutions w belong to the class W. Suppose for some values of parameters such solution w exists; then other meromorphic solutions form a one-parametric family (z-z0), z0∈C. Furthermore, each elliptic solution with pole at z=0 can be written as(20)wz=∑i=1s-1∑j=2q-1jβ-ijj-1!dj-2dzj-214℘′z+Ci℘z-Di2-℘z+∑i=1s-1β-i12℘′z+Ci℘z-Di+∑j=2q-1jβ-sjj-1!dj-2dzj-2℘z+β0,where β-ij are determined by (14), ∑i=1sβ-i1=0, and Ci2=4Di3-g2Di-g3.
Each rational function solutionw≔R(z) is expressed as(21)Rz=∑i=1s∑j=1qβijz-zij+β0,which has s(≤p) distinct poles of multiplicity q.
Each simply periodic solutionw≔R(η) is a rational function of η=eμz(μ∈C) and is expressed as (22)Rη=∑i=1s∑j=1qβijη-ηij+β0,which has s(≤p) distinct poles of multiplicity q.Lemma 4 (see [46, 47]).
Weierstrass elliptic functions℘(z) have an addition formula as below: (23)℘z-z0=-℘z0-℘z+14℘′z+℘′z0℘z-℘z02.When g2=g3=0, Weierstrass elliptic functions can be degenerated to rational functions according to (24)℘z,0,0=1z2.When Δ(g2,g3)=0, it can also be degenerated to simple periodic functions according to(25)℘z,3d2,-d3=2d-3d2coth2⁡3d2z.By the above definitions and lemmas, we now present the complex method as below for the convenience of readers.Step 1.
Substitute the transformationT:u(x,t)→w(z) defined by (x,t)→z into a given partial differential equation (PDE) to yield a nonlinear ordinary differential equation (ODE).Step 2.
Substitute (14) into the ODE to determine whether the weak p,q condition holds.Step 3.
Find out meromorphic solutionsw(z) of the ODE with a pole at z=0, in which we have m-1 integral constants.Step 4.
Obtain meromorphic solutionsw(z-z0) by Lemmas 3 and 4.Step 5.
Substituting the inverse transformationT-1 into the meromorphic solutions, we get the exact solutions for the original PDE.


## 3. Proof of Main Results

Proof of Theorem1.
Substituting (14) into (4) we have p=2, q=1, β-1=±6/a2, β0=lb/2a2, β1=-6((4a2a5-b2)l2+4λa1a2)/24a23/2, β2=-(6ba2a5-b3)l3+6lbλa1a2+12δa22/48a22 and β3 is an arbitrary constant.
Therefore, (4) is a second-order BBEq and satisfies weak 2,1 condition. Hence, by Lemma 3, we obtain that meromorphic solutions of (4) belong to W. We will show meromorphic solutions of (4) in the following.
By (21), we infer that the indeterminate rational solutions of (4) are (26)R1z=β11z+β12z-z1+β10,with pole at z=0.
SubstitutingR1(z) into (4), we have(27)R1,1z=±6a21z+lb2a2,where λ=-l2(4a2a5-b2)/4a1a2 and δ=-l3b3/24a22. (28)R1,2z=±6a21z-1z-z1-1z1+lb2a2,where λ=-(4a2a5-b2)l2z12+24a2/4a1a2z12 and δ=-l3b3z13-72lba2z1-966a23/2/24a22z13.
So the rational solutions of (4) are (29)wr,1z=±6a21z-z0+lb2a2,wr,2z=±6a21z-z0-1z-z0-z1-1z1+lb2a2,where z0∈C, z1≠0. λ=-l2(4a2a5-b2)/4a1a2, δ=-l3b3/24a22 in the former case, or λ=-(4a2a5-b2)l2z12+24a2/4a1a2z12, δ=-l3b3z13-72lba2z1-966a23/2/24a22z13 in the latter case.
To obtain simply periodic solutions, letη=eμz, and substitute w=R(η) into Eq. (4), then (30)μ2ηR′+η2R′′-a1λ+a5l2η+lb2η2-a23η3-δ=0.
Substituting(31)R2z=β21η-1+β22η-η1+β20,into (30), we obtain that (32)R2,1z=±6a2μ1η-1+12+lb2a2,R2,2z=±6a2μ1η-1-η1η-η1-η1+12η1-1+lb2a2,where λ=-(4a2a5-b2)l2+6a2μ2/4a1a2,δ=-l3b3+126μ3a23/2-18lba2μ2/24a22 in the former case, or λ=3μ2(η1+1)2/2(η1-1)2-l2b2/4a2, δ=3(η1+1)2((2a2/3μ+lb/2)η1+2a2/3μ-lb/2)μ2/2a2(η1-1)3-l3b3/24a23 in the latter case.
Insertingη=eμz into (32), we can get simply periodic solutions to (4) with pole at z=0(33)ws0,1z=±32a2μcoth⁡μ2z+lb2a2,ws0,2z=±32a2μcoth⁡μ2z-coth⁡μ2z-z1-coth⁡μ2z1+lb2a2,where λ=-(4a2a5-b2)l2+6a2μ2/4a1a2, δ=-l3b3+126μ3a23/2-18lba2μ2/24a22 in the former case, or λ=3μ2/2coth2⁡μ/2z1-l2b2/4a2, δ=3/2a2μ2coth2⁡μ/2z1(μcoth⁡μ/2z1+3/2a2lb/2)-l3b3/24a22 in the latter case.
So simply periodic solutions of (4) are (34)ws,1z=±32a2μcoth⁡μ2z-z0+lb2a2,ws,2z=±32a2μcoth⁡μ2z-z0-coth⁡μ2z-z0-z1-coth⁡μ2z1+lb2a2,where z0∈C, z1≠0. λ=-(4a2a5-b2)l2+6a2μ2/4a1a2, δ=-l3b3+126μ3a23/2-18lba2μ2/24a22 in the former case, or λ=3μ2/2coth2⁡μ/2z1-l2b2/4a2, δ=3/2a2μ2coth2⁡μ/2z1(μcoth⁡μ/2z1+3/2a2lb/2)-l3b3/24a22 in the latter case.
From (20), we have the indeterminate relations to elliptic solutions of (4) with pole at z=0(35)wd1z=β-12℘′z+C1℘z-D1+β30,where C12=4D13-g2D1-g3. Making use of Lemma 4 to wd1(z), and considering the results obtained above, we infer that β30=lb/2a2, g3=0, C1=D1=0. So we obtain (36)wd1z=±32a2℘′z℘z+lb2a2,where g3=0.
Thus, the elliptic function solutions of (4) are(37)wdz=±32a2℘′z-z0,g2,0℘z-z0,g2,0+lb2a2,where z0∈C, g3=0, g2 is arbitrary. Applying the addition formula, we can rewrite it as(38)wdz=±32a2-℘+E4℘E2+4℘2E+2℘′F-℘g2-Eg212E2-g2℘+4E3-3Eg2℘′+4℘3+12E℘2-3g2℘-Eg2F+lb2a2,where g3=0, F2=4E3-g2E, E and g2 are arbitrary.Proof of Theorem2.
By Theorem1, we can obtain the rational function solutions of (3) which are (39)vr,1z=∫wr,1zdz=∫±6a21z-z0+lb2a2dz=±6a2ln⁡z-z0+lb2a2z-z0+c1,vr,2z=∫wr,2zdz=∫±6a21z-z0-1z-z0-z1-1z1+lb2a2dz=±6a2ln⁡z-z0z-z1-z0+lb2a2∓6a21z1z-z0+c2,where z0∈C, z1≠0, c1 and c2 are integral constants.
The simply periodic solutions of (3) are(40)vs,1z=∫ws,1zdz=∫±32a2μcoth⁡μ2z-z0+lb2a2dz=∓32a2μln⁡coth2⁡μ2z-z0-1+lb2a2z-z0+c3,vs,2z=∫ws,2zdz=∫±32a2μcoth⁡μ2z-z0-coth⁡μ2z-z0-z1-coth⁡μ2z1dz=∓32a2μln⁡coth2⁡μ2z-z0-1±32a2μln⁡coth2⁡μ2z-z0-z1-1+lb2a2∓32a2μcoth⁡μ2z1z-z0+c4,where z0∈C, z1≠0, c3 and c4 are integral constants.
The elliptic function solutions of (3) are(41)vd,2z=∫wdzdz=∫±32a2℘′z-z0℘z-z0+lb2a2dz=±32a2ln⁡℘z-z0+lb2a2z-z0+c5=±32a2ln⁡-℘z+14℘′z+G℘z-H2-H+lb2a2z-z0+c5,where c5 is the integral constant, G2=4H3-g2H, g3=0.


## 4. Computer Simulations

In this section, we illustrate our main results by some computer simulations. We carry out further analysis to the properties of the new solutions as in the following figures.(1) By employing the complex method, we are able to obtain the rational solutionswr,1(z) and wr,2(z) of (4). Figure 1 shows shape of solutions wr,2(z) for a2=6, b=-24, l=1, λ=-1, z0=0, and z1=1 within the interval -5≤x,y≤5. Note that they have one generation pole which are showed by Figure 1.Figure 1
The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding towr,2(z), (a) t=-5, (b) t=0, and (c) t=5.
(a) (b) (c)(2) By applying the complex method, we achieve the simply periodic solutionsws,1(z) and ws,2(z) of (4). The solutions ws,1(z) and ws,2(z) come from the hyperbolic function. Figure 2 shows the shape of solutions ws,1(z) for a2=6, b=-24, l=1, λ=-1, z0=0, and μ=1 within the interval -2π≤x,y≤2π.Figure 2
The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding tows,2(z), (a) t=-5, (b) t=0, and (c) t=5.
(a) (b) (c)(3) By using the complex method, we are able to get the rational solutionsvr,1(z) and vr,2(z) of (3). Figure 3 shows the shape of solutions vr,2(z) for a2=6, b=6, l=1, λ=-1, z0=0, z1=6, and c2=0 within the interval -5≤x,y≤5. Note that they have one generation pole which are showed by Figure 3.Figure 3
The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding tovr,2(z), (a) t=-5, (b) t=0, and (c) t=5.
(a) (b) (c)(4) By employing the complex method, we obtain the simply periodic solutionsvs,1(z) and vs,2(z) of (3). The solutions vs,1(z) and vs,2(z) are in terms of the hyperbolic function solution. Figure 4 shows the shape of solutions vs,1(z) for a2=6, b=6, l=1, λ=-1, z0=0, μ=1, and c3=0 within the interval -2π≤x,y≤2π. It may be observed from Figure 4 that when t increases, there would be a delay for the appearance of the peak within the natural topology of traveling wave solution.Figure 4
The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding tovs,1(z), (a) t=-5, (b) t=0, and (c) t=5.
(a) (b) (c)


## 5. Conclusions

In summary, we have utilized the complex method to construct exact solutions of the nonlinear evolution equation. We first show that meromorphic solutions of the (2 + 1)-dimensional Jaulent-Miodek equation belong to the classW, and then we obtain the exact traveling wave solutions for this equation. To our knowledge, the solutions in this paper have not been reported in former literature. The simply periodic solutions ws,2(z), vs,2(z) and the rational solutions wr,2(z), vr,2(z) are not only new but also not degenerated successively by the elliptic function solutions. We expand the results in [32, 33].Based on the previous works [32, 33], the complex method allows us to confirm that meromorphic solutions of the differential equation belong to the class W easily. By the indeterminate forms of the solutions, we can find meromorphic solutions w(z) for the differential equation with a pole at z=0; then we are able to obtain all meromorphic solutions w(z-z0), z0∈C for the differential equation with an arbitrary pole. The results demonstrate that the applied method is direct and efficient method, which allows us to do tedious and complicated algebraic calculation. We can apply the idea of this study to other nonlinear evolution equations.



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