Research Article Bifurcation of Traveling Wave Solutions for (2+1) … · 2019. 7. 31. · Research...

Research ArticleBifurcation of Traveling Wave Solutions for (2+1)-DimensionalNonlinear Models Generated by the Jaulent-Miodek Hierarchy

Yanping Ran,1,2 Jing Li,1 Xin Li,1 and Zheng Tian1

1College of Applied Science, Beijing University of Technology, Beijing 100124, China2School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China

Correspondence should be addressed to Jing Li; [email protected]

Received 27 June 2014; Accepted 15 July 2014

Academic Editor: Yonghui Xia

Copyright © 2015 Yanping Ran 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.

Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by thebifurcation method of planar dynamical systems.The bifurcation regions in different subsets of the parameters space are obtained.According to the different phase portraits in different regions, we obtain kink (antikink) wave solutions, solitary wave solutions,and periodic wave solutions for the third of these models by dynamical systemmethod. Furthermore, the explicit exact expressionsof these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.

1. Introduction

In [1–4], four (2+1)-dimensional nonlinear models gen-erated by the Jaulent-Miodek hierarchy were developed.These nonlinear models are completely integrable evolutionequations. There are many approaches to investigate nonlin-ear evolution equation, for example, the inverse scatteringmethod, the Bäcklund transformation method, the Darbouxtransformation method, the Hirota bilinear method [1–3, 5–8], and the dynamical systems method [9–11]. The Hirotabilinear method [3] is used to formally derive the multiplekink solutions and multiple singular kink solutions of thesemodels. By applying the direct symmetry method [4], groupinvariant solutions and some new exact solutions of the(2+1)-dimensional Jaulent-Miodek equation are obtained.Dynamical systems method is a very effective method toresearch qualitative behavior for traveling wave solutions ofthese completely integrable evolution equations. In [11], onlyconsidering bifurcation parametric 𝑐, some exact travelingwave solutions are given by applying themethod of dynamicalsystems for these models. In this paper, all wave solutionsare given by the method of dynamical systems under moregeneral parametric conditions. Some computer symbolicsystems such as Maple and Mathmatic allow us to performcomplicated and tedious calculations.

Four (2+1)-dimensional nonlinear models generated bythe Jaulent-Miodek hierarchy [3] are given by

𝑤𝑡= − (𝑤

𝑥𝑥− 2𝑤3)𝑥−

3

2

(𝑤𝑥𝜕−1

𝑥𝑤𝑦+ 𝑤𝑤𝑦) ,

𝑤𝑡=

1

2

(𝑤𝑥𝑥− 2𝑤3)𝑥−

3

2

(−

1

4

𝜕−1

𝑥𝑤𝑦𝑦+ 𝑤𝑤𝑦) ,

𝑤𝑡=

1

4


3

4

(

1

4

𝜕−1

𝑥𝑤𝑦𝑦+ 𝑤𝑥𝜕−1

𝑥𝑤𝑦) ,

𝑤𝑡= 2 (𝑤

𝑥𝑥− 2𝑤3)𝑥−

3

4

(𝜕−1

𝑥𝑤𝑦𝑦− 2𝑤𝑥𝜕−1

𝑥𝑤𝑦− 6𝑤𝑤

𝑦) ,

(1)

where 𝜕−1𝑥

is the inverse of 𝜕𝑥with 𝜕

𝑥𝜕−1

𝑥= 𝜕−1

𝑥𝜕𝑥= 1 and

𝜕−1

𝑥= ∫

𝑥

−∞

𝑓 (𝑡) 𝑑𝑡. (2)

We will study the third model given by

𝑤𝑡=

1

4


3

4

(

1

4

𝜕−1

𝑥𝑤𝑦𝑦+ 𝑤𝑥𝜕−1

𝑥𝑤𝑦) . (3)

By introducing the potential

𝑤 (𝑥, 𝑦, 𝑡) = 𝑢𝑥(𝑥, 𝑦, 𝑡) , (4)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 820916, 14 pageshttp://dx.doi.org/10.1155/2015/820916

2 Abstract and Applied Analysis

to remove the integral term in the system (3), we obtain thefollowing equation

𝑢𝑥𝑡+

1

4

𝑢𝑥𝑥𝑥𝑥

−

2

3

𝑢2

𝑥𝑢𝑥𝑥+

3

16

𝑢𝑦𝑦+

3

4

𝑢𝑥𝑥𝑢𝑦= 0. (5)

We are interested in thewave solutions of the system (3) inthis paper. Motivated by [9], we obtain dynamical propertiesof (11) and different wave solutions of the system (3) indetail. This paper is organized as follows. In Section 2, weestablish the traveling wave equation (3) for the third modelof (1). Furthermore, we obtain the first integral of dynamicalgoverning equation of the system (11). Then, we analyze thebifurcation behaviors of the system (11). Phase portraits inthe different subsets of parameter space will be presented inSection 3. In Section 4, using the information of the phaseportraits in Section 3, we analyze all the possible travelingwave solutions of the system (11). Some explicit parametricrepresentations of traveling wave solutions of (3) and thesystem (11) are also obtained. The final section includes briefsummary, future plans, and potential fields of applications.

2. Traveling Wave Equation for the System (3)

We assume that the traveling wave transform of the system(3) is in the form

𝑢 (𝑥, 𝑦, 𝑡) = Ψ (𝜉) ,

𝜉 = 𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡,

(6)

where 𝑐 is propagating wave velocity. Let 𝑘 = 𝑟, 𝑐 = 1, thetraveling wave transform of (6) is equivalent to 𝜉 = 𝑥 +𝑦 − 𝑐𝑡 [11]. So, our traveling wave transform is more general.According to physical meaning of traveling wave solutions ofthe system (3), we always assume that 𝑐 > 0, 𝑘 ̸= 0, and𝑟 ̸= 0. Now, substituting (6) into (5), we have the travelingwave equation

−𝑘𝑐𝑢𝜉𝜉+

𝑘4

4

𝑢𝜉𝜉𝜉𝜉

−

3

2

𝑘4𝑢2

𝜉𝑢𝜉𝜉+

3

16

𝑟2𝑢𝜉𝜉+

3

4

𝑘2𝑟𝑢𝜉𝜉𝑢𝜉= 0.

(7)

Integrating (7) with respect to 𝜉 once, we have

−16𝑘𝑐𝑢𝜉+ 4𝑘4𝑢𝜉𝜉𝜉− 8𝑘4𝑢3

𝜉+ 3𝑟2𝑢𝜉+ 6𝑘2𝑟𝑢2

𝜉= 0. (8)

Setting 𝑢𝜉= 𝜑, (8) becomes

−16𝑘𝑐𝜑 + 4𝑘4𝜑𝜉𝜉− 8𝑘4𝜑3+ 3𝑟2𝜑 + 6𝑘

2𝑟𝜑2= 0. (9)

Furthermore, (8) can be rewritten as

4𝑘4𝜑𝜉𝜉+ (3𝑟2− 16𝑘𝑐) 𝜑 + 6𝑘

2𝑟𝜑2− 8𝑘4𝜑3= 0. (10)

Letting 𝜑 = 𝑦, then we have the following planar system

𝑑𝜑

𝑑𝜉

= 𝑦,

𝑑𝑦

𝑑𝜉

= −

(3𝑟2− 16𝑘𝑐)

4𝑘4

𝜑 −

3𝑟

2𝑘2𝜑2+ 2𝜑3.

(11)

Obviously, the above system (11) is aHamiltonian systemwithHamiltonian function

𝐻(𝜑, 𝑦) =

𝑦2

2

+

(3𝑟2− 16𝑘𝑐)

8𝑘4

𝜑2+

𝑟

6𝑘2𝜑3−

𝜑4

2

.(12)

In order to research the system (11), let 𝑎 = −(3𝑟2 −16𝑘𝑐)/4𝑘

4, 𝑏 = −3𝑟/2𝑘2; the system (11) becomes

𝑑𝜑

𝑑𝜉

= 𝑦,

𝑑𝑦

𝑑𝜉

= 𝑎𝜑 + 𝑏𝜑2+ 2𝜑3.

(13)

The Hamiltonian function of (13) is

𝐻(𝜑, 𝑦) =

1

2

𝑦2−

𝑎

2

𝜑2−

1

3

𝑏𝜑3−

1

2

𝜑4. (14)

3. The Bifurcation Analysis of the System (11)

In this section, our aim is to study the traveling wave solu-tions of the system (11) by applying bifurcation method andqualitative theory of dynamical systems [9, 10]. Throughsome special phase orbits, we obtain smooth periodic wavesolutions, solitary wave solutions, kink and antikink wavesolutions, and so on. Fixing 𝑐, we discuss the phase portraitof the system (11) along with the changes of parameters 𝑟and 𝑘 so as to study traveling wave solutions of the system(11). Further more, through the traveling wave solutions ofthe system (11) and the potential relation (4), traveling wavesolutions of the system (3) will be obtained.

3.1. Phase Portraits and Qualitative Analysis of the System (11).In order to investigate the phase portrait of the system (11),we set

𝑓 (𝜑) = 𝑎𝜑 + 𝑏𝜑2+ 2𝜑3= 𝜑 (𝑎 + 𝑏𝜑 + 2𝜑

2) . (15)

Let Δ = (33𝑟2 − 128𝑘𝑐)/4𝑘4. Obviously, 𝑓(𝜑) has atleast one zero point (𝜑

0, 𝑓(𝜑0)) = (0, 0). The number of the

singular points of the system (11) may be decided by the signof Δ. Obviously, the system (11) has only one trivial singularpoint (0, 0). Thus the other singular points of the system (11)are given as follows. (1) When Δ < 0, the system (11) hasonly one trivial singular point (0, 0); (2) when Δ > 0, thesystem (11) has two singular points (𝜑

1,2, 0), where 𝜑

1= (3𝑟 +

2𝑘2√Δ)/8𝑘

2> 𝜑2= (3𝑟 − 2𝑘

2√Δ)/8𝑘

2; (3) when Δ = 0,the system has a second-order singular point (𝜑

3, 0), where

𝜑3= 𝜑1= 𝜑2.

We notice that the Jacobian of linearized system of thesystem (11) at the singular points is given by

𝐽 (𝜑𝑖, 0) = −𝑓

(𝜑𝑖) , (𝑖 = 0, 1, 2, 3) . (16)

Abstract and Applied Analysis 3

Thus, the characteristic values of linearized system of thesystem (11) at (𝜑

𝑖, 0) are 𝜆 = ±√𝑓(𝜑

𝑖). From the qualitative

theory of dynamical system, we know that

(i) if 𝑓(𝜑𝑖) > 0, (𝜑

𝑖, 0) is a saddle point;

(ii) if 𝑓(𝜑𝑖) < 0, (𝜑

𝑖, 0) is a center point;

(iii) if 𝑓(𝜑𝑖) = 0, (𝜑

𝑖, 0) is a degenerate saddle point.

Let

𝐻(𝜑, 𝑦) = ℎ, (17)

where ℎ is Hamiltonian value. When 𝑟2 ≥ 4𝑘𝑐,

𝐻(𝜑, 0) =

𝑎

2

𝜑2+

𝑏

3

𝜑3+

1

2

𝜑4= 0 (18)

has four real roots.It is well known that the planar Hamiltonian system is

determined by its potential energy level curve and its singularpoint in the form of (𝜑∗, 0). So, we are interested in lookingfor the possible zeros of (15) and determining whether thereare heteroclinic orbits, homoclinic orbits, periodic orbits atdifferent singular points.

In order to find the heteroclinic orbits and the homoclinicorbits of the system (11), let

𝑓(𝜑) = 𝑎 + 2𝑏𝜑 + 6𝜑

2= 0. (19)

From (19), we can get the following expressions of its roots:

𝜑∗

1=

3𝑟 + √27𝑟2− 96𝑘𝑐

12𝑘2

,

𝜑∗

2=

3𝑟 − √27𝑟2− 96𝑘𝑐

12𝑘2

.

(20)

Substituting (20) into (15), we can get

𝑓 (𝜑∗

1) = − ((3𝑟 + √27𝑟

2− 96𝑘𝑐)

× (15𝑟2− 64𝑘𝑐 + 𝑟√27𝑟

2− 96𝑘𝑐)) (288𝑘

6)

−1

,

𝑓 (𝜑∗

2) = − ((−3𝑟 + √27𝑟

2− 96𝑘𝑐)

× (−15𝑟2− 64𝑘𝑐 + 𝑟√27𝑟

2− 96𝑘𝑐)) (288𝑘

6)

−1

,

(21)

−𝑓 (𝜑∗

1) − 𝑓 (𝜑

∗

2) = −

𝑟 (−𝑟2+ 4𝑘𝑐)

2𝑘6

.(22)

Theorem 1. When 𝑘 > 0, 𝑐 > 0, from (22), one has thefollowing.

(i) When 𝑟 = ±2√𝑘𝑐, there are two heteroclinic orbitsformed by the saddle points (±2√𝑘𝑐, 0).

(ii) When 𝑟 ∈ (−4√3𝑘𝑐/3, −2√𝑘𝑐) ∪ (2√𝑘𝑐, 4√3𝑘𝑐/3),there are no heteroclinic orbits, while there are homo-clinic orbits formed by other saddle points except fortwo saddle points (±2√𝑘𝑐, 0).

Proof. When 𝑟 = ±2√𝑘𝑐, we have 𝜑0,2= −√𝑘𝑐/𝑘

2< 0 and

𝑓(−√𝑘𝑐/𝑘

2) < 0. According to the qualitative theory of

dynamical system, (𝜑0,2, 0) are saddle points. Furthermore,

when 𝑟 = ±2√𝑘𝑐, −𝑓(𝜑∗1) = 𝑓(𝜑

∗

2) holds. Similarly, if 𝑟 ∈

(−4√3𝑘𝑐/3, −2√𝑘𝑐)∪ (2√𝑘𝑐, 4√3𝑘𝑐/3), we have that (𝜑1,2, 0)

is the saddle point and −𝑓(𝜑∗1) ̸= 𝑓(𝜑

∗

2) holds. Applying

Theorems 1 and 2 [12], Theorem 1 is proved.

In order to give the details of the bifurcation, if 𝑐 > 0,𝑘 > 0, we can obtain the following six bifurcation boundaries:

𝐿1: 𝑟 = −

4√3𝑘𝑐

3

,

𝐿2: 𝑟 = −2√𝑘𝑐,

𝐿3: 𝑟 = −

8√66𝑘𝑐

33

,

𝐿4: 𝑟 =

8√66𝑘𝑐

33

,

𝐿5: 𝑟 = 2√𝑘𝑐,

𝐿6: 𝑟 =

4√3𝑘𝑐

3

.

(23)

All these bifurcation boundaries divide the parameterspace into seven regions (see Figure 1(a)) in which differentphase portraits exist. All the corresponding phase portraitswill be shown in Figure 2.

If 𝑐 > 0, 𝑘 < 0, there is one bifurcation boundary:

𝐿7: 𝑟 = 0. (24)

In this case, the corresponding phase portraits in twobifurcation regions 𝐹

7and 𝐹

8(see Figure 1(b)) will be shown

in Figure 3.Assuming that the following conditions hold:

𝑘 > 0, 𝑐 > 0, Δ ≥ 0. (𝐼)

Therefore, we can obtain the phase portraits of the system (11)in Figure 2.

Set

ℎ𝑖= 𝐻 (𝜑

𝑖, 𝑦) . (25)

According to Figure 2, we obtain Case 1 as follows.

Case 1. Suppose that 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0, in additionto one of conditions (1)–(12), we can obtain the sign of 𝑓(𝜑

𝑖)

and the relation among ℎ(𝜑𝑖, 0) by choosing suitable 𝑟, 𝑘, and

𝑐, respectively.

(1) When 𝑟 < −4√3𝑘𝑐/3, the fact is that ℎ0< ℎ1< ℎ2

exists and the system (11) has two saddles at (𝜑1,2, 0) and

a center at (0, 0) determined by (16). When ℎ ∈ (ℎ0, ℎ1),

the system (11) has a family of periodic orbits in which theperiodic orbit Γ

1is included (see Figure 2(a)). When ℎ ∈

(−∞, ℎ0) ∪ (ℎ2,∞), periodic orbits become open curves.


L1

L2L3L4

L5

L6

k−5

−10

−15

5

10

15

0

1 2 3 4 5

(a) 𝑘 > 0, 𝑐 > 0, Δ ≥ 0

L7

−1

1−1

−2

−3

−4

−5

−0.5 0.50

(b) 𝑘 < 0, 𝑐 > 0

Figure 1: Transition boundaries on (𝑘 − 𝑟) plane of system (10).

(2) When 𝑟 = −4√3𝑘𝑐/3, the coefficient of 𝜑 vanishes.Both singular points (𝜑

0,1, 0) are degenerated to (𝜑

0, 0) and

(𝜑2, 0) becoming a saddle point in the system (11) (see

Figure 2(b)). In the system (11), all the level curves areopen.

(3) When −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐, the fact is that ℎ1<

ℎ0< ℎ2exists and the system (11) has two saddles at (𝜑

0,2, 0)

and a center at (𝜑1, 0). When ℎ = ℎ

0, the system (11) has

homoclinics orbit Γ4and a special orbit Γ

5(see Figure 2(c)).

When ℎ ∈ (ℎ1, ℎ0), the system (11) has a family of periodic

orbits in which the periodic orbit Γ6is included. When ℎ →

ℎ−

0, periodic orbits become the homoclinic orbit Γ

4. When

ℎ ∈ (−∞, ℎ1), periodic orbits become open curves.

(4) When 𝑟 = −2√𝑘𝑐, the system (11) has two saddles at(𝜑0,2, 0) and a center at (𝜑

1, 0), where 𝜑

2= 2𝜑1= −√𝑘𝑐/𝑘

2,ℎ1= −9𝑐

2/4𝑘4< ℎ0= ℎ2. When ℎ = ℎ


heteroclinic orbits consisting of Γ10

and Γ11, which connects

two saddles (𝜑0,2, 0) (see Figure 2(d)). When ℎ ∈ (ℎ

1, ℎ0),

the system (11) has a family of periodic orbits in which theperiodic orbit Γ

12is included. But when ℎ → ℎ−

0, periodic

orbits become the heteroclinic orbit Γ10and the orbit Γ

11.

(5) When −2√𝑘𝑐 < 𝑟 < −8√66𝑘𝑐/33, we can obtain ℎ2<

ℎ1< ℎ0and the system (11) has two saddles at (𝜑

0,2, 0) and a

center at (𝜑1, 0) (see Figure 2(e)).The system (11) has a family

of open curves.(6) When 𝑟 = −8√66𝑘𝑐/33, both singular points (𝜑

1,2, 0)

are degenerated to (−2√66𝑘𝑐/11, 0); (𝜑0, 0) becomes a saddle

in the system (11) (see Figure 2(f)). The system (11) has afamily of open curves.

(7) When 𝑟 = 8√66𝑘𝑐/33, both singular points (𝜑1,2, 0)

are degenerated to (2√66𝑘𝑐/11, 0), (𝜑0, 0) becomes a saddle

in the system (11) (see Figure 2(g)). The system (11) has afamily of open curves.

(8)When 8√66𝑘𝑐/33 < 𝑟 < 2√𝑘𝑐, the system (11) has twosaddles at (𝜑

0,1, 0) and a center at (𝜑

2, 0). The system (11) has

a family of open curves (see Figure 2(h)).(9) When 𝑟 = 2√𝑘𝑐, the system (11) has two saddles at

(𝜑0,2, 0) and a center at (𝜑

1, 0), where 𝜑

2= 2𝜑1= −√𝑘𝑐/𝑘

2,ℎ2= −9𝑐

2/4𝑘4< ℎ0= ℎ1. When ℎ = ℎ


heteroclinic orbits consisting of Γ13

and Γ14, which connects

two saddles (𝜑0,2, 0). When ℎ ∈ (ℎ

1, ℎ0), the system (11) has

a family of periodic orbits in which the periodic orbit Γ15

isincluded (see Figure 2(i)). But when ℎ → ℎ+

0, periodic orbits

become the heteroclinic orbit Γ13and the orbit Γ

14.

(10) When 2√𝑘𝑐 < 𝑟 < 4√3𝑘𝑐/3, the system (11) has twosaddles at (𝜑

0,1, 0) and a center at (𝜑

2, 0) and ℎ

2< ℎ0< ℎ1

exists. When ℎ = ℎ0, the system (11) has homoclinics orbit

Γ7and a special orbit Γ

8. When ℎ ∈ (ℎ

0, ℎ1), the system (11)

has a family of periodic orbits in which the periodic orbit Γ9

is included (see Figure 2(j)). When ℎ → ℎ+0, periodic orbits

become the homoclinics orbit Γ7.When ℎ ∈ (ℎ

1,∞), periodic

orbits become open curves.(11) When 𝑟 = 4√3𝑘𝑐/3, the coefficient of 𝜑 vanishes.

Both singular points (𝜑0,2, 0) are degenerated to (𝜑

0, 0) and

(𝜑1, 0) becoming a saddle in the system (11) (see Figure 2(k)).

In the system (11), all the level curves are open.(12) When 𝑟 > 4√3𝑘𝑐/3, the system (11) has two saddles

at (𝜑1,2, 0) and a center at (0, 0) and ℎ

0< ℎ2< ℎ1exists.

When ℎ ∈ (ℎ0, ℎ2), the system (11) has a family of periodic

orbits in which the periodic orbit Γ16is included. When ℎ ∈

(−∞, ℎ0) ∪ (ℎ1,∞), periodic orbits become open curves (see

Figure 2(l)).


0

−2

−1

y 1

2

10−2−3 −1−4−5−6

𝜑

(a) 𝑟 < −4√3𝑘𝑐/3

y

0

−2

−4

−8

−6

4

6

8

2

10−2−3 −1−4−5 2

𝜑

(b) 𝑟 = −4√3𝑘𝑐/3

y

0

−2

−4

4

2

10−2−3 −1−4

𝜑

(c) −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐

0

−2

−1

y 1

2

0 0.5−1−1.5 −0.5−2−2.5

𝜑

(d) 𝑟 = −2√𝑘𝑐

0

−2

−1

y 1

2

0 0.5−1−1.5 −0.5−2−2.5

𝜑

(e) −2√𝑘𝑐 < 𝑟 < −8√66𝑘𝑐/33

0

−2

−1

y 1

2

0 0.5−1−1.5 −0.5−2−2.5

𝜑

(f) 𝑟 = −8√66𝑘𝑐/33

0

−2

−1

y 1

2

0 0.5 1 1.5 2 2.5−0.5

𝜑

(g) 𝑟 = 8√66𝑘𝑐/33

0

−2

−1

y 1

2

0 0.5 1 1.5 2 2.5−0.5

𝜑

(h) 8√66𝑘𝑐/33 < 𝑟 < 2√𝑘𝑐

0

−2

−1

y 1

2

0 0.5 1 1.5 2 2.5−0.5

𝜑

(i) 𝑟 = 2√𝑘𝑐

y

0

−2

−4

4

2

10−1 42 3

𝜑

(j) 2√𝑘𝑐 < 𝑟 < 4√3𝑘𝑐/3

10−1 4 52 3

𝜑

−2

y

0

−2

−4

−8

−6

4

6

8

2

(k) 𝑟 = 4√3𝑘𝑐/3

y

0

−2

−4

4

2

10−1 4 5 62 3

𝜑

(l) 𝑟 > 4√3𝑘𝑐/3

Figure 2: The bifurcation phase portraits in different regions of Figure 1(a) for the system (11).


y

𝜑

1 2 3 4−1−2−3−4−5

10

5

0

0

−5

−10

(a) 𝑟 < 0

y

𝜑

1 2 3 4 5−2 −1−3−4

10

5

0

−5

−10

(b) 𝑟 > 0

Figure 3: The bifurcation phase portraits in different regions of Figure 1(b) for the system (11).

1 2 3 4−1−2−3−4

0.2

0.1

00

−0.1

−0.2

𝜒

(a) 𝜑1(𝜉)

1 2 3 4−1−2−3−4

0.2

0.1

0

0

−0.1

−0.2

𝜉

(b) 𝑤1(𝜉) (𝑐1= 0)

Figure 4: The periodic wave solutions of the system (11) and the system (3) when 𝑟 < −4√3𝑘𝑐/3, 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0.

Assuming that the following conditions

𝑘 < 0, 𝑐 > 0 (𝐼𝐼)

hold, according to Figure 3, we obtain Case 2 as follows.

Case 2. Suppose that 𝑘 < 0, 𝑐 > 0, similarly, we have thefollowing.

(13)When 𝑟 < 0, the system (11) has two saddles at (𝜑1,2, 0)

and a center at (0, 0) and ℎ0< ℎ1< ℎ2exists. When ℎ ∈

(ℎ0, ℎ1)∪(ℎ1, ℎ2), the system (11) has a family of periodic orbits

in which the periodic orbit Γ17is included; under other cases,

periodic orbits become open curves (see Figure 4(a)).

(14) When 𝑟 > 0, the system (11) has two saddles at(𝜑1,2, 0) and a center at (0, 0) and ℎ

0< ℎ2< ℎ1exists. When

ℎ ∈ (ℎ0, ℎ2) ∪ (ℎ2, ℎ1), the system (11) has a family of periodic

orbits in which the periodic orbit Γ18is included; under other

cases, periodic orbits become open curves (see Figure 4(b)).

4. Smooth Solitary Wave Solutions, PeriodicWave Solutions, and Kink Wave Solutionsfor the System (11) and the System (3)

In this section, we will seek all traveling wave solutions whichcorrespond to the special bounded phase orbits of the system(11) in Section 3.The explicit expressions of the system (3) are


also obtained by all traveling wave solutions of the system (11)and the relation (4).

4.1. Smooth Solitary Wave Solutions, Periodic Wave Solutions,and Kink and Antikink Wave Solutions of the System (11).From the qualitative theory of dynamical system, we knowthat a smooth solitary wave solution of a partial differ-ential system corresponds to a smooth homoclinic orbitof a traveling wave equation. A periodic orbit of travelingwave equation corresponds to a periodic traveling wavesolution of a partial differential system. Similarly, a smoothheteroclinic orbit of traveling wave equation corresponds to asmooth kink (antikink) wave solution of a partial differentialsystem.

According to the above analysis, in this section, we con-sider the existence and the explicit exact expressions ofsmooth periodic wave solutions, smooth solitary wave solu-tions, and smooth kink (antikink) wave solutions for thesystem (11) and the system (3) under the parameter conditions(𝐼) and (𝐼𝐼).

Firstly, we consider the existence of smooth periodic wavesolutions under parameter conditions (𝐼) and (𝐼𝐼).

Proposition 2. (i) When 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0, thesystem (11) has a family of smooth periodic wave solutions (seeFigure 2), which correspond to 𝐻(𝜑, 𝑦) = ℎ, ℎ ∈ 𝐼, where 𝐼 isone of intervals in (1), (3), (4), (9), (10), and (12) of Case 1.

(1) When 𝑟 < −4√3𝑘𝑐/3, ℎ ∈ (ℎ0, ℎ1), where ℎ

0< ℎ1< ℎ2

in Case 1(1) (see Figure 2(a)).

(2) When −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐, ℎ ∈ (ℎ1, ℎ0), where

ℎ1< ℎ0< ℎ2in Case 1(3) (see Figure 2(c)).

(3) When 𝑟 = −2√𝑘𝑐, ℎ ∈ (ℎ1, ℎ0), where ℎ

1< ℎ0= ℎ2in

Case 1(4) (see Figure 2(d)).

(4) When 𝑟 = 2√𝑘𝑐, ℎ ∈ (ℎ2, ℎ0), where ℎ

2< ℎ0= ℎ1in

Case 1(9) (see Figure 2(i)).

(5) When 2√𝑘𝑐 < 𝑟 < 4√3𝑘𝑐/3, ℎ ∈ (ℎ0, ℎ1), where ℎ

2<

ℎ0< ℎ1in Case 1(10) (see Figure 2(j)).

(6) When 𝑟 > 4√3𝑘𝑐/3, ℎ ∈ (ℎ0, ℎ2), where ℎ

0< ℎ1< ℎ2

in Case 1(12) (see Figure 2(l)).

(ii) When 𝑘 < 0, 𝑐 > 0, the system (11) has a familyof smooth periodic wave solutions (see Figure 2), which corre-spond to𝐻(𝜑, 𝑦) = ℎ, ℎ ∈ 𝐼, where 𝐼 is one of the intervals in(13) and (14) of Case 2.

(1) When 𝑟 < 0, ℎ ∈ (ℎ0, ℎ1)∪(ℎ1, ℎ2), where ℎ

0< ℎ1< ℎ2

in Case 2(13) (see Figure 3(a)).

(2) When 𝑟 < 0, ℎ ∈ (ℎ0, ℎ2)∪(ℎ2, ℎ1), where ℎ

0< ℎ2< ℎ1

in Case 2(14) (see Figure 3(b)).

Secondly, we discuss the existence of solitary wave solu-tions under group (I). We can summarize the results for thesystem (11) from Figures 2(c) and 2(j).

Proposition 3. Under conditions (𝐼), one has following results.

(i) When −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐, the system (11) hasa smooth solitary wave solution of valley type, whichcorresponds to the orbit Γ

4of𝐻(𝜑, 𝑦) = ℎ

0.

(ii) When 2√k𝑐 < 𝑟 < 4√3𝑘𝑐/3, the system (11) hasa smooth solitary wave solution of peak type, whichcorresponds to the orbit Γ

19of𝐻(𝜑, 𝑦) = ℎ

0.

Finally, we mention the conditions of existence for kinkwave solutions of the system (11).

Proposition 4. When conditions (𝐼) hold, the system (11) hassmooth kink (antkink) under one of the following conditions:

(1) 𝑟 = −2√kc, the system (11) has smooth kink whichcorresponds to the orbits Γ

9and Γ

10of 𝐻(𝜑, 𝑦) = ℎ

0

(see Figure 2(d));

(2) 𝑟 = 2√kc, the system (11) has smooth kink whichcorresponds to the orbits Γ

14and Γ15

of 𝐻(𝜑, 𝑦) = ℎ0

(see Figure 2(i)).

4.2. Exact Traveling Wave Solutions of the System (11) and theSystem (3). Firstly, wewill obtain some explicit expressions oftraveling wave solutions for the system (11) when conditions(𝐼) and (𝐼𝐼) hold. Furthermore, using potential (4) for thesystem (3), its exact traveling wave solutions are given asfollows.

We only choose one of the periodic orbits to calculateperiodic wave solutions.

(1) Periodic wave solutions for the system (11) and thesystem (3).

There are periodic orbits such as Γ1, Γ6, Γ9, Γ12, Γ15, Γ16,

Γ17, and Γ

18(see Figures 2(a), 2(c), 2(d), 2(i), 2(j), 2(l), 3(a),

and 3(b)), which correspond to periodic wave solutions forthe system (11). We only choose one of the periodic orbits(see Figure 2(a)) to calculate periodic wave solutions. Thismethod can be used for other periodic orbits.

When 𝑟 < −4√3𝑘𝑐/3 (see Figure 2(a)), we notice thatthere are periodic orbit Γ

1and two special orbits Γ

2, Γ3passing

the points (𝜑4, 0), (𝜑

5, 0), (𝜑

6, 0), and (𝜑

7, 0). In the (𝜑, 𝑦)-

plane the expressions of the orbits are given as

𝑦 = ±√−

(3𝑟2− 16𝑘𝑐)

8𝑘4

𝜑2−

27𝑟

𝑘2𝜑3+ 𝜑4+ ℎ

= ±√(𝜑 − 𝜑4) (𝜑 − 𝜑

5) (𝜑 − 𝜑

6) (𝜑 − 𝜑

7),

(26)

where 𝜑4< 𝜑5< 0 < 𝜑

6< 𝜑7.

Substituting (26) into 𝑑𝜑/𝑑𝜉 = 𝑦 and integrating themalong Γ

1, Γ2, and Γ

3, it follows that

±∫

𝜑

𝜑5

1

√(𝜑 − 𝜑4) (𝜑 − 𝜑

5) (𝜑 − 𝜑

6) (𝜑 − 𝜑

7)

𝑑𝑠 = ∫

𝜉

0

𝑑𝑠.

(27)


Completing the above integral, we obtain one of the periodictraveling wave solutions (see Figure 4(a)) of (26):

𝜑1(𝜉)

= 𝜑4+

(𝜑5− 𝜑4) 𝑠𝑛 (𝜉, 𝜑

4, 𝜑5, 𝜑6, 𝜑7)

1 − ((𝜑6− 𝜑5) / (𝜑6− 𝜑4)) 𝑠𝑛 (𝜉, 𝜑

4, 𝜑5, 𝜑6, 𝜑7)

,

(28)

where 𝑠𝑛(𝜉, 𝜑4, 𝜑5, 𝜑6, 𝜑7) = 𝑠𝑛

2(√2(𝜑

5− 𝜑7)(𝜑4− 𝜑6)𝜉/2,

(𝜑6− 𝜑5)(𝜑7− 𝜑4)/(𝜑7− 𝜑5)(𝜑6− 𝜑4)).

Noting (6), we obtain the the following exact periodicwave solutions of the system (11) from (28):

𝑞1(𝑥, 𝑦, 𝑡)

= 𝜑4+

(𝜑5− 𝜑4) 𝑠𝑛 (𝑥, 𝑦, 𝑡, 𝜑

4, 𝜑5, 𝜑6, 𝜑7)

1 − ((𝜑6− 𝜑5) / (𝜑6− 𝜑4)) 𝑠𝑛 (𝑥, 𝑦, 𝑡, 𝜑

4, 𝜑5, 𝜑6, 𝜑7)

,

(29)

where 𝑠𝑛(𝑥,𝑦, 𝑡,𝜑4,𝜑5,𝜑6,𝜑7) = 𝑠𝑛

2(2√(𝜑

5− 𝜑7)(𝜑4− 𝜑6)(𝑘𝑥+

𝑟𝑦− 𝑐𝑡)/2, (𝜑6−𝜑5)(𝜑7−𝜑4)/(𝜑7−𝜑5)(𝜑6−𝜑4)). 𝑞1(𝑥, 𝑦, 𝑡) is

one of the smooth periodic wave solutions of the system (11).Since 𝑢

1(𝜉) = ∫ 𝜑

1(𝜉)𝑑𝜉, integrating (28) about 𝜉, by (6),

we can obtain one of smooth wave solutions 𝑢1(𝑥, 𝑦, 𝑡) of sys-

tem (5). Applying the potential (4), the periodicwave solutionfor the system (3) is obtained as follows:

𝑤1(𝑥, 𝑦, 𝑡) = 𝑘𝑞

1(𝑥, 𝑦, 𝑡) + 𝑐

1, (30)

where 𝑐1is a constant.

Noting (6), the periodic traveling wave solution𝑤1(𝜉) for

the system (3) is obtained (see Figure 4(b)).

Remark 5. In [11], some periodic wave solutions of the system(11) are obtained, but the periodic wave solutions of thesystem (3) are not given. The periodic wave solutions of thesystem (3) cannot be derived by themethod [3]. In this paper,we obtain all periodic wave solutions of the system (11) andthe system (3).

(2) Solitary wave solutions for the system (11) and thesystem (3).

When −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐, from the phase portrait(see Figure 2(c)), we notice that there are a homoclinic orbitΓ4and a special orbit Γ

5passing the points (𝜑

8, 0), (𝜑

9, 0), and

(0, 0). In (𝜑, 𝑦)-plane, the expressions of the orbits are givenas

𝑦 = ±√𝜑2(𝜑 − 𝜑

8) (𝜑 − 𝜑

9), (31)

when𝐻(𝜑, 𝑦) = 0, where 𝜑8= (𝑟 − 2√𝑟

2− 4𝑘𝑐)/2𝑘

2< 𝜑9=

(𝑟 + 2√𝑟2− 4𝑘𝑐)/2𝑘

2< 0.


4and Γ5, we have

±∫

𝜑

𝜑9

1

√𝜑2(𝜑 − 𝜑

8) (𝜑 − 𝜑

9)

𝑑𝑠 = ∫

𝜉

0

𝑑𝑠. (32)

Completing the above integral, we obtain the following soli-tary wave solution (see Figure 5(a)) of the system (11):

𝜑2(𝜉) =

2𝜑8𝜑9

(𝜑8− 𝜑9) cosh ((√2/2)√𝜑8𝜑9𝜉) + 𝜑8 + 𝜑9

.

(33)

Noting (6), we obtain the following exact solitary wavesolutions (see Figure 5(a)) of the system (11) from (33):

𝑞2(𝑥, 𝑦, 𝑡)

= −

2𝑎𝑘2

2√𝑟2− 4𝑘𝑐 cosh ((√2𝑎/2) (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)) − 𝑟

,

(34)

where 𝑞2(𝑥, 𝑦, 𝑡) is a solitary wave solution of the system (11).

Since 𝑢(𝜉) = ∫𝜑(𝜉)𝑑𝜉, integrating (33) about 𝜉, we canobtain𝑢2(𝜉)

= 2√2tanh−1((𝑟 + 2√𝑟

2− 4𝑘𝑐) tanh (√2𝑎𝜉/4)2𝑘2√𝑎

) .

(35)

According to (6), wave solutions of traveling wave equation(7) from (5) are able to obtain

𝑢2(𝑥, 𝑦, 𝑡)

= 2√2tanh−1 ((𝑟 + 2√𝑟2 − 4𝑘𝑐)(√2𝑎 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

4

)

× tanh(√2𝑎 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

4

)

× (2𝑘2√𝑎)

−1

) ,

(36)

where 𝑢2(𝑥, 𝑦, 𝑡) is one of the smooth wave solutions of (5).

We substitute (36) into the potential𝑤(𝑥, 𝑦, 𝑡) = 𝑢𝑥(𝑥, 𝑦, 𝑡)

as defined in (4) to obtain

𝑤2(𝑥, 𝑦, 𝑡) = − (2𝑘

3𝑎 (𝑟 + 2√𝑟

2− 4𝑘𝑐))

× ((5𝑟2+ 4𝑟√𝑟

2− 4𝑘𝑐 − 4𝑘

4𝑎 − 16𝑘𝑐)

× cosh(√2𝑎 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

4

)

2

−4𝑟√𝑟2− 4𝑘𝑐 + 5𝑟

2+ 16𝑘𝑐)

−1

,

(37)

where 𝑤2(𝑥, 𝑦, 𝑡) is a solitary wave solution of the system (3).

Using travelingwave transform (6), the solitary wave solution𝑤2(𝜉) of the system (3) is obtained (see Figure 5(b)).


1 2 3 40−1−2−3−4

y

𝜉−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

(a) 𝜑2(𝜉)

1 2 3 40−1−2−3−4

y

𝜉−0.1

−0.2

−0.3

−0.4

−0.5

−0.6

−0.7

(b) 𝑤2(𝜉)

Figure 5: The solitary wave of the system (11) and the system (3) when −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐, 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0.

When 2√𝑘𝑐 < 𝑟 < 4√3𝑘𝑐/3 (see Figure 2(j)), the expres-sions of the homoclinic orbit Γ

7and the special orbit Γ

8

passing the points (𝜑11, 0), (0, 0), and (𝜑

12, 0) are given as in

(𝜑, 𝑦)-plane:

𝑦 = ±√𝜑2(𝜑 − 𝜑

11) (𝜑 − 𝜑

12), (38)

when𝐻(𝜑, 𝑦) = 0, where 0 < 𝜑11= (𝑟 − 2√𝑟

2− 4𝑘𝑐)/2𝑘

2<

𝜑12= (𝑟 + 2√𝑟

2− 4𝑘𝑐)/2𝑘

2.Substituting (38) into 𝑑𝜑/𝑑𝜉 = 𝑦 and integrating them

along Γ19and Γ20, we have

±∫

𝜑11

𝜑

1

√𝜑2(𝜑 − 𝜑

11) (𝜑 − 𝜑

12)

𝑑𝑠 = ∫

𝜉

0

𝑑𝑠. (39)

Completing the above integral, we obtain the following soli-tary wave solution (see Figure 6(a)) of the system (11):

𝜑3(𝜉) =

2𝜑8𝜑9

(𝜑9− 𝜑8) cosh ((√2/2)√𝜑8𝜑9𝜉) + 𝜑8 + 𝜑9

.

(40)Noting (6), we obtain the following exact wave solutions

of the system (11) from (40)𝑞3(𝑥, 𝑦, 𝑡)

=

2𝑘2𝑎

2√𝑟2− 4𝑘𝑐 cosh ((√2𝑎/2) (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)) + 𝑟

,

(41)

where 𝑞3(𝑥, 𝑦, 𝑡) is a solitary wave solution of the system (11).

Since 𝑢(𝜉) = ∫ 𝜑(𝜉)𝑑𝜉, integrating (40) about 𝜉, we canobtain

𝑢3(𝜉) = −2√2tanh−1(

(−𝑟 + 2√𝑟2− 4𝑘𝑐) tanh (√2𝑎𝜉/4)2𝑘2√𝑎

) .

(42)

According to (6), one of the smooth wave solutions of (5) isable to obtain

𝑢3(𝑥, 𝑦, 𝑡)

= −2√2tanh−1 (((−𝑟 + 2√𝑟2 − 4𝑘𝑐)

× tanh(√2𝑎 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

4

))

× (2𝑘2√𝑎)

−1

) .

(43)

Substitute (42) into the potential 𝑤(𝑥, 𝑦, 𝑡) = 𝑢𝑥(𝑥, 𝑦, 𝑡)

as defined in (4) to obtain

𝑤3(𝑥, 𝑦, 𝑡) = (2𝑘

3𝑎 (𝑟 − 2√𝑟

2− 4𝑘𝑐))

× ((−5𝑟2+ 4𝑟√𝑟

2− 4𝑘𝑐 + 4𝑘

2𝑎 + 16𝑘𝑐)

× (cosh(√2𝑎 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

4

))

2

−4𝑟√𝑟2− 4𝑘𝑐 + 5𝑟

2− 16𝑘𝑐)

−1

,

(44)

where 𝑤3(𝑥, 𝑦, 𝑡) is a solitary wave solution of the system (3).

Applying (6), the solitary wave solution 𝑤3(𝜉) is obtained for

the system (3) (see Figure 6(b)).


1 2 3 40−1−2−3−4

y

𝜉

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) 𝜑3(𝜉)

1 2 3 40−1−2−3−4

y

𝜉

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) 𝑤3(𝜉)

Figure 6: The solitary wave of the system (11) and the system (3) when 2√𝑘𝑐 < 𝑟 < 4√3𝑘𝑐/3, 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0.

Remark 6. In [11], only one solitary wave solution of peaktype wave solutions of the system (3) is obtained. However,the solitary wave solutions of system (3) cannot be foundby Hirota’s bilinear method [3]. Fortunately, we obtain allsolitary wave solutions of the system (11) and the system (3).

(3) Kink (antikink) wave solutions for the system (11) andthe system (3).

When 𝑟 = −2√𝑘𝑐 (see Figure 2(d)), in (𝜑, 𝑦)-plane, theexpressions of the heteroclinic orbits Γ

10and Γ11passing the

points (𝜑13, 0), (0, 0) are given as

𝑦 = ±√𝑐𝜑2

𝑘3+ 2

√𝑘𝑐

𝑘2𝜑3+

1

2

𝜑4,

(45)

when𝐻(𝜑, 𝑦) = 0, where 𝜑13= −√𝑘𝑐/𝑘

2< 0.


10and Γ11, we have

∫

𝜑

𝜑13

1

√𝜑2(𝜑 + √𝑘𝑐/𝑘

2)

2

𝑑𝑠 =

√2

2

∫

𝜉

0

𝑑𝑠, (46a)

−∫

𝜑

𝜑13

1

√𝜑2(𝜑 + √𝑘𝑐/𝑘

2)

2

𝑑𝑠 =

√2

2

∫

𝜉

0

𝑑𝑠. (46b)

Completing the above integral (46a), we obtain the followingkink wave solution of the system (11) (see Figure 7(a)):

𝜑4(𝜉) = −

√𝑘𝑐 (1 − tanh (√2𝑘𝑐𝜉/2𝑘2))2𝑘2

.(47)

Noting (6), we obtain the exact wave solution of the system(11) from (47). Consider

𝑞4(𝑥, 𝑦, 𝑡) = −

√𝑘𝑐 (1 − tanh (√2𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡) /2𝑘2))2𝑘2

,

(48)

where 𝑞4(𝑥, 𝑦, 𝑡) is a kink wave solution of the system (11).

Since 𝑢(𝜉) = ∫ 𝜑(𝜉)𝑑𝜉, integrating (47) about 𝜉, we canobtain

𝑢4(𝜉)

= −(

1

2𝑘2

√𝑘𝑐𝜉 +

√2

4

× ( ln(1 + tanh(√2𝑘𝑐𝜉

2𝑘2))

+

√2

4

ln(1 − tanh(√2𝑘𝑐𝜉

2𝑘2)))) .

(49)

According to (6), kink wave solutions of (5) are able to obtain

𝑢4(𝑥, 𝑦, 𝑡)

= −(

1

2𝑘2

√𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

+

√2

4

( ln(− tanh(√2𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

2𝑘2

) − 1)

+

√2

4

ln(1 − tanh(√2𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

2𝑘2

)))) .

(50)


2 40−2−4

y

𝜉

−0.5

−1

−1.5

−2

(a) 𝜑4(𝜉) = −(√𝑘𝑐(1 − tanh(√2𝑘𝑐𝜉/2𝑘2)))/2𝑘2

2 40−2−4

y

𝜉

−0.5

−1

−1.5

−2

(b) 𝑤4(𝜉) = √𝑘𝑐(−1 + tanh(√2𝑘𝑐𝜉/2𝑘2))/2𝑘

2 40−2−4

y

𝜉

−0.5

−1

−1.5

−2

(c) 𝜑4 (𝜉) = √𝑘𝑐(1 + tanh(√2𝑘𝑐𝜉/2𝑘2))/ − 2𝑘2

2 40−2−4

y

𝜉

−0.5

−1

−1.5

−2

(d) 𝑤4 (𝜉) = √𝑘𝑐(1 + tanh(√2𝑘𝑐𝜉/2𝑘2))/ − 2𝑘

Figure 7: The kink (antikink) wave solutions of the system (11) and the system (3) when 𝑟 = −2√𝑘𝑐, 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0.

Using the potential (4), the kink wave solution for thesystem (3) are obtained as follows:

𝑤4(𝑥, 𝑦, 𝑡) = −

√𝑘𝑐 (1 − tanh (√2𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡) /2𝑘2))2𝑘

,

(51)

where 𝑤4(𝑥, 𝑦, 𝑡) is the smooth kink wave solution of the

system (3). The smooth kink wave solution 𝑤4(𝜉) of the sys-

tem (3) is obtained from (51) (see Figure 7(b)). Analogously,completing the above integral (46b), we have the followingantikink wave solution

𝜑4 (𝜉) = −

√𝑘𝑐 (1 + tanh (√2𝑘𝑐𝜉/2𝑘2))2𝑘2

, (52)

𝑤4 (𝑥, 𝑦, 𝑡) = −

√𝑘𝑐 (1 + tanh (√2𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡) /2𝑘2))2𝑘

,

(53)

for the system (11) and the system (3), respectively (seeFigures 7(c) and 7(d)).

When 𝑟 = 2√𝑘𝑐 (see Figure 2(i)), in (𝜑, 𝑦)-plane, theexpressions of the heteroclinic orbits Γ

13and Γ14passing the

points (0, 0), (𝜑14, 0) are given as

𝑦 = ±√𝑐𝜑2

𝑘3− 2

√𝑘𝑐

𝑘2𝜑3+

1

2

𝜑4,

(54)

when𝐻(𝜑, 𝑦) = 0, 𝜑14= √𝑘𝑐/𝑘

2> 0.



13and Γ14, we have

∫

𝜑14

𝜑

√2

√𝜑2(𝜑 − √𝑘𝑐/𝑘

2)

2

𝑑𝑠 = ∫

𝜉

0

𝑑𝑠, (55a)

−∫

𝜑14

𝜑

√2

√𝜑2(𝜑 − √𝑘𝑐/𝑘

2)

2

𝑑𝑠 = ∫

𝜉

0

𝑑𝑠. (55b)

Completing the above integral (55a), we obtain the exact kinkwave solution of the system (11) (see Figure 8(a)):

𝜑5(𝜉) =

√𝑘𝑐 (1 + tanh (√2𝑘𝑐𝜉/2𝑘2))2𝑘2

.(56)

Noting (6), we obtain the following exact wave solution of thesystem (11) from (56):

𝑞5(𝑥, 𝑦, 𝑡)

=

√𝑘𝑐 (1 + tanh (√𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡) /2𝑘))2𝑘2

,

(57)

where 𝑞5(𝑥, 𝑦, 𝑡) is a kink wave solution of the system

(11).Since 𝑢(𝜉) = ∫ 𝜑(𝜉)𝑑𝜉, integrating (56) about 𝜉, we can

obtain

𝑢5(𝜉) = (

√𝑘𝑐

2𝑘2𝜉 −

√2

4

× ( ln(tanh(√𝑘𝑐𝜉

2𝑘

) − 1)

−

√2

4

ln(1 + tanh(√2𝑘𝑐𝜉

2𝑘2)))) .

(58)

According to (6), the traveling wave solution of (5) is able toobtain

𝑢5(𝑥, 𝑦, 𝑡)

= (

√𝑘𝑐

2𝑘2(𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

−

√2

4

( ln(tanh(√𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

2𝑘

) − 1)

−

√2

4

ln(1 + tanh(√2𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡)

2𝑘2

)))).

(59)

Applying the potential (4), the kink wave solution of thesystem (3) is obtained as follows:

𝑤5(𝑥, 𝑦, 𝑡)

=

√𝑘𝑐 (1 + tanh (√𝑘𝑐 (𝑘𝑥 + 𝑟𝑦 − 𝑐𝑡) /2𝑘))2𝑘

,

(60)

where 𝑤5(𝑥, 𝑦, 𝑡) is the smooth kink wave solution of the

system (3).Noting (6), we can obtain the smooth kink wave solution

𝑤5(𝜉) of the system (3) (see Figure 8(b)). Analogously, com-

pleting the above integral (55b), we obtain the exact antikinkwave solution:

𝜑5 (𝜉) =

√𝑘𝑐 (1 − tanh (√2𝑘𝑐𝜉/2𝑘2))2𝑘2

,

𝑤5 (𝜉) =

√𝑘𝑐 (1 − tanh (√𝑘𝑐 (𝜉) /2𝑘))2𝑘

,

(61)

for the system (11) and the system (3), respectively (seeFigures 8(c) and 8(d)).

Remark 7. In [3], applying the necessary condition for thekink waves to exist, multiple kink solutions and multiplesingular kink solutions of the system (3) are formally derived.By the special traveling wave transform in [11], no kink(antikink) solutions of the system (11) and system (3) areobtained. In this paper, not considering the necessary con-dition for the kink waves to exist [3], we obtain all kink andantikink wave solutions of the system (11) and the system (3)by the bifurcation method of dynamical systems.

From (1) to (12), we can obtain three theorems aboutthe exact periodic wave solutions and smooth solitary wavesolutions and kink (antikink) wave solutions for the system(11) and the system (3).

Theorem 8. When conditions (𝐼) or (𝐼𝐼) hold, one can obtainsome representative smooth exact periodic wave solutions of thesystem (11) from the periodic obits (see Figures 2(a), 2(c), 2(i),2(j), and 2(l)) and Figures 3(a) and 3(b)), which correspondto the representative smooth exact periodic wave solutions ofthe system (3), such as the periodic wave solution 𝑞

1(𝑥, 𝑦, 𝑡)

of the system (11) corresponding to the periodic wave solution𝑤1(𝑥, 𝑦, 𝑡) of the system (3).

Theorem 9. Under conditions (𝐼), the following results hold.

(1) When −4√3𝑘𝑐/3 < 𝑟 < −2√𝑘𝑐, the system (11)has an exact solitary wave solution 𝑞

2(𝑥, 𝑦, 𝑡), which

corresponds to the solitary wave solutions𝑤2(𝑥, 𝑦, 𝑡) of

the system (3).(2) When 2√𝑘𝑐 < 𝑟 < 4√3𝑘𝑐/3, the system (11) has exact

solitary wave solutions 𝑞3(𝑥, 𝑦, 𝑡), which correspond to

the solitary wave solutions𝑤3(𝑥, 𝑦, 𝑡) of the system (3).

Theorem 10. Under conditions (𝐼), the following results hold.

(1) When 𝑟 = −2√𝑘𝑐, the system (11) has an exact smoothkink wave solution 𝑞

4(𝑥, 𝑦, 𝑡), which corresponds to


2 40−2−4

y

𝜉

0.5

1

1.5

2

(a) 𝜑5(𝜉) = √𝑘𝑐(1 + tanh(√2𝑘𝑐𝜉/2𝑘2))/2𝑘2

2 40−2−4

y

𝜉

0.5

1

1.5

2

(b) 𝑤5(𝜉) = √𝑘𝑐(1 + tanh(√𝑘𝑐𝜉/2𝑘))/2𝑘

2 40−2−4

y

𝜉

0.5

1

1.5

2

(c) 𝜑5 (𝜉) = √𝑘𝑐(1 − tanh(√2𝑘𝑐𝜉/2𝑘2))/2𝑘2

2 40−2−4

y

𝜉

0.5

1

1.5

2

(d) 𝑤5 (𝜉) = √𝑘𝑐(1 − tanh(√𝑘𝑐𝜉/2𝑘))/2𝑘

Figure 8: The kink (antikink) wave solutions of system (11) and (3) when 𝑟 = 2√𝑘𝑐, 𝑘 > 0, 𝑐 > 0, and Δ ≥ 0.

the kink wave solution 𝑤4(𝑥, 𝑦, 𝑡) of the system (3),

respectively.

(2) When 𝑟 = 2√𝑘𝑐, the system (11) has an exact smoothkink wave solution 𝑞

5(𝑥, 𝑦, 𝑡), which corresponds to the

kink wave solution 𝑤5(𝑥, 𝑦, 𝑡) of the system (3).

5. Discussion

In this work we obtain all wave solutions of the completeintegrability of the (2+1)-dimensional nonlinear evolutionequation, the third model, by dynamical systems method.This method can be used for the remaining three models.By determining the necessary condition for the complete

integrability of these models in [3], multiple kink solutionsand multiple singular kink solutions were formally derivedfor the third model. Compared to the method in [3], thenecessary condition which is among the coefficients of thespatial variables is not necessary in our method. In [11], onlysolitary wave solutions are obtained by a special travelingwave transform. To our knowledge, we completely obtain allsolitary wave solutions and kink (antikink) wave solutionsfor these models by the bifurcation method of dynamicalsystem.The idea of applying themethod of dynamical systemto research the complete integrability can be used for othermodels.This will be examined in forthcoming works. For theremaining three models, we can follow the same approach toderive all wave solutions for them.


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledge the support of theNationalNatural Science Foundation of China throughGrantnos. 11072007, 11372014, and 11290152, the Natural ScienceFoundation of Beijing through Grant no. 1122001, the Inter-national Science and Technology Cooperation Program ofChina through Grant no. 2014DFR61080, and the ResearchFund for theDoctoral Program ofHigher Education of Chinathrough Grant no. 20131103120027. All authors wish to thankprofessor Li Jibin formany valuable suggestions leading to theimprovement of this paper.

References

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