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Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy

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Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the bifurcation 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 system method. Furthermore, the explicit exact expressions of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.
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Research Article
Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional
Nonlinear Models Generated by the Jaulent-Miodek Hierarchy
Yanping Ran,1,2 Jing Li,1Xin Li,1and Zheng Tian1
1College of Applied Science, Beijing University of Technology, Beijing 100124, China
2School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China
Correspondence should be addressed to Jing Li; leejing@bjut.edu.cn
Received  June ; Accepted  July 
Academic Editor: Yonghui Xia
Copyright ©  Yanping Ran et al. is 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 (+)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the
bifurcation method of planar dynamical systems. e bifurcation regions in dierent subsets of the parameters space are obtained.
According to the dierent phase portraits in dierent regions, we obtain kink (antikink) wave solutions, solitary wave solutions,
and periodic wave solutions for the third of these models by dynamical system method. Furthermore, the explicit exact expressions
of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.
1. Introduction
In [], four (+)-dimensional nonlinear models gen-
erated by the Jaulent-Miodek hierarchy were developed.
ese nonlinear models are completely integrable evolution
equations. ere are many approaches to investigate nonlin-
ear evolution equation, for example, the inverse scattering
method, the B¨
acklund transformation method, the Darboux
transformation method, the Hirota bilinear method [,
], and the dynamical systems method []. e Hirota
bilinear method []isusedtoformallyderivethemultiple
kink solutions and multiple singular kink solutions of these
models. By applying the direct symmetry method [], group
invariant solutions and some new exact solutions of the
(+)-dimensional Jaulent-Miodek equation are obtained.
Dynamical systems method is a very eective method to
research qualitative behavior for traveling wave solutions of
these completely integrable evolution equations. In [], only
considering bifurcation parametric , some exact traveling
wave solutions are given by applying the method of dynamical
systems for these models. In this paper, all wave solutions
are given by the method of dynamical systems under more
general parametric conditions. Some computer symbolic
systems such as Maple and Mathmatic allow us to perform
complicated and tedious calculations.
Four (+)-dimensional nonlinear models generated by
the Jaulent-Miodek hierarchy []aregivenby
𝑡=−𝑥𝑥 −23𝑥3
2𝑥−1
𝑥𝑦+𝑦,
𝑡=1
2𝑥𝑥 −23𝑥3
2−1
4−1
𝑥𝑦𝑦 +𝑦,
𝑡=1
4𝑥𝑥 −23𝑥3
41
4−1
𝑥𝑦𝑦 +𝑥−1
𝑥𝑦,
𝑡=2𝑥𝑥 −23𝑥3
4−1
𝑥𝑦𝑦 −2𝑥−1
𝑥𝑦−6𝑦,
()
where −1
𝑥is the inverse of 𝑥with 𝑥−1
𝑥=−1
𝑥𝑥=1and
−1
𝑥=𝑥
−∞ (). ()
We will study the third model given by
𝑡=1
4𝑥𝑥 −23𝑥3
41
4−1
𝑥𝑦𝑦 +𝑥−1
𝑥𝑦. ()
By introducing the potential
,,=𝑥,,,()
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2015, Article ID 820916, 14 pages
http://dx.doi.org/10.1155/2015/820916
Abstract and Applied Analysis
to remove the integral term in the system (),weobtainthe
following equation
𝑥𝑡 +1
4𝑥𝑥𝑥𝑥 2
32
𝑥𝑥𝑥 +3
16𝑦𝑦 +3
4𝑥𝑥𝑦=0. ()
We are interested in the wave solutions of the system () in
this paper. Motivated by [], we obtain dynamical properties
of () and dierent wave solutions of the system () in
detail. is paper is organized as follows. In Section ,we
establish the traveling wave equation () for the third model
of ().Furthermore,weobtaintherstintegralofdynamical
governing equation of the system ().en,weanalyzethe
bifurcation behaviors of the system ().Phaseportraitsin
the dierent subsets of parameter space will be presented in
Section .InSection , using the information of the phase
portraits in Section ,weanalyzeallthepossibletraveling
wave solutions of the system (). Some explicit parametric
representations of traveling wave solutions of () and the
system () are also obtained. e nal section includes brief
summary, future plans, and potential elds of applications.
2. Traveling Wave Equation for the System (3)
We assume that the traveling wave transform of the system
() is in the form ,,=Ψ(),
=+, ()
where is propagating wave velocity. Let =,=1,the
traveling wave transform of () is equivalent to =+
−[]. So, our traveling wave transform is more general.
According to physical meaning of traveling wave solutions of
the system (), we always assume that >0, =0,and
 =0.Now,substituting() into (),wehavethetraveling
wave equation
−𝜉𝜉 +4
4𝜉𝜉𝜉𝜉 3
242
𝜉𝜉𝜉 +3
162𝜉𝜉 +3
42𝜉𝜉𝜉=0.
()
Integrating () with respect to once, we have
−16𝜉+44𝜉𝜉𝜉 −843
𝜉+32𝜉+622
𝜉=0. ()
Setting 𝜉=,() becomes
−16+44𝜉𝜉 −843+32+622=0. ()
Furthermore, () canberewrittenas
44𝜉𝜉 +3216+622−843=0. ()
Letting 󸀠=,thenwehavethefollowingplanarsystem

 =,

 =−3216
44− 3
222+23.()
Obviously, the above system () is a Hamiltonian system with
Hamiltonian function
,=2
2+3216
842+
6234
2.()
In order to research the system (),let=−(3
2
16)/44,=−3/22;thesystem() becomes

 =,

 =+2+23.()
e Hamiltonian function of () is
,=1
22
221
331
24.()
3. The Bifurcation Analysis of the System (11)
In this section, our aim is to study the traveling wave solu-
tions of the system () by applying bifurcation method and
qualitative theory of dynamical systems [,]. rough
some special phase orbits, we obtain smooth periodic wave
solutions, solitary wave solutions, kink and antikink wave
solutions,andsoon.Fixing,wediscussthephaseportrait
of the system () along with the changes of parameters
and so as to study traveling wave solutions of the system
(). Further more, through the traveling wave solutions of
the system () and the potential relation (), traveling wave
solutions of the system () will be obtained.
3.1. Phase Portraits and Qualitative Analysis of the System (11).
In order to investigate the phase portrait of the system (),
we set
=+2+23=++22. ()
Let  = (332− 128)/44.Obviously,() has at
least one zero point (0,(0)) = (0,0).enumberofthe
singular points of the system () may be decided by the sign
of .Obviously,thesystem() has only one trivial singular
point (0,0). us the other singular points of the system ()
are given as follows. () When <0,thesystem() has
only one trivial singular point (0,0);()when>0,the
system () has two singular points (1,2,0),where1=(3+
22)/82>
2=(32
2)/82;()when=0,
the system has a second-order singular point (3,0),where
3=1=2.
We notice that the Jacobian of linearized system of the
system () at the singular points is given by
𝑖,0=󸀠𝑖, (=0,1,2,3).()
Abstract and Applied Analysis
us, the characteristic values of linearized system of the
system () at (𝑖,0)are =±
󸀠(𝑖).Fromthequalitative
theory of dynamical system, we know that
(i) if 󸀠(𝑖)>0,(𝑖,0)is a saddle point;
(ii) if 󸀠(𝑖)<0,(𝑖,0)is a center point;
(iii) if 󸀠(𝑖)=0,(𝑖,0)is a degenerate saddle point.
Let ,=, ()
where is Hamiltonian value. When 2≥4,
,0=
22+
33+1
24=0 ()
has four real roots.
It is well known that the planar Hamiltonian system is
determined by its potential energy level curve and its singular
point in the form of (,0).So,weareinterestedinlooking
for the possible zeros of () and determining whether there
are heteroclinic orbits, homoclinic orbits, periodic orbits at
dierent singular points.
In order to nd the heteroclinic orbits and the homoclinic
orbits of the system (),let
󸀠=+2+62=0. ()
From (), we can get the following expressions of its roots:
1=3+27296
122,
2=327296
122.()
Substituting () into (),wecanget

1=3+27296
×15264+272962886−1 ,

2=−3+27296
×−15264+272962886−1 ,
()
−
1−
2=−2+4
26.()
eorem 1. When >0,>0,from(22), one has the
following.
(i) When =±2
, there are two heteroclinic orbits
formed by the saddle points (±2,0).
(ii) When  ∈ (−43/3,−2) (2,43/3),
there are no heteroclinic orbits, while there are homo-
clinic orbits formed by other saddle points except for
two saddle points (±2,0).
Proof. When =±2
,wehave0,2 =−
/2<0and
󸀠(−/2)<0. According to the qualitative theory of
dynamical system, (0,2,0)are saddle points. Furthermore,
when =±2
,−(
1)=(
2)holds. Similarly, if ∈
(−43/3,−2)(2,43/3),wehavethat(1,2,0)
is the saddle point and −(
1) =(
2)holds. Applying
eorems  and  [], eorem  is proved.
In order to give the details of the bifurcation, if >0,
>0,wecanobtainthefollowingsixbifurcationboundaries:
1:=−43
3,
2:=−2,
3:=−866
33 ,
4:=866
33 ,
5:=2,
6:=43
3.
()
All these bifurcation boundaries divide the parameter
space into seven regions (see Figure (a))inwhichdierent
phase portraits exist. All the corresponding phase portraits
will be shown in Figure .
If >0,<0, there is one bifurcation boundary:
7:=0. ()
In this case, the corresponding phase portraits in two
bifurcation regions 7and 8(see Figure (b))willbeshown
in Figure .
Assuming that the following conditions hold:
>0, >0, ≥0. ()
erefore, we can obtain the phase portraits of the system ()
in Figure .
Set
𝑖=𝑖,.()
According to Figure ,weobtainCaseas follows.
Case 1. Suppose that >0,>0,and≥0, in addition
to one of conditions ()–(), we can obtain the sign of 󸀠(𝑖)
and the relation among (𝑖,0)by choosing suitable ,,and
,respectively.
() When <−4
3/3,thefactisthat0<
1<
2
exists and the system () has two saddles at (1,2,0)and
acenterat(0,0) determined by ().When∈(
0,1),
the system () has a family of periodic orbits in which the
periodic orbit Γ1is included (see Figure (a)). When ∈
(−∞,0)∪(2,∞), periodic orbits become open curves.
Abstract and Applied Analysis
L1
L2
L3
L4
L5
L6
k
−5
−10
−15
5
10
15
0
12345
(a) 𝑘>0,𝑐>0,Δ≥0
L7
−1
1−1
−2
−3
−4
−5
−0.5 0.50
(b) 𝑘<0,𝑐>0
F : Transition boundaries on (−)planeofsystem().
() When =−4
3/3, the coecient of vanishes.
Both singular points (0,1,0)are degenerated to (0,0)and
(2,0) becomingasaddlepointinthesystem() (see
Figure (b)). In the system (),allthelevelcurvesare
open.
() When −43/3 <  < −2,thefactisthat1<
0<2exists and the system () has two saddles at (0,2,0)
and a center at (1,0).When=
0,thesystem() has
homoclinics orbit Γ4and a special orbit Γ5(see Figure (c)).
When ∈(
1,0),thesystem() has a family of periodic
orbits in which the periodic orbit Γ6is included. When →
0, periodic orbits become the homoclinic orbit Γ4.When
(−∞,1), periodic orbits become open curves.
() When =−2
,thesystem() has two saddles at
(0,2,0)and a center at (1,0),where2=2
1=−
/2,
1=−92/44<
0=
2.When=
0,thesystem() has
heteroclinic orbits consisting of Γ10 and Γ11,whichconnects
two saddles (0,2,0)(see Figure (d)). When ∈(
1,0),
the system () has a family of periodic orbits in which the
periodic orbit Γ12 is included. But when →
0, periodic
orbits become the heteroclinic orbit Γ10 and the orbit Γ11.
() When −2<<−866/33,wecanobtain2<
1<0and the system () has two saddles at (0,2,0)and a
center at (1,0)(see Figure (e)). e system () has a family
of open curves.
() When =−866/33, both singular points (1,2,0)
are degenerated to (−266/11,0);(0,0)becomes a saddle
in the system () (see Figure (f)). e system () has a
family of open curves.
() When =8
66/33, both singular points (1,2,0)
are degenerated to (266/11,0),(0,0)becomes a saddle
in the system () (see Figure (g)). e system () has a
family of open curves.
() When 866/33<<2,thesystem() has two
saddles at (0,1,0)and a center at (2,0).esystem() has
afamilyofopencurves(seeFigure (h)).
() When =2
,thesystem() has two saddles at
(0,2,0)and a center at (1,0),where2=2
1=−
/2,
2=−92/44<
0=
1.When=
0,thesystem() has
heteroclinic orbits consisting of Γ13 and Γ14,whichconnects
two saddles (0,2,0).When∈(
1,0),thesystem() has
a family of periodic orbits in which the periodic orbit Γ15 is
included (see Figure (i)). But when →
+
0, periodic orbits
become the heteroclinic orbit Γ13 and the orbit Γ14.
() When 2<<43/3,thesystem() has two
saddles at (0,1,0)and a center at (2,0)and 2<
0<
1
exists. When =
0,thesystem() has homoclinics orbit
Γ7and a special orbit Γ8.When∈(
0,1),thesystem()
has a family of periodic orbits in which the periodic orbit Γ9
is included (see Figure (j)). When →
+
0, periodic orbits
become the homoclinics orbit Γ7.When∈(1,∞), periodic
orbits become open curves.
() When =4
3/3, the coecient of vanishes.
Both singular points (0,2,0)are degenerated to (0,0)and
(1,0)becoming a saddle in the system () (see Figure (k)).
In the system (),allthelevelcurvesareopen.
() When >43/3,thesystem() has two saddles
at (1,2,0)and a center at (0,0) and 0<
2<
1exists.
When ∈(
0,2),thesystem() has a family of periodic
orbits in which the periodic orbit Γ16 is included. When ∈
(−∞,0)∪(1,∞), periodic orbits become open curves (see
Figure (l)).
Abstract and Applied Analysis
0
−2
−1
y1
2
1
0−2−3 −1−4−5−6
𝜑
(a) 𝑟<−4
3𝑘𝑐/3
y
0
−2
−4
−8
−6
4
6
8
2
1
0−2−3 −1−4−5 2
𝜑
(b) 𝑟=−4
3𝑘𝑐/3
y
0
−2
−4
4
2
1
0−2−3 −1−4
𝜑
(c) −43𝑘𝑐/3 < 𝑟 < −2𝑘𝑐
0
−2
−1
y1
2
0 0.5−11.5 −0.5−2−2.5
𝜑
(d) 𝑟=−2
𝑘𝑐
0
−2
−1
y1
2
0 0.5−1−1.5 −0.5−22.5
𝜑
(e) −2𝑘𝑐<𝑟<−8
66𝑘𝑐/33
0
−2
−1
y1
2
0 0.5−1−1.5 −0.5−2−2.5
𝜑
(f) 𝑟=−8
66𝑘𝑐/33
0
−2
−1
y1
2
0 0.5 1 1.5 2 2.5−0.5
𝜑
(g) 𝑟=8
66𝑘𝑐/33
0
−2
−1
y1
2
0 0.5 1 1.5 2 2.5−0.5
𝜑
(h) 866𝑘𝑐/33 < 𝑟 < 2𝑘𝑐
0
−2
−1
y1
2
0 0.5 1 1.5 2 2.5−0.5
𝜑
(i) 𝑟=2
𝑘𝑐
y
0
−2
−4
4
2
1
0−1 4
23
𝜑
(j) 2𝑘𝑐<𝑟<4
3𝑘𝑐/3
1
0−1 4 523
𝜑
−2
y
0
−2
−4
−8
−6
4
6
8
2
(k) 𝑟=4
3𝑘𝑐/3
y
0
−2
−4
4
2
1
01 45623
𝜑
(l) 𝑟>4
3𝑘𝑐/3
F : e bifurcation phase portraits in dierent regions of Figure (a) for the system ().
Abstract and Applied Analysis
y
𝜑
1234
−1−2−3−4−5
10
5
0
0
−5
−10
(a) 𝑟<0
y
𝜑
12345−2 −1−3−4
10
5
0
−5
−10
(b) 𝑟>0
F : e bifurcation phase portraits in dierent regions of Figure (b) for the system ().
1234−1−2−3−4
0.2
0.1
0
0
−0.1
−0.2
𝜒
(a) 𝜑1(𝜉)
1234−1−2−3−4
0.2
0.1
0
0
−0.1
−0.2
𝜉
(b) 𝑤1(𝜉) (𝑐1=0)
F : e periodic wave solutions of the system () and the system () when <−43/3,>0,>0,and≥0.
Assuming that the following conditions
<0, >0 ()
hold, according to Figure ,weobtainCaseas follows.
Case 2. Suppose that <0,>0,similarly,wehavethe
following.
() When <0,thesystem() hastwosaddlesat(1,2,0)
and a center at (0,0)and 0<
1<
2exists. When ∈
(0,1)∪(1,2),thesystem() has a family of periodic orbits
in which the periodic orbit Γ17 is included; under other cases,
periodic orbits become open curves (see Figure (a)).
() When >0,thesystem() has two saddles at
(1,2,0)and a center at (0,0)and 0<2<1exists. When
∈(0,2)∪(2,1),thesystem() has a family of periodic
orbits in which the periodic orbit Γ18 is included; under other
cases, periodic orbits become open curves (see Figure (b)).
4. Smooth Solitary Wave Solutions, Periodic
Wave Solutions, and Kink Wave Solutions
for the System (11) and the System (3)
In this section, we will seek all traveling wave solutions which
correspond to the special bounded phase orbits of the system
() in Section . e explicit expressions of the system () are
Abstract and Applied Analysis
also obtained by all traveling wave solutions of the system ()
and the relation ().
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 know
that a smooth solitary wave solution of a partial dier-
ential system corresponds to a smooth homoclinic orbit
of a traveling wave equation. A periodic orbit of traveling
wave equation corresponds to a periodic traveling wave
solution of a partial dierential system. Similarly, a smooth
heteroclinic orbit of traveling wave equation corresponds to a
smooth kink (antikink) wave solution of a partial dierential
system.
According to the above analysis, in this section, we con-
sider the existence and the explicit exact expressions of
smooth periodic wave solutions, smooth solitary wave solu-
tions, and smooth kink (antikink) wave solutions for the
system () and the system () under the parameter conditions
()and ().
Firstly, we consider the existence of smooth periodic wave
solutions under parameter conditions ()and ().
Proposition 2. (i) When >0,>0,and≥0,the
system (11) has a family of smooth periodic wave solutions (see
Figure 2), which correspond to (,) = ,∈,whereis
one of intervals in (1), (3), (4), (9), (10), and (12) of Case 1.
() When <−43/3,∈(0,1),where0<1<2
in Case 1(1) (see Figure 2(a)).
() When −43/3 <  < −2,∈(
1,0),where
1<0<2in Case 1(3) (see Figure 2(c)).
() When =−2,∈(1,0),where1<0=2in
Case 1(4) (see Figure 2(d)).
() When =2
,∈(
2,0),where2<
0=
1in
Case 1(9) (see Figure 2(i)).
() When 2<<43/3,∈(0,1),where2<
0<1in Case 1(10) (see Figure 2(j)).
() When >43/3,∈(0,2),where0<1<2
in Case 1(12) (see Figure 2(l)).
(ii) When <0,>0,thesystem(11) has a family
of smooth periodic wave solutions (see Figure 2), which corre-
spond to (,)= ,∈,whereis one of the intervals in
(13) and (14) of Case 2.
() When <0,∈(0,1)∪(1,2),where0<1<2
in Case 2(13) (see Figure 3(a)).
() When <0,∈(0,2)∪(2,1),where0<2<1
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 the
system () from Figures (c) and (j).
Proposition 3. Under conditions (),onehasfollowingresults.
(i) When −43/3 <  < −2,thesystem(11) has
a smooth solitary wave solution of valley type, which
corresponds to the orbit Γ4of (,)=0.
(ii) When 2k<<4
3/3,thesystem(11) has
a smooth solitary wave solution of peak type, which
corresponds to the orbit Γ19 of (,)=0.
Finally, we mention the conditions of existence for kink
wave solutions of the system ().
Proposition 4. When conditions ()hold, the system (11) has
smooth kink (antkink) under one of the following conditions:
() =−2
kc, the system (11) has smooth kink which
corresponds to the orbits Γ9and Γ10 of (,) = 0
(see Figure 2(d));
() =2
kc, the system (11) has smooth kink which
corresponds to the orbits Γ14 and Γ15 of (,) = 0
(see Figure 2(i)).
4.2. Exact Traveling Wave Solutions of the System (11) and the
System (3).Firstly, we will obtain some explicit expressions of
traveling wave solutions for the system () when conditions
()and ()hold. Furthermore, using potential () for the
system (), its exact traveling wave solutions are given as
follows.
We only choose one of the periodic orbits to calculate
periodic wave solutions.
() Periodic wave solutions for the system () and the
system ().
ere are periodic orbits such as Γ1,Γ6,Γ9,Γ12,Γ15,Γ16,
Γ17,andΓ18 (see Figures (a),(c),(d),(i),(j),(l),(a),
and (b)), which correspond to periodic wave solutions for
the system ().Weonlychooseoneoftheperiodicorbits
(see Figure (a)) to calculate periodic wave solutions. is
method can be used for other periodic orbits.
When <−4
3/3 (see Figure (a)), we notice that
there are periodic orbit Γ1and two special orbits Γ2,Γ3passing
the points (4,0),(5,0),(6,0),and(7,0).Inthe(,)-
plane the expressions of the orbits are given as
=±3216
84227
23+4+
4−5−6−7,
()
where 4<5<0<6<7.
Substituting () into / = and integrating them
along Γ1,Γ2,andΓ3,itfollowsthat
±𝜑
𝜑51
4−5−6−7=𝜉
0.
()
Abstract and Applied Analysis
Completing the above integral, we obtain one of the periodic
traveling wave solutions (see Figure (a))of():
1()
=4+5−4,4,5,6,7
1−6−5/6−4,4,5,6,7,
()
where (,4,5,6,7)=
2(2(5−7)(4−6)/2,
(6−5)(7−4)/(7−5)(6−4)).
Noting (),weobtainthethefollowingexactperiodic
wave solutions of the system () from ():
1,,
=4+5−4,,,4,5,6,7
16−5/6−4,,,4,5,6,7,
()
where (,,,4,5,6,7)=2(2(5−7)(4−6)(+
)/2,(6−5)(7−4)/(7−5)(6−4)).1(,,)is
one of the smooth periodic wave solutions of the system ().
Since 1()= 1(),integrating() about ,by(),
we can obtain one of smooth wave solutions 1(,,)of sys-
tem (). Applying the potential (),theperiodicwavesolution
for the system () is obtained as follows:
1,,=1,,+1,()
where 1is a constant.
Noting (), the periodic traveling wave solution 1()for
the system () is obtained (see Figure (b)).
Remark 5. In [], some periodic wave solutions of the system
() are obtained, but the periodic wave solutions of the
system () are not given. e periodic wave solutions of the
system () cannot be derived by the method []. In this paper,
we obtain all periodic wave solutions of the system () and
the system ().
() Solitary wave solutions for the system () and the
system ().
When −43/3 <  < −2,fromthephaseportrait
(see Figure (c)), we notice that there are a homoclinic orbit
Γ4and a special orbit Γ5passing the points (8,0),(9,0),and
(0,0).In(,)-plane, the expressions of the orbits are given
as
=±28−9,()
when (,) =0,where8=(−22−4)/22<9=
(+22−4)/22<0.
Substituting () into / = and integrating them
along Γ4and Γ5,wehave
±𝜑
𝜑91
28−9=𝜉
0. ()
Completing the above integral, we obtain the following soli-
tary wave solution (see Figure (a))ofthesystem():
2()=289
8−9cosh 2/289+8+9.
()
Noting (), we obtain the following exact solitary wave
solutions (see Figure (a))ofthesystem() from ():
2,,
=− 22
22−4cosh 2/2+,
()
where 2(,,)is a solitary wave solution of the system ().
Since () = ∫(),integrating() about ,wecan
obtain
2()
=22tanh−1 +22−4tanh 2/4
22.
()
According to (), wave solutions of traveling wave equation
() from () are able to obtain
2,,
=22tanh−1 +22−42+−
4
×tanh 2+−
4
×22−1,
()
where 2(,,)is one of the smooth wave solutions of ().
We substitute () into the potential (,,)=𝑥(,,)
as dened in () to obtain
2,,=−23+22−4
×52+42−4−4416
×cosh 2+−
42
−42−4+52+16−1 ,
()
where 2(,,)is a solitary wave solution of the system ().
Using traveling wave transform (), the solitary wave solution
2()of the system () is obtained (see Figure (b)).
Abstract and Applied Analysis
1234
0−1−2−3
−4
y
𝜉
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
(a) 𝜑2(𝜉)
12340−1−2−3−4
y
𝜉
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
(b) 𝑤2(𝜉)
F : e solitary wave of the system () and the system () when −43/3< <−2,>0,>0,and≥0.
When 2<<43/3(see Figure (j)), the expres-
sions of the homoclinic orbit Γ7and the special orbit Γ8
passing the points (11,0),(0,0),and(12,0)are given as in
(,)-plane:
=±211−12, ()
when (,) =0,where0<
11 =(−22−4)/22<
12 =(+22−4)/22.
Substituting () into / = and integrating them
along Γ19 and Γ20,wehave
±𝜑11
𝜑1
211−12=𝜉
0. ()
Completing the above integral, we obtain the following soli-
tary wave solution (see Figure (a))ofthesystem():
3()=289
9−8cosh 2/289+8+9.
()
Noting (), we obtain the following exact wave solutions
of the system () from ()
3,,
=22
22−4cosh 2/2+−+,()
where 3(,,)is a solitary wave solution of the system ().
Since () = ∫(),integrating() about ,wecan
obtain
3()=−22tanh−1 −+22−4tanh 2/4
22.
()
According to (), one of the smooth wave solutions of () is
able to obtain
3,,
=−22tanh−1 −+22−4
×tanh 2+−
4
×22−1 .
()
Substitute () into the potential (,,) = 𝑥(,,)
as dened in () to obtain
3,,=23−22−4
×52+42−4+42+16
×cosh 2+
42
−42−4+5216−1 ,
()
where 3(,,)is a solitary wave solution of the system ().
Applying (), the solitary wave solution 3()is obtained for
the system () (see Figure (b)).
 Abstract and Applied Analysis
1234
0−1−2−3
−4
y
𝜉
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) 𝜑3(𝜉)
12340−1−2−3−4
y
𝜉
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(b) 𝑤3(𝜉)
F : e solitary wave of the system () and the system () when 2<<43/3,>0,>0,and≥0.
Remark 6. In [], only one solitary wave solution of peak
type wave solutions of the system () is obtained. However,
the solitary wave solutions of system () cannot be found
by Hirota’s bilinear method []. Fortunately, we obtain all
solitary wave solutions of the system () and the system ().
() Kink (antikink) wave solutions for the system () and
the system ().
When =−2
(see Figure (d)), in (,)-plane, the
expressions of the heteroclinic orbits Γ10 and Γ11 passing the
points (13,0),(0,0)are given as
=±2
3+2
23+1
24,()
when (,)=0,where13 =−/2<0.
Substituting () into / = and integrating them
along Γ10 and Γ11,wehave
𝜑
𝜑13 1
2+/22= 2
2𝜉
0, (a)
𝜑
𝜑13 1
2+/22= 2
2𝜉
0. (b)
Completing the above integral (a),weobtainthefollowing
kink wave solution of the system () (see Figure (a)):
4()=−1tanh 2/22
22.()
Noting (),weobtaintheexactwavesolutionofthesystem
() from (). Consider
4,,=1tanh 2+−/22
22,
()
where 4(,,)is a kink wave solution of the system ().
Since () = ∫(),integrating() about ,wecan
obtain
4()
=−1
22+ 2
4
×ln 1+tanh 2
22
+2
4ln 1tanh 2
22.
()
According to (), kink wave solutions of () are able to obtain
4,,
=−1
22+−
+2
4ln −tanh 2+−
22−1
+2
4ln 1 − tanh 2+−
22.
()
Abstract and Applied Analysis 
240−2−4
y
𝜉
−0.5
−1
−1.5
−2
(a) 𝜑4(𝜉) = −(𝑘𝑐(1 − tanh(2𝑘𝑐𝜉/2𝑘2)))/2𝑘2
240−2−4
y
𝜉
−0.5
−1
−1.5
−2
(b) 𝑤4(𝜉) = 𝑘𝑐(−1 + tanh(2𝑘𝑐𝜉/2𝑘2))/2𝑘
240−2−4
y
𝜉
−0.5
−1
−1.5
−2
(c) 𝜑4󸀠(𝜉) = 𝑘𝑐(1 + tanh(2𝑘𝑐𝜉/2𝑘2))/ − 2𝑘2
240−2−4
y
𝜉
−0.5
−1
−1.5
−2
(d) 𝑤4󸀠(𝜉) = 𝑘𝑐(1 + tanh(2𝑘𝑐𝜉/2𝑘2))/ − 2𝑘
F : e kink (antikink) wave solutions of the system () and the system () when =−2,>0,>0,and≥0.
Using t he potential (), the kink wave solution for the
system () are obtained as follows:
4,,=1tanh 2+−/22
2 ,
()
where 4(,,)is the smooth kink wave solution of the
system ().esmoothkinkwavesolution4()of the sys-
tem () is obtained from () (see Figure (b)). Analogously,
completing the above integral (b),wehavethefollowing
antikink wave solution
4󸀠()=−1+tanh 2/22
22,()
4󸀠,,=1+tanh 2+/22
2 ,
()
for the system () and the system (),respectively(see
Figures (c) and (d)).
When =2
 (see Figure (i)), in (,)-plane, the
expressions of the heteroclinic orbits Γ13 and Γ14 passing the
points (0,0),(14,0)are given as
=±2
3−2
23+1
24,()
when (,)=0,14 =/2>0.
 Abstract and Applied Analysis
Substituting () into / = and integrating them
along Γ13 and Γ14,wehave
𝜑14
𝜑2
2/22=𝜉
0, (a)
𝜑14
𝜑2
2/22=𝜉
0. (b)
Completing the above integral (a),weobtaintheexactkink
wave solution of the system () (see Figure (a)):
5()=1+tanh 2/22
22.()
Noting (), we obtain the following exact wave solution of the
system () from ():
5,,
=1+tanh +/2
22,()
where 5(,,) is a kink wave solution of the system
().
Since () = ∫(),integrating() about ,wecan
obtain
5()=
22−2
4
×ln tanh 
2 −1
2
4ln 1+tanh 2
22.
()
According to (), the traveling wave solution of () is able to
obtain
5,,
=
22+−
2
4ln tanh +−
2 −1
2
4ln 1 + tanh 2+−
22.
()
Applying the potential (), the kink wave solution of the
system () is obtained as follows:
5,,
=1+tanh +/2
2 ,()
where 5(,,)is the smooth kink wave solution of the
system ().
Noting (), we can obtain the smooth kink wave solution
5()of the system () (see Figure (b)). Analogously, com-
pleting the above integral (b), we obtain the exact antikink
wave solution:
5󸀠()=1tanh 2/22
22,
5󸀠()=1tanh ()/2
2 ,
()
for the system () and the system (),respectively(see
Figures (c) and (d)).
Remark 7. In [],applyingthenecessaryconditionforthe
kink waves to exist, multiple kink solutions and multiple
singular kink solutions of the system () are formally derived.
By the special traveling wave transform in [], no kink
(antikink) solutions of the system () and system () are
obtained. In this paper, not considering the necessary con-
dition for the kink waves to exist [], we obtain all kink and
antikink wave solutions of the system () and the system ()
by the bifurcation method of dynamical systems.
From () to (), we can obtain three theorems about
the exact periodic wave solutions and smooth solitary wave
solutions and kink (antikink) wave solutions for the system
() and the system ().
eorem 8. When conditions ()or ()hold, one can obtain
some representative smooth exact periodic wave solutions of the
system (11) from the periodic obits (see Figures 2(a),2(c),2(i),
2(j),and2(l))andFigures3(a) and 3(b)), which correspond
to the representative smooth exact periodic wave solutions of
the system (3), such as the periodic wave solution 1(,,)
of the system (11) corresponding to the periodic wave solution
1(,,)of the system (3).
eorem 9. Under conditions (), the following results hold.
() When −43/3 <  < −2,thesystem(11)
has an exact solitary wave solution 2(,,),which
corresponds to the solitary wave solutions 2(,,)of
the system (3).
() When 2<<43/3,thesystem(11) has exact
solitary wave solutions 3(,,), which correspond to
the solitary wave solutions 3(,,)of the system (3).
eorem 10. Under conditions (), the following results hold.
() When =−2,thesystem(11) has an exact smooth
kink wave solution 4(,,), which corresponds to
Abstract and Applied Analysis 
240−2−4
y
𝜉
0.5
1
1.5
2
(a) 𝜑5(𝜉) = 𝑘𝑐(1 + tanh(2𝑘𝑐𝜉/2𝑘2))/2𝑘2
240−2−4
y
𝜉
0.5
1
1.5
2
(b) 𝑤5(𝜉) = 𝑘𝑐(1 + tanh(𝑘𝑐𝜉/2𝑘))/2𝑘
240−2−4
y
𝜉
0.5
1
1.5
2
(c) 𝜑5󸀠(𝜉) = 𝑘𝑐(1 − tanh(2𝑘𝑐𝜉/2𝑘2))/2𝑘2
240−2−4
y
𝜉
0.5
1
1.5
2
(d) 𝑤5󸀠(𝜉) = 𝑘𝑐(1 − tanh(𝑘𝑐𝜉/2𝑘))/2𝑘
F : e kink (antikink) wave solutions of system () and () when =2,>0,>0,and≥0.
the kink wave solution 4(,,)of the system (3),
respectively.
() When =2
,thesystem(11) has an exact smooth
kink wave solution 5(,,), which corresponds to the
kink wave solution 5(,,)of the system (3).
5. Discussion
In this work we obtain all wave solutions of the complete
integrability of the (+)-dimensional nonlinear evolution
equation, the third model, by dynamical systems method.
is method can be used for the remaining three models.
By determining the necessary condition for the complete
integrability of these models in [], multiple kink solutions
and multiple singular kink solutions were formally derived
for the third model. Compared to the method in [], the
necessary condition which is among the coecients of the
spatial variables is not necessary in our method. In [], only
solitary wave solutions are obtained by a special traveling
wave transform. To our knowledge, we completely obtain all
solitary wave solutions and kink (antikink) wave solutions
for these models by the bifurcation method of dynamical
system. e idea of applying the method of dynamical system
toresearchthecompleteintegrabilitycanbeusedforother
models. is will be examined in forthcoming works. For the
remaining three models, we can follow the same approach to
derive all wave solutions for them.
 Abstract and Applied Analysis
Conflict of Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
e authors gratefully acknowledge the support of the
National Natural Science Foundation of China through Grant
nos. , , and , the Natural Science
Foundation of Beijing through Grant no. , the Inter-
national Science and Technology Cooperation Program of
China through Grant no. DFR, and the Research
Fund for the Doctoral Program of Higher Education of China
through Grant no. . All authors wish to thank
professorLiJibinformanyvaluablesuggestionsleadingtothe
improvement of this paper.
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... Bifurcation Method to Analysis of Traveling Wave Solutions 305 Four (2+1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy [1][2][3][4] are extended in [5], which are ...
... The system(1) is completely integrable evolution equations. There are many methods to be used in travel wave soutions of nonlinear evolution equations, such as the inverse scattering method, the Bäcklund transformation method, algebraic-geometric method, the Darboux transformation method, multiple exp-function method [6], the Hirota bilinear method [1-3, 5, 7-9] and dynamical systems method [4,[10][11][12]. The Hirota bilinear method is used to formally derive the multiple kink solutions and multiple singular kink solutions of the (2+1)-dimensional nonlinear models [1], and multiple soliton solutions for the system (1) [5]. ...
... The Hirota bilinear method is used to formally derive the multiple kink solutions and multiple singular kink solutions of the (2+1)-dimensional nonlinear models [1], and multiple soliton solutions for the system (1) [5]. By the bifurcation method of the dynamical systems, some new exact solutions of the (2+1)-dimensional nonlinear models are obtained in [4]. In this paper, we will study the third model given ...
... where α is constant [1] . The traveling wave system of (1.1) is derived [5] . Wazwaz [1] has obtained by Hirota bilinear method multiple soliton solutions which were formally derived. ...
... Wazwaz [1] has obtained by Hirota bilinear method multiple soliton solutions which were formally derived. In this paper, we research the travel wave solutions of (1.1) by bifucation method of dynamical systems [3,5] . ...
... The traveling wave solutions of (1.1) corresponding to peridioc wave solutions and solitary wave solutions, have been found in [5] completely. In this paper, we will research other traveling wave solutions which closely related to kink wave solutions of (1.2). ...
... where α is constant [1] . The traveling wave system of (1.1) is derived [5] . Wazwaz [1] has obtained by Hirota bilinear method multiple soliton solutions which were formally derived. ...
... Wazwaz [1] has obtained by Hirota bilinear method multiple soliton solutions which were formally derived. In this paper, we research the travel wave solutions of (1.1) by bifucation method of dynamical systems [3,5] . ...
... .2) has a valley type solitary wave solution [5] 1 ...
... Multiple kink solutions, multiple singular kink solutions and multiple soliton solutions were formally derived [1] and the exact traveling wave solutions of (1.1) have been obtained [7] .To our knowledge the study for the exact traveling wave solutions of (1.4) in different subsets of 4-parameters space of the system (1.7), has not been considered yet. By using the method of the dynamical systems, we shall generally investigate wave solutions of (1.4) in this paper. ...
... , using the information of the phase portraits in [7] , we analyze all the possible travel wave solutions of the system (1.4), which correspond to periodic travel wave solutions of the system (1.1). Some explicit parametric representations of traveling wave solutions of (1.4) are obtained. ...
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