Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 869438, 19 pages
http://dx.doi.org/10.1155/2013/869438
Research Article
Some Explicit Expressions and Interesting
Bifurcation Phenomena for Nonlinear Waves in
Generalized Zakharov Equations
Shaoyong Li1,2 and Rui Liu1
1
2
Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
College of Mathematics and Information Sciences, Shaoguan University, Shaoguan, Guangdong 512005, China
Correspondence should be addressed to Rui Liu; scliurui@scut.edu.cn
Received 22 December 2012; Revised 7 February 2013; Accepted 17 February 2013
Academic Editor: Chun-Lei Tang
Copyright © 2013 S. Li and R. Liu. 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.
Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations 𝑢𝑡𝑡 −
𝑐𝑠2 𝑢𝑥𝑥 = 𝛽(|𝐸|2 )𝑥𝑥 , 𝑖𝐸𝑡 +𝛼𝐸𝑥𝑥 −𝛿1 𝑢𝐸+𝛿2 |𝐸|2 𝐸+𝛿3 |𝐸|4 𝐸 = 0, where 𝛼, 𝛽, 𝛿1 , 𝛿2 , 𝛿3 , and 𝑐𝑠 are real parameters, 𝐸 = 𝐸(𝑥, 𝑡) is a complex
function, and 𝑢 = 𝑢(𝑥, 𝑡) is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear
waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under
different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves,
symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of
kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena
are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The
second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric
solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated
from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric
solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that
the exp-function expressions include some results given by pioneers.
1. Introduction
Since the exact solutions to nonlinear wave equations help to
understand the characteristics of nonlinear equations, seeking exact solutions of nonlinear equations is an important
subject. For this purpose, there have been many methods
such as the Jacobi elliptic function method [1, 2], F-expansion
method [3, 4], and (𝐺󸀠 /𝐺)-expansion method [5, 6].
Recently, the bifurcation method of dynamical systems
[7–9] has been introduced to study the nonlinear partial
differential equations. Up to now, the method is widely used
in literatures such as [10–16].
In this paper, we consider the generalized Zakharov
equations [17], which read as
𝑢𝑡𝑡 − 𝑐𝑠2 𝑢𝑥𝑥 = 𝛽(|𝐸|2 )𝑥𝑥 ,
𝑖𝐸𝑡 + 𝛼𝐸𝑥𝑥 − 𝛿1 𝑢𝐸 + 𝛿2 |𝐸|2 𝐸 + 𝛿3 |𝐸|4 𝐸 = 0,
(1)
where 𝛼, 𝛽, 𝛿1 , 𝛿2 , 𝛿3 , and 𝑐𝑠 are real parameters. 𝐸 = 𝐸(𝑥, 𝑡)
is a complex function which represents the envelop of the
electric field, and 𝑢 = 𝑢(𝑥, 𝑡) is a real function which
represents the plasma density measured from its equilibrium
value. Huang and Zhang [17] used Fan’s direct algebraic
method to obtain some exact travelling wave solutions of (1)
as follows:
±
=
𝑢𝑎0
𝛽
2
(𝜑± ) ,
𝑐2 − 𝑐𝑠2 𝑎0
±
± 𝑖(𝛾𝑥−𝜔𝑡)
𝐸𝑎0
= 𝜑𝑎0
𝑒
,
±
=
𝑢𝑏0
𝛽
2
(𝜑± ) ,
𝑐2 − 𝑐𝑠2 𝑏0
±
± 𝑖(𝛾𝑥−𝜔𝑡)
𝐸𝑏0
= 𝜑𝑏0
𝑒
,
±
=
𝑢𝑐0
𝛽
2
(𝜑± ) ,
𝑐2 − 𝑐𝑠2 𝑐0
±
± 𝑖(𝛾𝑥−𝜔𝑡)
𝐸𝑐0
= 𝜑𝑐0
𝑒
,
2
Abstract and Applied Analysis
𝛽
2
(𝜑± ) ,
𝑐2 − 𝑐𝑠2 𝑑0
±
=
𝑢𝑑0
±
𝑢𝑒0
=
𝑐2
±
± 𝑖(𝛾𝑥−𝜔𝑡)
𝐸𝑑0
= 𝜑𝑑0
𝑒
,
𝛽
± 2
(𝜑𝑒0
),
2
− 𝑐𝑠
±
± 𝑖(𝛾𝑥−𝜔𝑡)
𝐸𝑒0
= 𝜑𝑒0
𝑒
,
(2)
where 𝛾 and 𝑤 are two constants and
±
𝜑𝑎0
= ±√
−12𝑝
3𝑞 −
√9𝑞2
− 48𝑝𝑟 cos (2√−𝑝𝜉)
,
(3)
±
= ±√
𝜑𝑏0
2𝑝
(−1 + tanh (√𝑝𝜉)),
𝑞
(4)
±
𝜑𝑐0
= ±√
2𝑝
(−1 − tanh (√𝑝𝜉)),
𝑞
(5)
±
= ±√
𝜑𝑑0
12𝑝
−3𝑞 + √9𝑞2 − 48𝑝𝑟 cosh (2√𝑝𝜉)
±
= ±√
𝜑𝑒0
2𝑝
(−1 − coth (√𝑝𝜉)),
𝑞
𝜉 = 𝑥 − 𝑐𝑡,
𝑝=
𝛼𝛾2 − 𝜔
,
𝛼
𝛽𝛿1
𝛿
− 2,
𝑞=
2
2
𝛼 (𝑐 − 𝑐𝑠 ) 𝛼
,
(6)
(7)
(8)
(9)
(10)
𝛿3
,
𝛼
(11)
𝑐 = 2𝛼𝛾.
(12)
𝑟=−
When 𝛼 = 1, 𝛽 = −1, 𝛿1 = −2, 𝛿2 = 2𝜆, 𝛿3 = 0, and 𝑐𝑠 =
1, (1) reduce to the equations
𝑢𝑡𝑡 − 𝑢𝑥𝑥 + (|𝐸|2 )𝑥𝑥 = 0,
𝑖𝐸𝑡 + 𝐸𝑥𝑥 + 2𝑢𝐸 + 2𝜆|𝐸|2 𝐸 = 0.
𝑖𝐸𝑡 + 𝐸𝑥𝑥 + 2𝑢𝐸 = 0.
2. Main Results
In this section, we state our main results. To relate conveniently, let us give some notations which will be used in the
latter statement and the derivations.
Let 𝑙𝑖 (𝑖 = 1, 2, . . . , 7) represent the following seven curves:
𝑙1 : 𝑝 =
𝑞2
4𝑟
(𝑟 > 0, 𝑞 < 0) ,
(15)
𝑙2 : 𝑝 =
3𝑞2
16𝑟
(𝑟 > 0, 𝑞 < 0) ,
(16)
𝑙3 : 𝑝 = 0
(𝑟 > 0, 𝑞 < 0) ,
(17)
𝑙4 : 𝑝 = 0
(𝑟 > 0, 𝑞 > 0) ,
(18)
𝑙5 : 𝑝 = 0
(𝑟 < 0, 𝑞 > 0) ,
(19)
𝑙6 : 𝑝 =
3𝑞2
16𝑟
(𝑟 < 0, 𝑞 > 0) ,
(20)
𝑙7 : 𝑝 =
𝑞2
4𝑟
(𝑟 < 0, 𝑞 < 0) .
(21)
(13)
El-Wakil et al. [18] used the extended Jacobi elliptic function
expansion method to obtain some Jacobi elliptic function
expression solutions of (13).
When 𝜆 = 0, (13) reduce to the equations
𝑢𝑡𝑡 − 𝑢𝑥𝑥 + (|𝐸|2 )𝑥𝑥 = 0,
In this paper, we investigate the nonlinear waves and the
bifurcation phenomena of (1). Coincidentally, under some
transformations, (1) reduce to a planar system (54) which is
similar to the planar system obtained by Feng and Li [16].
Many exact explicit parametric representations of solitary
waves, kink and antikink waves, and periodic waves were
obtained in [16], and their work is very important for the
𝜙6 model. In order to find the travelling wave solutions of
(1), here we consider (1) by using the bifurcation method
mentioned above; firstly, we obtain three types of explicit
nonlinear wave solutions, this is, the fractional expressions,
the trigonometric expressions, and the exp-function expressions. Secondly, we point out that these expressions represent symmetric and antisymmetric solitary waves, kink and
antikink waves, symmetric periodic and periodic-blow-up
waves, and 1-blow-up and 2-blow-up waves under different
parameter conditions. Thirdly, we reveal five kinds of interesting bifurcation phenomena mentioned in the abstract above.
The remainder of this paper is organized as follows. In
Section 2, we give some notations and state our main results.
In Section 3, we give derivations for our results. A brief
conclusion is given in Section 4.
(14)
By multisymplectic numerical method, Wang [19] proved
the preservation of discrete normal conservation law of (14)
theoretically and investigated the propagation and collision
behaviors of the solitary waves numerically. There are also
many other researchers studying (1) or its special case; for
more information, one can see [20–24].
Let 𝐴 𝑖 (𝑖 = 1, 2, . . . , 12) represent the regions surrounded
by the curves 𝑙𝑖 (𝑖 = 1, 2, . . . , 7) and the coordinate axes (see
Figure 1).
Let
𝛽
𝜑2 (𝜉) ,
𝑐2 − 𝑐𝑠2
(22)
𝐸 (𝑥, 𝑡) = 𝜑 (𝜉) 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
(23)
Δ = √𝑞2 − 4𝑝𝑟,
(24)
−𝑞 + Δ
,
2𝑟
(25)
𝑢 (𝑥, 𝑡) =
𝜑1 = √
Abstract and Applied Analysis
3
𝑝
𝑝
𝑙1
𝐴7
𝐴8
𝑙2
𝑂
𝐴5
𝐴4
𝑙5
𝐴 12
𝐴6
𝐴9
𝐴 11
𝐴 10
𝐴3
𝑙3
𝑞
𝑙4
𝑂
𝐴2
𝑙6
𝑞
𝐴1
𝑙7
(a) 𝑟 > 0
(b) 𝑟 < 0
Figure 1: The locations of the regions 𝐴 𝑖 (𝑖 = 1, 2, . . . , 12) and curves 𝑙𝑖 (𝑖 = 1, 2, . . . , 7).
𝜑2 = √
−𝑞 − Δ
,
2𝑟
(26)
𝐻 (𝜑, 𝑦) = ℎ,
(27)
where 𝐻(𝜑, 𝑦) is the first integral which will be given later
and ℎ is the integral constant.
In order to search for the solutions of (1) and studing the
bifurcation phenomena, we only need to get the solution 𝜑(𝜉)
according to (22) and (23). For convenience, throughout the
following work we only discuss the solution 𝜑(𝜉). Now let us
state our main results in the following Propositions 1, 2, and
3.
2.1. When the Orbit Γ Is Defined by 𝐻(𝜑,𝑦)=𝐻(0, 0)
Proposition 1. (i) For 𝑝 = 0, (1) have two fractional nonlinear
wave solutions
𝛽
2
𝐸𝑎± = 𝜑𝑎± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
𝑢𝑎± = 2 2 (𝜑𝑎± ) ,
(28)
𝑐 − 𝑐𝑠
These solutions have the following properties and wave shapes.
±
and 𝜑𝑏± are periodic(1) When (𝑞, 𝑝) ∈ 𝐴 1 or 𝐴 2 , then 𝜑𝑎0
blow-up waves (refer to Figure 2(a) or Figure 3(a)).
Specially, in region 𝐴 1 when 𝑝 → 0 − 0, the periodic±
become 1-blow-up waves 𝜑𝑎± , and
blow-up waves 𝜑𝑎0
for the varying process, see Figure 2, while the periodicblow-up waves 𝜑𝑏± become a trivial wave 𝜑 = 0. In
region 𝐴 2 when 𝑝 → 0−0, the periodic-blow-up waves
𝜑𝑏± become 2-blow-up waves 𝜑𝑎± , and for the varying
process, see Figure 3.
±
(2) When (𝑞, 𝑝) ∈ 𝐴 12 , then 𝜑𝑎0
and 𝜑𝑏± are symmetric
periodic waves (refer to Figure 5(a)). If 𝑝 → 0 − 0,
±
become symmetric
the symmetric periodic waves 𝜑𝑎0
±
solitary waves 𝜑𝑎 , and for the varying process, see
Figure 4. If 𝑝 → 3𝑞2 /16𝑟 + 0, the symmetric periodic
±
and 𝜑𝑏± become two trivial waves 𝜑 =
waves 𝜑𝑎0
±√−(3𝑞/4𝑟), and for the varying process, see Figure 5.
(iii) For 𝑝 > 0, (1) have four nonlinear wave solutions
where
6𝑞
𝜑𝑎± = ±√ 2 2
.
3𝑞 𝜉 − 4𝑟
(29)
If (𝑞, 𝑝) ∈ 𝑙4 , then 𝜑𝑎± are 1-blow-up waves (refer to Figure 2(d)).
If (𝑞, 𝑝) ∈ 𝑙3 , then 𝜑𝑎± are 2-blow-up waves (refer to
Figure 3(d)). If (𝑞, 𝑝) ∈ 𝑙5 , then 𝜑𝑎± are symmetric solitary
waves (refer to Figure 4(d)).
(ii) For 𝑝 < 0, (1) have two nonlinear wave solutions
𝑢𝑏± =
𝑐2
𝛽
2
(𝜑𝑏± ) ,
2
− 𝑐𝑠
𝐸𝑏± = 𝜑𝑏± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
(30)
where
𝜑𝑏± = ±√
−12𝑝
3𝑞 + √9𝑞2 − 48𝑝𝑟 cos (2√−𝑝𝜉)
𝑢𝑐± =
.
(31)
𝑢𝑑±
𝛽
2
(𝜑± ) ,
𝑐2 − 𝑐𝑠2 𝑐
𝛽
2
= 2 2 (𝜑𝑑± ) ,
𝑐 − 𝑐𝑠
𝐸𝑐± = 𝜑𝑐± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
(32)
𝐸𝑑±
=
𝜑𝑑± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
where
𝜑𝑐± = ±√
4𝜆𝑝
,
𝜆2 𝑒2√𝑝𝜉 + (𝜆 0 /12) 𝑒−2√𝑝𝜉 − 𝜆𝑞
(33)
4𝜆𝑝
,
𝜑𝑑± = ±√ 2 −2√𝑝𝜉
𝜆𝑒
+ (𝜆 0 /12) 𝑒2√𝑝𝜉 − 𝜆𝑞
𝜆 is a nonzero arbitrary real constant and 𝜆 0 = 3𝑞2 − 16𝑝𝑟.
Corresponding to 𝜆 > 0 or 𝜆 < 0, these solutions have the
following properties and wave shapes.
4
Abstract and Applied Analysis
𝜑
+
𝜑𝑎0
𝜑
+
𝜑𝑎0
0
+
𝜑𝑎0
0
𝜉
−
𝜑𝑎0
−
𝜑𝑎0
−
𝜑𝑎0
(b)
𝜑
𝜑
+
𝜑𝑎0
+
𝜑𝑎0
0
−
𝜑𝑎0
𝜉
−
𝜑𝑎0
(a)
+
𝜑𝑎0
+
𝜑𝑎0
+
𝜑𝑎0
0
𝜉
𝜉
−
𝜑𝑎0
−
𝜑𝑎0
(c)
−
𝜑𝑎0
(d)
±
Figure 2: (The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of 𝜑𝑎0
when (𝑞, 𝑝) ∈ 𝐴 1
−1
−2
−3
−4
and 𝑝 → 0 − 0, where 𝑟 = 1, 𝑞 = 4 and (a) 𝑝 = 0 − 10 , (b) 𝑝 = 0 − 10 , (c) 𝑝 = 0 − 10 , and (d) 𝑝 = 0 − 10 .
(1) For the case of 𝜆 > 0, there are four properties as
follows.
(1)a When (𝑞, 𝑝) ∈ 𝑙2 , that is, 𝑟 > 0, 𝑞 < 0, and
𝑟 = 3𝑞2 /16𝑝, then 𝜑𝑐± and 𝜑𝑑± become
±
= ±√
𝜑𝑐1
4𝑝
𝜆𝑒2√𝑝𝜉
−𝑞
,
(34)
±
= ±√
𝜑𝑑1
4𝑝
𝜆𝑒−2√𝑝𝜉 − 𝑞
,
which represent four low-kink waves (refer to
Figure 6(d) or Figure 7(d)). Specially, let 𝜆 =
±
±
±
±
and 𝜑𝑑1
become 𝜑𝑏0
and 𝜑𝑐0
.
−𝑞 > 0, then 𝜑𝑐1
(1)b If (𝑞, 𝑝) belongs to one of 𝐴 3 , 𝐴 7 , 𝐴 8 , and
𝜆 ≠ √𝜆 0 /12, then 𝜑𝑐± ≠ 𝜑𝑑± , and they represent four symmetric solitary waves (refer to
Figure 6(a)). Specially, when (𝑞, 𝑝) ∈ 𝐴 3 and
𝑝 → 3𝑞2 /16𝑟 − 0, then the four symmetric
solitary waves 𝜑𝑐± and 𝜑𝑑± become the four low±
±
and 𝜑𝑑1
, and for the varying
kink waves 𝜑𝑐1
process, see Figure 6.
(1)c If (𝑞, 𝑝) belongs to one of 𝐴 4 , 𝐴 5 , 𝐴 6 , 𝑙1 , and
𝜆 ≠ √𝜆 0 /12, then 𝜑𝑐± ≠ 𝜑𝑑± , and they represent
four 1-blow-up waves (refer to Figure 7(a)). Specially, when (𝑞, 𝑝) ∈ 𝐴 4 and 𝑝 → 3𝑞2 /16𝑟 + 0,
then the four 1-blow-up waves 𝜑𝑐± and 𝜑𝑑± become
±
±
and 𝜑𝑑1
, and for the
the four low-kink waves 𝜑𝑐1
varying process, see Figure 7.
(1)d If (𝑞, 𝑝) belongs to one of 𝐴 3 , 𝐴 7 , 𝐴 8 , and 𝜆 =
±
√𝜆 0 /12, then 𝜑𝑐± = 𝜑𝑑± = 𝜑𝑑0
. Specially, when
2
±
tend
(𝑞, 𝑝) ∈ 𝐴 3 and 𝑝 → 3𝑞 /16𝑟 − 0, then 𝜑𝑑0
to two trivial solutions 𝜑 = ±√−(3𝑞/4𝑟).
(2) For the case of 𝜆 < 0, there are three properties as
follows.
(2)a If (𝑞, 𝑝) ∈ 𝑙2 , that is, 𝑟 > 0, 𝑞 < 0, and
±
𝑟 = 3𝑞2 /16𝑝, then 𝜑𝑐± and 𝜑𝑑± become 𝜑𝑐1
and
±
𝜑𝑑1 which represent four 1-blow-up waves (refer
±
to Figure 8(d)). Specially, let 𝜆 = 𝑞 < 0, then 𝜑𝑐1
±
and 𝜑𝑑1 become the hyperbolic 1-blow-up wave
±
and
solutions 𝜑𝑒0
±
= ±√
𝜑𝑑2
2𝑝
(−1 + coth (√𝑝𝜉)).
𝑞
(35)
(2)b If (𝑞, 𝑝) ∈ 𝐴 3 and 𝜆 ≠ − √𝜆 0 /12, then 𝜑𝑐± ≠ 𝜑𝑑± ,
and they represent four 2-blow-up waves (refer to
Figure 8(a)). Specially, when 𝑝 → 3𝑞2 /16𝑟 − 0,
then the four 2-blow-up waves become four 1±
±
and 𝜑𝑑1
, and for the varying
blow-up waves 𝜑𝑐1
process, see Figure 8.
Abstract and Applied Analysis
5
𝜑
𝜑
𝜑𝑏+
𝜑𝑏+
0
0
𝜉
𝜑𝑏−
𝜉
𝜑𝑏−
𝜑𝑏−
(a)
𝜑𝑏−
(b)
𝜑
𝜑𝑏+
𝜑𝑏+
𝜑𝑏+
𝜑
𝜑𝑏+
0
𝜑𝑏+
0
𝜉
𝜑𝑏−
𝜉
𝜑𝑏−
𝜑𝑏−
(c)
(d)
Figure 3: (The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of 𝜑𝑏± when (𝑞, 𝑝) ∈ 𝐴 2
and 𝑝 → 0 − 0, where 𝑟 = 1, 𝑞 = −4 and (a) 𝑝 = 0 − 1, (b) 𝑝 = 0 − 0.5, (c) 𝑝 = 0 − 0.1, and (d) 𝑝 = 0 − 0.01.
(2)c If (𝑞, 𝑝) ∈ 𝐴 3 and 𝜆 = −√𝜆 0 /12, then 𝜑𝑐± =
±
of forms
𝜑𝑑± = 𝜑𝑐𝑑
where
𝜑𝑒± = ±
±
𝜑𝑐𝑑
= ±√
12𝑝
−3𝑞 −
√9𝑞2
− 48𝑝𝑟 cosh (2√𝑝𝜉)
,
2
𝛽
= 2 2 (𝜑𝑓± ) ,
𝑐 − 𝑐𝑠
𝐸𝑒± = 𝜑𝑒± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
(37)
𝐸𝑓±
=
𝜑𝑓± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
,
√((Δ − 𝑞) /2𝑟) (2Δ + 𝑞 − 2𝜇𝑟𝑒
)
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒−𝜃𝜉 + 4𝜇2 𝑟2 𝑒−2𝜃𝜉
,
𝜇 is a nonzero arbitrary real constant, Δ is given in (24), and
1
𝜃 = √ Δ (Δ − 𝑞).
𝑟
(39)
Let
Proposition 2. If (𝑞, 𝑝) belongs to one of the regions
𝐴 1 , 𝐴 2 , 𝐴 3 , 𝐴 4 , 𝐴 11 , and 𝐴 12 or curves 𝑙1 , 𝑙2 , 𝑙3 , 𝑙6 , and 𝑙7 , then
(1) have four nonlinear wave solutions
𝑢𝑓±
−𝜃𝜉
𝜑𝑓± = ±
2.2. When the Orbit Γ Is Defined by 𝐻(𝜑,𝑦)=𝐻(𝜑1 ,0)
𝛽
2
𝑢𝑒± = 2 2 (𝜑𝑒± ) ,
𝑐 − 𝑐𝑠
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒𝜃𝜉 + 4𝜇2 𝑟2 𝑒2𝜃𝜉
(38)
(36)
which represent hyperbolic blow-up waves. When
±
𝑝 → 3𝑞2 /16𝑟 − 0, then 𝜑𝑐𝑑
tend to two trivial
solutions 𝜑 = ±√−(3𝑞/4𝑟).
√((Δ − 𝑞) /2𝑟) (2Δ + 𝑞 − 2𝜇𝑟𝑒𝜃𝜉 )
𝜇0 =
𝑞 + 2Δ
.
2𝑟
(40)
About 𝜇0 , one has the following fact:
> 0 for (𝑞, 𝑝) ∈ 𝐴 1 , 𝐴 2 , 𝐴 3 , 𝑙3 ,
{
{
𝜇0 {= 0 for (𝑞, 𝑝) ∈ 𝑙2 ,
{
{< 0 for (𝑞, 𝑝) ∈ 𝐴 4 , 𝐴 11 , 𝐴 12 , 𝑙1 , 𝑙6 , 𝑙7 .
(41)
Corresponding to 𝜇 > 0 and 𝜇 < 0, these solutions have the
following properties and wave shapes.
6
Abstract and Applied Analysis
𝜑
𝜑
+
𝜑𝑎0
+
𝜑𝑎0
0
0
𝜉
−
𝜑𝑎0
+
𝜑𝑎0
+
𝜑𝑎0
𝜉
−
𝜑𝑎0
−
𝜑𝑎0
−
𝜑𝑎0
(a)
(b)
𝜑
𝜑
+
𝜑𝑎0
+
𝜑𝑎0
+
𝜑𝑎0
0
−
𝜑𝑎0
0
𝜉
−
𝜑𝑎0
𝜉
−
𝜑𝑎0
(c)
(d)
±
Figure 4: (The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of 𝜑𝑎0
when
(𝑞, 𝑝) ∈ 𝐴 12 and 𝑝 → 0 − 0, where 𝑟 = −1, 𝑞 = 4 and (a) 𝑝 = 0 − 0.5, (b) 𝑝 = 0 − 0.1, (c) 𝑝 = 0 − 0.01, and (d) 𝑝 = 0 − 0.001.
(1o ) For the case of 𝜇 > 0, there are six properties as follows.
(1 )a If (𝑞, 𝑝) ∈ 𝑙2 , that is, 𝑟 > 0, 𝑞 < 0, and 𝑟 =
3𝑞2 /16𝑝, then 𝜑𝑒± and 𝜑𝑓± become
o
∓
= ∓√
𝜑𝑒1
∓
𝜑𝑓1
4𝜇𝑝
16𝑝𝑒−2√𝑝𝜉
= ∓√
− 𝜇𝑞
4𝜇𝑝
16𝑝𝑒2√𝑝𝜉 − 𝜇𝑞
,
,
(42)
(43)
which represent four low-kink waves (refer to
Figure 9(d)). Specially, let 𝜇 = −(16𝑝/𝑞) > 0,
∓
∓
∓
∓
= 𝜑𝑐0
and 𝜑𝑓1
= 𝜑𝑏0
.
then 𝜑𝑒1
(1o )b If (𝑞, 𝑝) ∈ 𝐴 1 , 𝐴 2 , 𝐴 3 , and 𝜇 ≠ |𝜇0 |, then 𝜑𝑒± ≠ 𝜑𝑓± ,
and they represent four tall-kink waves (refer to
Figure 9(a)). Specially, when (𝑞, 𝑝) ∈ 𝐴 3 and
∓
𝑝 → 3𝑞2 /16𝑟 − 0, then 𝜑𝑒± and 𝜑𝑓± become 𝜑𝑒1
∓
and 𝜑𝑓1
, and for the varying process, see Figure 9.
(1o )c If (𝑞, 𝑝) ∈ 𝐴 4 , 𝐴 11 , 𝐴 12 , 𝑙6 , and 𝜇 ≠ |𝜇0 |, then
𝜑𝑒± ≠ 𝜑𝑓± . When (𝑞, 𝑝) ∈ 𝐴 4 , they represent four antisymmetric solitary waves (refer to
Figure 10(a)). When (𝑞, 𝑝) ∈ 𝐴 11 , 𝐴 12 , and 𝑙6 ,
they represent four symmetric solitary waves
(refer to Figure 11(a)). Specially, when (𝑞, 𝑝) ∈
𝐴 4 , if 𝑝 → 3𝑞2 /16𝑟 + 0, then 𝜑𝑒± and 𝜑𝑓±
∓
∓
become 𝜑𝑒1
and 𝜑𝑓1
, and for the varying process,
see Figure 10. When (𝑞, 𝑝) ∈ 𝐴 11 and 𝑝 →
𝑞2 /4𝑟 + 0, then 𝜑𝑒± and 𝜑𝑓± tend to two trivial
solutions 𝜑 = ±√−(𝑞/2𝑟), and for the varying
process, see Figure 11.
o
(1 )d If (𝑞, 𝑝) ∈ 𝐴 3 and 𝜇 = |𝜇0 |, then 𝜑𝑒± = 𝜑𝑓∓ = 𝜑𝑔∓
of forms
𝜑𝑔∓ = ∓
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) sinh ((𝜃/2) 𝜉)
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
,
(44)
which represent two tall-kink waves and tend to a
trivial wave 𝜑 = 0 when 𝑝 → 3𝑞2 /16𝑟 − 0.
o
(1 )e If (𝑞, 𝑝) ∈ 𝐴 4 and 𝜇 = |𝜇0 |, then 𝜑𝑒± = 𝜑𝑓± = 𝜑ℎ∓
of forms
𝜑ℎ∓ = ∓
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) cosh ((𝜃/2) 𝜉)
√𝑞 − 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
,
(45)
which represent two antisymmetric solitary
waves, tend to a trivial wave 𝜑 = 0 when
𝑝 → 3𝑞2 /16𝑟 + 0, and tend to two trivial waves
𝜑 = ±√−(𝑞/2𝑟) when 𝑝 → 𝑞2 /4𝑟 − 0.
o
(1 )f If (𝑞, 𝑝) ∈ 𝐴 11 , 𝐴 12 , 𝑙6 and 𝜇 = |𝜇0 |, then 𝜑𝑒± =
𝜑𝑓± = 𝜑ℎ± which represent two symmetric solitary
waves. Specially, when (𝑞, 𝑝) ∈ 𝐴 11 and 𝑝 →
𝑞2 /4𝑟 − 0, then 𝜑ℎ± tend to two trivial waves 𝜑 =
±√−(𝑞/2𝑟).
Abstract and Applied Analysis
7
𝜑
𝜑
+
𝜑𝑎0
+
𝜑𝑎0
𝜑𝑏+
0
0
𝜉
−
𝜑𝑎0
𝜑𝑏+
𝜑𝑏−
𝜉
−
𝜑𝑎0
(a)
𝜑𝑏−
(b)
𝜑
𝜑
+
𝜑𝑎0
+
𝜑𝑎0
= 𝜑𝑏+
𝜑𝑏+
0
0
𝜉
−
𝜑𝑎0
𝜑𝑏−
𝜉
−
𝜑𝑎0
= 𝜑𝑏−
(c)
(d)
±
Figure 5: (The symmetric periodic waves become two trivial waves.) The varying process for the figures of 𝜑𝑎0
and 𝜑𝑏± when (𝑞, 𝑝) ∈ 𝐴 12 and
2
2
−1
−2
𝑝 → 3𝑞 /16𝑟 + 0, where 𝑟 = −1, 𝑞 = 4, and 𝑙6 : 𝑝 = 3𝑞 /16𝑟 = −3 and (a) 𝑝 = −3 + 10 , (b) 𝑝 = −3 + 10 , (c) 𝑝 = −3 + 10−3 , and (d)
𝑝 = −3 + 10−6 .
(2o ) For the case of 𝜇 < 0, there are five properties as follows.
±
±
(2o )a If (𝑞, 𝑝) ∈ 𝑙2 , then 𝜑𝑒1
and 𝜑𝑓1
represent four 1blow-up waves (refer to Figure 12(d)). Specially,
±
±
∓
= 𝜑𝑒0
and 𝜑𝑓± = 𝜑𝑑2
.
let 𝜇 = 16𝑝/𝑞 < 0, then 𝜑𝑒1
(2o )b If (𝑞, 𝑝) ∈ 𝐴 1 , 𝐴 2 , 𝐴 3 , 𝐴 4 and 𝜇 ≠ − |𝜇0 |,
then 𝜑𝑒± ≠ 𝜑𝑓± , and they represent four pairs
of 1-blow-up waves (refer to Figure 12(a)).
In particular, when (𝑞, 𝑝)
∈
𝐴 4,
if 𝑝 → 3𝑞2 /16𝑟 + 0, then the four pairs of
1-blow-up waves become two pairs of 1-blow-up
waves, and the varying process is displayed in
Figure 12. If 𝑝 → 𝑞2 /4𝑟 − 0, then the four pairs
of 1-blow-up waves become two trivial waves
𝜑 = ±√−(𝑞/2𝑟).
(2o )c If (𝑞, 𝑝) ∈ 𝐴 11 , 𝐴 12 , 𝑙6 and 𝜇 ≠ − |𝜇0 |, then
𝜑𝑒± ≠ 𝜑𝑓± , and they represent four tall-kink waves.
(2o )d If 𝜇 = −|𝜇0 |, when (𝑞, 𝑝) ∈ 𝐴 3 , then 𝜑𝑒± = 𝜑𝑓± =
𝜑ℎ∓ which represent two pairs of 1-blow-up waves.
When (𝑞, 𝑝) ∈ 𝐴 4 , then 𝜑𝑒± = 𝜑𝑓∓ = 𝜑𝑔∓ which
represent two pairs of 1-blow-up waves.
(2o )e If (𝑞, 𝑝) ∈ 𝐴 11 , 𝐴 12 , 𝑙6 and 𝜇 = −|𝜇0 |, then 𝜑𝑒± =
𝜑𝑓∓ = 𝜑𝑔∓ which represent two tall-kink waves.
2.3. When the Orbit Γ Is Defined by 𝐻(𝜑,𝑦) = 𝐻(𝜑2 , 0)
Proposition 3. (i) If (𝑞, 𝑝) belongs to one of 𝐴 3 , 𝐴 4 , 𝐴 11 , and
𝑙2 , then (1) have four nonlinear wave solutions
𝛽
2
𝑢𝑖± = 2 2 (𝜑𝑖± ) ,
𝐸𝑖± = 𝜑𝑖± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
𝑐 − 𝑐𝑠
(46)
2
𝛽
𝑢𝑗± = 2 2 (𝜑𝑗± ) ,
𝐸𝑗± = 𝜑𝑗± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
𝑐 − 𝑐𝑠
where
𝜑𝑖± = ±√
(𝑞 + Δ) (2Δ − 𝑞)
𝜂
cos ( 𝜉) ,
2
𝑟 (𝑞 + 4Δ + (𝑞 − 2Δ) cos (𝜂𝜉))
(47)
𝜑𝑗± = ±√
(𝑞 + Δ) (2Δ − 𝑞)
𝜂
sin ( 𝜉) ,
2
𝑟 (𝑞 + 4Δ − (𝑞 − 2Δ) cos (𝜂𝜉))
(48)
1
(49)
𝜂 = √ (4𝑝𝑟 − 𝑞 (𝑞 + Δ)).
𝑟
When (𝑞, 𝑝) ∈ 𝐴 3 , 𝐴 4 , or 𝑙2 , then 𝜑𝑖± and 𝜑𝑗± represent
periodic blow-up wave solutions (refer to Figure 13(a)). When
(𝑞, 𝑝) ∈ 𝐴 11 , then 𝜑𝑖± and 𝜑𝑗± represent periodic wave solutions
(refer to Figure 14(a)).
(ii) If (𝑞, 𝑝) ∈ 𝑙1 or 𝑙7 , then (1) have two fractional wave
solutions
𝛽
2
𝐸𝑘± = 𝜑𝑘± 𝑒𝑖(𝛾𝑥−𝜔𝑡) ,
𝑢𝑘± = 2 2 (𝜑𝑘± ) ,
(50)
𝑐 − 𝑐𝑠
8
Abstract and Applied Analysis
𝜑
𝜑
𝜑𝑐+
𝜑𝑑+
𝜑𝑐+
𝜑𝑑+
0
0
𝜉
𝜑𝑐−
𝜑𝑑−
𝜉
𝜑𝑐−
𝜑𝑑−
(a)
(b)
𝜑
𝜑
𝜑𝑐+
𝜑𝑐+
𝜑𝑑+
0
𝜑𝑑+
0
𝜉
𝜑𝑑−
𝜑𝑐−
𝜉
𝜑𝑐−
𝜑𝑑−
(c)
(d)
Figure 6: (The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of 𝜑𝑐± and 𝜑𝑑±
when (𝑞, 𝑝) ∈ 𝐴 3 , 𝜆 > 0, 𝜆 ≠ √𝜆 0 /12 and 𝑝 → 3𝑞2 /16𝑟 − 0, where 𝑟 = 1, 𝑞 = −4, 𝜆 = 4, and 𝑙2 : 𝑝 = 3𝑞2 /16𝑟 = 3 and (a) 𝑝 = 3 − 10−1 , (b)
𝑝 = 3 − 10−3 , (c) 𝑝 = 3 − 10−7 , and (d) 𝑝 = 3 − 10−9 .
where
𝜑𝑘± = ±√
−𝑞
𝑞𝜉
.
2𝑟 √𝑞2 𝜉2 − 12𝑟
(51)
𝑙1 , 𝜑𝑘±
represent two 1-blow-up waves (refer
When (𝑞, 𝑝) ∈
to Figure 13(d)). When (𝑞, 𝑝) ∈ 𝑙7 , 𝜑𝑘± represent two tall-kink
waves (refer to Figure 14(d)).
(iii) If (𝑞, 𝑝) ∈ 𝐴 4 and 𝑝 → 𝑞2 /4𝑟 − 0, then the
periodic blow-up waves 𝜑𝑖± tend to two trivial solutions 𝜑 =
±√−(𝑞/2𝑟), while 𝜑𝑗± tend to the 1-blow-up waves 𝜑𝑘± , and the
varying process is displayed in Figure 13. If (𝑞, 𝑝) ∈ 𝐴 11 and
𝑝 → 𝑞2 /4𝑟 + 0, then the periodic waves 𝜑𝑖± tend to two trivial
solutions 𝜑 = ±√−(𝑞/2𝑟), while 𝜑𝑗± tend to the tall-kink waves
𝜑𝑘± , and for the varying process, see Figure 14.
3. The Derivations of Main Results
To derive our results, substituting (23) and 𝑢(𝑥, 𝑡) = 𝑢(𝜉) with
𝜉 = 𝑥 − 𝑐𝑡 into (1), it follows that
󸀠󸀠
(𝑐2 − 𝑐𝑠2 ) 𝑢󸀠󸀠 = 𝛽(𝜑2 ) ,
𝛼𝜑󸀠󸀠 + 𝑖 (2𝛼𝛾 − 𝑐) 𝜑󸀠 + (𝑤 − 𝛼𝛾2 ) 𝜑
− 𝛿1 𝑢𝜑 + 𝛿2 𝜑3 + 𝛿3 𝜑5 = 0.
(52)
Integrating the first equation of (52) twice with respect
to 𝜉 and taking the integral constants to zero, we get (22).
Substituting (22) into the second equation of (52) and letting
𝑐 = 2𝛼𝛾, it follows that
𝜑󸀠󸀠 = 𝑝𝜑 + 𝑞𝜑3 + 𝑟𝜑5 .
(53)
Form (53) we obtain the planar system
𝑑𝜑
= 𝑦,
𝑑𝜉
𝑑𝑦
= 𝑝𝜑 + 𝑞𝜑3 + 𝑟𝜑5 ,
𝑑𝜉
(54)
with the first integral
𝑞
𝑟
𝐻 (𝜑, 𝑦) = 𝑦2 − (𝑝𝜑2 + 𝜑4 + 𝜑6 ) = ℎ.
2
3
(55)
According to the qualitative theory, we obtain the bifurcation phase portraits of system (54) as Figures 15 and 16.
Using the information given by Figures 15 and 16, we give
derivations to Propositions 1, 2, and 3, respectively.
3.1. The Derivations to Proposition 1. When the orbit Γ is
defined by 𝐻(𝜑, 𝑦) = 𝐻(0, 0), from (55) we obtain
𝑞
𝑟
𝑦 = ±√ 𝑝𝜑2 + 𝜑4 + 𝜑6 .
2
3
(56)
Abstract and Applied Analysis
9
𝜑
𝜑𝑑+
𝜑
𝜑𝑑+
𝜑𝑐+
𝜑𝑐+
0
0
𝜉
𝜑𝑑−
𝜑𝑐−
𝜑𝑐−
𝜑𝑑−
(a)
(b)
𝜑
𝜑𝑐+
𝜉
𝜑
𝜑𝑑+
0
𝜑𝑑+
𝜑𝑐+
0
𝜉
𝜑𝑐−
𝜉
𝜑𝑐−
𝜑𝑑−
𝜑𝑑−
(c)
(d)
Figure 7: (The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of 𝜑𝑐± and 𝜑𝑑± when
(𝑞, 𝑝) ∈ 𝐴 4 , 𝜆 > 0, 𝜆 ≠ √𝜆 0 /12, and 𝑝 → 3𝑞2 /16𝑟 + 0, where 𝑟 = 1, 𝑞 = −4, 𝜆 = 4, and 𝑙2 : 𝑝 = 3𝑞2 /16𝑟 = 3 and (a) 𝑝 = 3 + 10−1 , (b)
𝑝 = 3 + 10−3 , (c) 𝑝 = 3 + 10−6 , and (d) 𝑝 = 3 + 10−9 .
Substituting (56) into the first equation of (54) and
integrating it, we have
∫
𝑙
𝜑
𝑑𝑠
√𝑝𝑠2 + (𝑞/2) 𝑠4 + (𝑟/3) 𝑠6
= 𝜉,
(57)
Thus, we have
lim 𝜑±
𝑝 → 0−0 𝑎0
where 𝑙 is an arbitrary constant or ±∞.
For 𝑝 = 0, letting 𝑙 = √−(3𝑞/2𝑟) or ±∞ and completing
the above integral, it follows that
𝜑 = ±√
6𝑞
,
3𝑞2 𝜉2 − 4𝑟
= lim ± √
−12𝑝
𝑓 (𝑝)
= ±√ lim
−12
(𝑝)
𝑝 → 0−0
𝑝 → 0−0 𝑓󸀠
= ± ( lim (2√9𝑞2 − 48𝑝𝑟
𝑝 → 0−0
× ( (3𝑞2 − 16𝑝𝑟) 𝜉2
(58)
× (sin (2√−𝑝𝜉) /2√−𝑝𝜉)
which yields 𝜑𝑎± as (29).
For 𝑝 < 0, completing (57) and solving for 𝜑, it follows
that
𝜑 = ±√
−12𝑝
3𝑞 −
√9𝑞2
− 48𝑝𝑟 sin (𝜆 + 2√−𝑝𝜉)
,
(59)
where 𝜆 = 𝜆(𝑙) is an arbitrary real constant. Let 𝜆 = ±(𝜋/2),
±
and 𝜑𝑏± as (3) and
respectively, and we obtain the solutions 𝜑𝑎0
(31).
When (𝑞, 𝑝) ∈ 𝐴 1 , that is, 𝑟 > 0, 𝑞 > 0, and 𝑝 < 0, let
𝑓 (𝑝) = 3𝑞 − √9𝑞2 − 48𝑝𝑟 cos (2√−𝑝𝜉) ,
(60)
𝑔 (𝑝) = 3𝑞 + √9𝑞2 − 48𝑝𝑟 cos (2√−𝑝𝜉) .
−1
1/2
−4𝑟 cos (2√−𝑝𝜉) ) ) )
= ±√
= 𝜑𝑎±
lim 𝜑±
𝑝 → 0−0 𝑏
6𝑞
− 4𝑟
(see (29)) ,
3𝑞2 𝜉2
= lim ± √
𝑝 → 0−0
= ±√ lim
−12𝑝
𝑔 (𝑝)
−12𝑝
𝑝 → 0−0 𝑔 (𝑝)
= 0.
(61)
10
Abstract and Applied Analysis
𝜑
𝜑
𝜑𝑑+
𝜑𝑐+
𝜑𝑑+
𝜑𝑐+
0
0
𝜉
𝜑𝑐−
𝜉
𝜑𝑑−
𝜑𝑐−
𝜑𝑑−
(a)
(b)
𝜑
𝜑
𝜑𝑑+
𝜑𝑐+
𝜑𝑑+
𝜑𝑐+
0
0
𝜉
𝜑𝑐−
𝜉
𝜑𝑑−
𝜑𝑐−
(c)
𝜑𝑑−
(d)
Figure 8: (The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of 𝜑𝑐± and 𝜑𝑑± when (𝑞, 𝑝) ∈ 𝐴 3 , 𝜆 <
0, 𝜆 ≠ − √𝜆 0 /12, and 𝑝 → 3𝑞2 /16𝑟 − 0, where 𝑟 = 1, 𝑞 = −4, 𝜆 = −4, and 𝑙2 : 𝑝 = 3𝑞2 /16𝑟 = 3 and (a) 𝑝 = 3 − 10−1 , (b) 𝑝 = 3 − 10−3 , (c)
𝑝 = 3 − 10−6 , and (d) 𝑝 = 3 − 10−9 .
When (𝑞, 𝑝) ∈ 𝐴 2 , that is, 𝑟 > 0, 𝑞 < 0, and 𝑝 < 0,
similarly we get
lim 𝜑±
𝑝 → 0−0 𝑏
= 𝜑𝑎± .
(62)
When (𝑞, 𝑝) ∈ 𝐴 12 , that is, 𝑟 < 0, 𝑞 > 0, and 𝑝 < 0, if
𝑝 → 3𝑞2 /16𝑟 + 0, then 𝑓(𝑝) and 𝑔(𝑝) → 0.
Thus,
±
(or 𝜑𝑏± ) = ±√ −
lim 𝜑𝑎0
𝑝 → 0−0
For the case of 𝜆 > 0. When (𝑞, 𝑝) ∈ 𝑙2 , that is, 𝑟 > 0, 𝑞 <
0 and 𝑟 = 3𝑞2 /16𝑝. In (33) letting 𝑟 = 3𝑞2 /16𝑝, we get (34).
Furthermore, in (34) letting 𝜆 = −𝑞 > 0, it follows that
3𝑞
.
4𝑟
+ 1)
4𝑝𝑒−√𝑝𝜉
−𝑞 (𝑒√𝑝𝜉 + 𝑒−√𝑝𝜉 )
= ±√
2𝑝
𝑒√𝑝𝜉 − 𝑒−√𝑝𝜉
(1 − √𝑝𝜉 −√𝑝𝜉 )
−𝑞
𝑒
+𝑒
= ±√
2𝑝
(1 − tanh (√𝑝𝜉))
−𝑞
= ±√
2𝑝
(−1 + tanh (√𝑝𝜉))
𝑞
(64)
where 𝜆 = 𝜆(𝑙) is an arbitrary real constant.
Note that if 𝜑(𝜉) is a solution of (53), so is 𝜑(−𝜉). Thus,
from (64) we obtain the solutions 𝜑𝑐± and 𝜑𝑑± as in (33).
4𝑝
−𝑞 (𝑒2√𝑝𝜉
= ±√
(63)
For 𝑝 > 0, completing (57) and solving for 𝜑, it follows
that
4𝜆𝑝𝑒2√𝑝𝜉
,
𝜑 = ±√ 2 4√𝑝𝜉
𝜆𝑒
+ (𝑞2 /4 − 4𝑝𝑟/3) − 𝜆𝑞𝑒2√𝑝𝜉
±
𝜑𝑐1
= ±√
±
= 𝜑𝑏0
(see (4)) ,
Abstract and Applied Analysis
11
𝜑
𝜑
𝜑𝑓+
𝜑𝑒+
𝜑𝑒+
0
𝜑𝑒−
𝜑𝑓+
0
𝜉
𝜉
𝜑𝑓−
𝜑𝑒−
𝜑𝑓−
(a)
(b)
𝜑
𝜑
𝜑𝑓−
𝜑𝑒−
𝜑𝑓+
𝜑𝑒+
0
𝜑𝑒−
𝜑𝑓−
0
𝜉
𝜉
𝜑𝑓+
𝜑𝑒+
(c)
(d)
Figure 9: (The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of 𝜑𝑒± and 𝜑𝑓± when
𝜇 > 0, 𝜇 ≠ |𝜇0 |, (𝑞, 𝑝) ∈ 𝐴 3 , and 𝑝 → 3𝑞2 /16𝑟 − 0, where 𝑟 = 1, 𝑞 = −4, 𝑙2 : 𝑝 = 3𝑞2 /16𝑟 = 3, and 𝜇 = 4 and (a) 𝑝 = 3 − 0.5, (b) 𝑝 = 3 − 10−2 ,
(c) 𝑝 = 3 − 10−4 , and (d) 𝑝 = 3 − 10−6 .
±
𝜑𝑑1
= ±√
For the case of 𝜆 < 0, similarly we can obtain (7), (35),
and (36), and here we omit the process. Hereto, we have
completed the derivations for Proposition 1.
4𝑝
−𝑞 (𝑒−2√𝑝𝜉
+ 1)
= ±√
4𝑝𝑒√𝑝𝜉
−𝑞 (𝑒√𝑝𝜉 + 𝑒−√𝑝𝜉 )
= ±√
2𝑝
𝑒√𝑝𝜉 − 𝑒−√𝑝𝜉
(1 + √𝑝𝜉 −√𝑝𝜉 )
−𝑞
𝑒
+𝑒
= ±√
2𝑝
(1 + tanh (√𝑝𝜉))
−𝑞
= ±√
2𝑝
±
(−1 − tanh (√𝑝𝜉)) = 𝜑𝑐0
(see (5)) .
𝑞
3.2. The Derivations to Proposition 2. When the orbit Γ is
defined by 𝐻(𝜑, 𝑦) = 𝐻(𝜑1 , 0), from (55) we obtain
𝑞
𝑟
𝑦 = ±√ 𝑝𝜑2 + 𝜑4 + 𝜑6 + 𝐻 (𝜑1 , 0)
2
3
𝑟
2
= ±√ (𝜑12 − 𝜑2 ) (𝜑2 + 𝜇0 ),
3
(65)
When 𝜆 = √𝜆 0 /12, by (33) it follows that
𝜑𝑐±
=
𝜑𝑑±
= ±√
= ±√
−6𝑞 + √9𝑞2 − 48𝑝𝑟 (𝑒√𝑝𝜉 + 𝑒−√𝑝𝜉 )
−3𝑞 +
𝜑
𝑑𝑠
𝑚
√(𝑟/3) (𝜑12 − 𝑠2 )2 (𝑠2 + 𝜇0 )
= 𝜉,
(68)
where 𝑚 is an arbitrary constant or ±∞.
Completing the above integral and solving for 𝜑, it follows
that
12𝑝
√9𝑞2
where 𝜑1 and 𝜇0 are given in (25) and (40). Substituting (67)
into the first equation of (54) and integrating it, we have
∫
24𝑝
(67)
− 48𝑝𝑟 cosh (2√𝑝𝜉)
±
= 𝜑𝑑0
(see (6)) .
(66)
𝜑 = ±√ 𝜑12 −
𝜇2 𝑒2𝜃𝜉
4𝑎1 𝜇𝑒𝜃𝜉
,
− 2𝜇𝑏1 𝑒𝜃𝜉 + 𝑏12 − 4𝑎1
(69)
12
Abstract and Applied Analysis
𝜑
𝜑
𝜑𝑒−
𝜑𝑒−
𝜑𝑓−
0
0
𝜉
𝜑𝑓+
𝜑𝑒+
𝜑𝑓−
𝜉
𝜑𝑓+
𝜑𝑒+
(a)
(b)
𝜑
𝜑
𝜑𝑓−
𝜑𝑒−
𝜑𝑒−
𝜑𝑓−
0
𝜑𝑒+
𝜑𝑓+
0
𝜉
𝜉
𝜑𝑓+
(c)
𝜑𝑒+
(d)
Figure 10: (The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of 𝜑𝑒±
and 𝜑𝑓± when 𝜇 > 0, 𝜇 ≠ |𝜇0 |, (𝑞, 𝑝) ∈ 𝐴 4 , and 𝑝 → 3𝑞2 /16𝑟 + 0, where 𝑟 = 1, 𝑞 = −4, 𝑙2 : 𝑝 = 3𝑞2 /16𝑟 = 3, and 𝜇 = 4 and (a) 𝑝 = 3 + 10−1 ,
(b) 𝑝 = 3 + 10−3 , (c) 𝑝 = 3 + 10−5 , and (d) 𝑝 = 3 + 10−7 .
where 𝜃 is given in (39), 𝜇 = 𝜇(𝑚) is an arbitrary constant,
and
3Δ (Δ − 𝑞)
,
𝑎1 =
4𝑟2
𝑞 − 4Δ
𝑏1 =
.
2𝑟
−4𝑝
= ∓√
𝑞 (1 + 𝑒−2√𝑝𝜉 )
= ∓√
2𝑝
(−1 − tanh (√𝑝𝜉))
𝑞
(71)
If (𝑞, 𝑝) ∈ 𝐴 3 and 𝜇 = |𝜇0 |, that is, 𝑟 > 0, 𝜇 = 𝜇0 > 0, and
2Δ + 𝑞 > 0, we have
𝜑𝑒±
=±
=±
=±
∓
= 𝜑𝑐0
(see (5)) ,
∓
𝜑𝑓1
−4𝑝
= ∓√
𝑞 (1 + 𝑒2√𝑝𝜉 )
2𝑝
(−1 + tanh (√𝑝𝜉))
𝑞
∓
= 𝜑𝑏0
(see (4)) .
(70)
Similar to the derivations for 𝜑𝑐± and 𝜑𝑑± , we get 𝜑𝑒± and 𝜑𝑓±
(see (38)) from (69).
For the case of 𝜇 > 0, when (𝑞, 𝑝) ∈ 𝑙2 , that is, 𝑟 > 0, 𝑞 <
0, and 𝑟 = 3𝑞2 /16𝑝, then 𝜃 = 2√𝑝 and 𝑞 + 2Δ = 0. From
∓
(38) and (41), it is easy to check that 𝜑𝑒± and 𝜑𝑓± become 𝜑𝑒1
∓
and 𝜑𝑓1 (see (42)), respectively. Furthermore, in (42) letting
𝜇 = −(16𝑝/𝑞), it follows that
∓
𝜑𝑒1
= ∓√
=±
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞 − 2𝜇𝑟𝑒𝜃𝜉 )
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒𝜃𝜉 + 4𝜇2 𝑟2 𝑒2𝜃𝜉
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞) (1 − 𝑒𝜃𝜉 )
√(𝑞 + 2Δ)2 + 2 (𝑞 + 2Δ) (4Δ − 𝑞) 𝑒𝜃𝜉 + (𝑞 + 2Δ)2 𝑒2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (1 − 𝑒𝜃𝜉 )
√𝑞 + 2Δ + 2 (4Δ − 𝑞) 𝑒𝜃𝜉 + (𝑞 + 2Δ) 𝑒2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (𝑒−𝜃𝜉/2 − 𝑒𝜃𝜉/2 )
√2 (4Δ − 𝑞) + (𝑞 + 2Δ) (𝑒−𝜃𝜉 + 𝑒𝜃𝜉 )
Abstract and Applied Analysis
13
𝜑
𝜑𝑓+
𝜑𝑒+
0
𝜑𝑒+
0
𝜉
𝜑𝑒−
𝜑𝑓−
𝜑𝑓+
𝜑
𝜑𝑓+
𝜉
𝜑𝑓−
𝜑𝑒−
(a)
(b)
𝜑
𝜑
𝜑𝑒+
𝜑𝑒+ = 𝜑𝑓+
0
0
𝜉
𝜉
𝜑𝑒− = 𝜑𝑓−
𝜑𝑓−
𝜑𝑒−
(c)
(d)
Figure 11: (The four symmetric solitary waves become two trivial waves.) The varying process for the figures of 𝜑𝑒± and 𝜑𝑓± when 𝜇 >
0, 𝜇 ≠ |𝜇0 |, (𝑞, 𝑝) ∈ 𝐴 11 , and 𝑝 → 𝑞2 /4𝑟 + 0, where 𝑟 = −1, 𝑞 = 4, 𝑙7 : 𝑝 = 𝑞2 /4𝑟 = −4, and 𝜇 = 1 and (a) 𝑝 = −3.5, (b) 𝑝 = −4 + 10−1 , (c)
𝑝 = −4 + 10−2 , and (d) 𝑝 = −4 + 10−5 .
=∓
If (𝑞, 𝑝) ∈ 𝐴 4 and 𝜇 = |𝜇0 |, that is, 𝑟 > 0, 𝜇 = −𝜇0 > 0,
and 2Δ + 𝑞 < 0, we have
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) sinh ((𝜃/2) 𝜉)
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
𝜑𝑒±
= 𝜑𝑔∓ (see (44)) ,
𝜑𝑓±
=±
=±
=±
=±
=±
=±
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞 − 2𝜇𝑟𝑒−𝜃𝜉 )
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒−𝜃𝜉 + 4𝜇2 𝑟2 𝑒−2𝜃𝜉
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞) (1 − 𝑒
2
√(𝑞 + 2Δ) + 2 (𝑞 + 2Δ) (4Δ −
𝑞) 𝑒−𝜃𝜉
−𝜃𝜉
)
+ (𝑞 +
=±
2
2Δ) 𝑒−2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (1 − 𝑒−𝜃𝜉 )
=∓
√𝑞 + 2Δ + 2 (4Δ − 𝑞) 𝑒−𝜃𝜉 + (𝑞 + 2Δ) 𝑒−2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (𝑒𝜃𝜉/2 − 𝑒−𝜃𝜉/2 )
=∓
√2 (4Δ − 𝑞) + (𝑞 + 2Δ) (𝑒−𝜃𝜉 + 𝑒𝜃𝜉 )
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) sinh ((𝜃/2) 𝜉)
=∓
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
= 𝜑𝑔± (see (44)) .
(72)
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞 − 2𝜇𝑟𝑒𝜃𝜉 )
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒𝜃𝜉 + 4𝜇2 𝑟2 𝑒2𝜃𝜉
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞) (1 + 𝑒𝜃𝜉 )
√(𝑞 + 2Δ)2 + 2 (𝑞 + 2Δ) (4Δ − 𝑞) 𝑒𝜃𝜉 + (𝑞 + 2Δ)2 𝑒2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (1 + 𝑒𝜃𝜉 )
√𝑞 + 2Δ + 2 (4Δ − 𝑞) 𝑒𝜃𝜉 + (𝑞 + 2Δ) 𝑒2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (𝑒−𝜃𝜉/2 + 𝑒𝜃𝜉/2 )
√2 (4Δ − 𝑞) + (𝑞 + 2Δ) (𝑒−𝜃𝜉 + 𝑒𝜃𝜉 )
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) cosh ((𝜃/2) 𝜉)
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
= 𝜑ℎ∓ (see (45)) ,
14
Abstract and Applied Analysis
𝜑
𝜑𝑒−
𝜑𝑓+
𝜑
𝜑𝑒+
𝜑𝑓−
𝜑𝑒−
0
𝜑𝑒+
𝜑𝑒−
𝜑𝑓−
𝜑𝑒−
𝜑𝑒+
𝜑𝑓+
𝜑𝑓−
𝜑𝑒+
(b)
𝜑
𝜑
𝜑𝑒+
𝜑𝑓+
𝜑𝑓−
𝜑𝑓−
𝜑𝑓+
𝜑𝑒+
0
𝜉
𝜑𝑒−
𝜉
𝜑𝑒−
(a)
𝜑𝑓+
𝜑𝑓−
0
𝜉
0
𝜑𝑒+
𝜑𝑓+
𝜑𝑓−
𝜑𝑓+
(c)
𝜉
𝜑𝑒−
(d)
Figure 12: (The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of
𝜑𝑒± and 𝜑𝑓± when 𝜇 < 0, 𝜇 ≠ − |𝜇0 |, (𝑞, 𝑝) ∈ 𝐴 4 , and 𝑝 → 3𝑞2 /16𝑟 + 0, where 𝑟 = 1, 𝑞 = −4, 𝑙2 : 𝑝 = 3𝑞2 /16𝑟 = 3, and 𝜇 = −4 and (a)
𝑝 = 3 + 10−2 , (b) 𝑝 = 3 + 10−4 , (c) 𝑝 = 3 + 10−6 , and (d) 𝑝 = 3 + 10−9 .
𝜑𝑓±
=±
=±
=∓
If (𝑞, 𝑝) ∈ 𝐴 11 , 𝐴 12 , 𝑙6 and 𝜇 = |𝜇0 |, that is, 𝑟 < 0, 𝜇 =
−𝜇0 > 0, and 2Δ + 𝑞 > 0, we have
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞 − 2𝜇𝑟𝑒−𝜃𝜉 )
𝜑𝑒±
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒−𝜃𝜉 + 4𝜇2 𝑟2 𝑒−2𝜃𝜉
=±
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞) (1 + 𝑒−𝜃𝜉 )
√(𝑞 + 2Δ)2 + 2 (𝑞 + 2Δ) (4Δ − 𝑞) 𝑒−𝜃𝜉 + (𝑞 + 2Δ)2 𝑒−2𝜃𝜉
=±
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (1 + 𝑒−𝜃𝜉 )
√𝑞 + 2Δ + 2 (4Δ − 𝑞) 𝑒−𝜃𝜉 + (𝑞 + 2Δ) 𝑒−2𝜃𝜉
=±
=∓
=∓
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞 − 2𝜇𝑟𝑒𝜃𝜉 )
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒𝜃𝜉 + 4𝜇2 𝑟2 𝑒2𝜃𝜉
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞) (1 + 𝑒𝜃𝜉 )
√(𝑞 + 2Δ)2 + 2 (𝑞 + 2Δ) (4Δ − 𝑞) 𝑒𝜃𝜉 + (𝑞 + 2Δ)2 𝑒2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (1 + 𝑒𝜃𝜉 )
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (𝑒−𝜃𝜉/2 + 𝑒𝜃𝜉/2 )
√𝑞 + 2Δ + 2 (4Δ − 𝑞) 𝑒𝜃𝜉 + (𝑞 + 2Δ) 𝑒2𝜃𝜉
√2 (4Δ − 𝑞) + (𝑞 + 2Δ) (𝑒−𝜃𝜉 + 𝑒𝜃𝜉 )
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (𝑒−𝜃𝜉/2 + 𝑒𝜃𝜉/2 )
=±
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) cosh ((𝜃/2) 𝜉)
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
=±
= 𝜑ℎ∓ (see (45)) .
(73)
√2 (4Δ − 𝑞) + (𝑞 + 2Δ) (𝑒−𝜃𝜉 + 𝑒𝜃𝜉 )
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) cosh ((𝜃/2) 𝜉)
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
= 𝜑ℎ± (see (45)) ,
Abstract and Applied Analysis
15
𝜑
𝜑𝑗+
𝜑𝑗+
𝜑𝑗−
0
𝜑𝑗−
𝜑
𝜑𝑗+
𝜑𝑗−
0
𝜉
𝜑𝑗+
𝜑𝑗−
𝜑𝑗+
(b)
𝜑
𝜑
𝜑𝑗+
𝜑𝑗−
0
𝜑𝑗+
𝜉
𝜑𝑗−
𝜑𝑗+
𝜑𝑗+
(a)
𝜑𝑗−
𝜑𝑗+
𝜑𝑗−
𝜑𝑗+
0
𝜉
𝜑𝑗−
𝜑𝑗−
𝜑𝑗+
(c)
𝜉
(d)
Figure 13: (The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of 𝜑𝑗± when (𝑞, 𝑝) ∈ 𝐴 4
and 𝑝 → 𝑞2 /4𝑟 − 0, where 𝑟 = 1, 𝑞 = −4, and 𝑙1 : 𝑝 = 𝑞2 /4𝑟 = 4 and (a) 𝑝 = 4 − 10−1 , (b) 𝑝 = 4 − 10−2 , (c) 𝑝 = 4 − 10−4 , and (d) 𝑝 = 4 − 10−6 .
𝜑𝑓±
=±
=±
=±
=±
=±
the process. Hereto, we have completed the derivations for
Proposition 2.
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞 − 2𝜇𝑟𝑒−𝜃𝜉 )
3.3. The Derivations to Proposition 3. When the orbit Γ is
defined by 𝐻(𝜑, 𝑦) = 𝐻(𝜑2 , 0), firstly, if (𝑞, 𝑝) belongs to one
of 𝐴 3 , 𝐴 4 , 𝐴 11 , and 𝑙2 , then from (55) we obtain
√(2Δ + 𝑞)2 + 4𝜇𝑟 (4Δ − 𝑞) 𝑒−𝜃𝜉 + 4𝜇2 𝑟2 𝑒−2𝜃𝜉
√(Δ − 𝑞) /2𝑟 (2Δ + 𝑞) (1 + 𝑒−𝜃𝜉 )
√(𝑞 + 2Δ)2 + 2 (𝑞 + 2Δ) (4Δ − 𝑞) 𝑒−𝜃𝜉 + (𝑞 + 2Δ)2 𝑒−2𝜃𝜉
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (1 + 𝑒−𝜃𝜉 )
𝑞
𝑟
𝑦 = ±√ 𝑝𝜑2 + 𝜑4 + 𝜑6 + 𝐻 (𝜑2 , 0)
2
3
𝑟
2
= ±√ (𝜑2 − 𝜑22 ) (𝜑2 + 𝛿0 ),
3
√𝑞 + 2Δ + 2 (4Δ − 𝑞) 𝑒−𝜃𝜉 + (𝑞 + 2Δ) 𝑒−2𝜃𝜉
(75)
where 𝜑2 is given in (16) and 𝛿0 = (𝑞 − 2Δ)/2𝑟. Substituting
(75) into the first equation of (54) and integrating it, we have
√(1/2𝑟) (Δ − 𝑞) (2Δ + 𝑞) (𝑒−𝜃𝜉/2 + 𝑒𝜃𝜉/2 )
√2 (4Δ − 𝑞) + (𝑞 + 2Δ) (𝑒−𝜃𝜉 + 𝑒𝜃𝜉 )
∫
√(1/𝑟) (Δ − 𝑞) (2Δ + 𝑞) cosh ((𝜃/2) 𝜉)
𝜑
𝑛
√−𝑞 + 4Δ + (𝑞 + 2Δ) cosh (𝜃𝜉)
= 𝜑ℎ± (see (45)) .
(74)
For the case of 𝜇 < 0, similarly we can obtain the
relations of the solutions 𝜑𝑒± , 𝜑𝑓± , 𝜑𝑔± , and 𝜑ℎ± , and here we omit
𝑑𝑠
√(𝑟/3) (𝑠2 − 𝜑22 )2 (𝑠2 + 𝛿0 )
= 𝜉,
(76)
where 𝑛 is an arbitrary constant or ±∞.
Completing the above integral and solving for 𝜑, it follows
that
𝜑 = ±√𝜑22 −
2𝑎2
,
𝑏2 + 𝛿0 sin (𝛿 + 𝜂𝜉)
(77)
16
Abstract and Applied Analysis
𝜑
𝜑
𝜑𝑗−
0
0
𝜉
𝜑𝑗−
𝜑𝑗+
𝜑𝑗+
𝜑𝑗−
𝜑𝑗+
𝜉
𝜑𝑗+
𝜑𝑗−
(a)
(b)
𝜑
𝜑
𝜑𝑗+
𝜑𝑗+
0
0
𝜉
𝜉
𝜑𝑗−
𝜑𝑗−
(c)
(d)
Figure 14: (The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of 𝜑𝑗± when (𝑞, 𝑝) ∈ 𝐴 11 and
𝑝 → 𝑞2 /4𝑟 + 0, where 𝑟 = −1, 𝑞 = 4, and 𝑙7 : 𝑝 = 𝑞2 /4𝑟 = −4 and (a) 𝑝 = −3.5, (b) 𝑝 = −4 + 10−1 , (c) 𝑝 = −4 + 10−3 , and (d) 𝑝 = −4 + 10−5 .
where 𝜂 is given in (49), 𝛿 = 𝛿(𝑛) is an arbitrary constant,
and
𝑎2 =
3Δ (Δ + 𝑞)
,
4𝑟2
𝑏2 = −
𝑞 + 4Δ
.
2𝑟
When 𝑝 → 𝑞2 /4𝑟, it follows that
Δ = √𝑞2 − 4𝑝𝑟 󳨀→ 0,
(78)
In (77) letting 𝛿 = ±(𝜋/2), respectively, we obtain the
solutions 𝜑𝑖± and 𝜑𝑗± as (47) and (48).
Secondly, if (𝑞, 𝑝) ∈ 𝑙1 or 𝑙7 , that is, Δ = 0 and 𝜑12 = −𝛿0 =
−(𝑞/2𝑟), thus from (75), we obtain
Δ (Δ + 𝑞)
1
𝜂 = √ (4𝑝𝑟 − 𝑞 (𝑞 + Δ)) = √ −
󳨀→ 0,
𝑟
𝑟
(81)
cos (𝜂𝜉) = 1 −
=1+
𝑞 3
𝑟
𝑦 = ±√ (𝜑2 + ) .
3
2𝑟
which yields 𝜑𝑘± as in (51).
𝜂𝜉
𝜂𝜉
+
+ ⋅⋅⋅
2
4!
Δ (Δ + 𝑞) 2
𝜉 + 𝑜 (Δ2 ) .
2𝑟
Thus, we have
lim2
𝑝 → (𝑞 /4𝑟)
−𝑞
𝑞𝜉
,
2𝑟 √𝑞2 𝜉2 − 12𝑟
4 4
(79)
Similarly, we have
𝜑 = ±√
2 2
(80)
𝜑𝑖±
= ± lim2 √
𝑝 → 𝑞 /4𝑟
= ±√
(𝑞 + Δ) (2Δ − 𝑞)
𝜂
cos ( 𝜉)
2
𝑟 (𝑞 + 4Δ + (𝑞 − 2Δ) cos (𝜂𝜉))
𝑞
−𝑞2
= ±√ − ,
2𝑟𝑞
2𝑟
Abstract and Applied Analysis
17
𝑝
𝑙1
𝑦
𝑦
𝜑2
−𝜑1 −𝜑2
𝑙2
𝜑1
𝜑
𝜑
𝑦
𝜑
𝑦
𝑦
−𝜑1 −𝜑2
𝜑2
𝜑1
𝜑
𝑦
𝜑
−𝜑1
𝑦
−𝜑1 −𝜑2
𝜑2
𝜑1
𝜑1
𝜑
𝐴4
𝑦
𝑙3
𝜑
−𝜑1
𝜑1
𝐴5
𝐴6
𝐴3
𝜑
𝑂
𝐴2
𝐴1
𝑦
𝑙4
𝑞
𝜑
𝑦
𝑦
−𝜑1
−𝜑1
𝜑1
𝜑1
𝑦
𝜑
𝜑
−𝜑1
𝜑1
𝜑
Figure 15: The bifurcation phase portraits of system (54) for 𝑟 > 0.
2
lim 𝜑𝑗±
=
𝑝 → 𝑞2 /4𝑟
lim2
𝑝 → 𝑞 /4𝑟
(𝑞 + Δ) (2Δ − 𝑞)
𝜂
= lim2 ± √
sin ( 𝜉)
2
𝑝 → 𝑞 /4𝑟
𝑟 (𝑞 + 4Δ − (𝑞 − 2Δ) cos (𝜂𝜉))
=
=
±√
lim
± ( (− (𝑞 + Δ) (2Δ − 𝑞)
𝑝 → 𝑞2 /4𝑟
(Δ + 𝑞) (2Δ − 𝑞) 2
𝜉 + 𝑜 (Δ))
2𝑟
× (2𝑟 [6 − (𝑞 − 2Δ) (1 +
(𝑞 + Δ) (2Δ − 𝑞) (1 − cos (𝜂𝜉))
2𝑟 (𝑞 + 4Δ − (𝑞 − 2Δ) cos (𝜂𝜉))
lim2
𝑝 → 𝑞 /4𝑟
± ((−
−1
1/2
+𝑜 (Δ) ] ) )
Δ (Δ + 𝑞) 2
𝜉
2𝑟
= ±√
2
+𝑜 (Δ) ) × (2𝑟 [𝑞 + 4Δ − (𝑞 − 2Δ)
× (1 +
Δ (Δ + 𝑞) 2
𝜉)
2𝑟
Δ (Δ + 𝑞) 2
𝜉)
2𝑟
−1
+𝑜 (Δ2 ) ] ) )
1/2
= ±√
(𝑞3 /2𝑟) 𝜉2
2𝑟 (6 − (𝑞2 /2𝑟) 𝜉2 )
−𝑞
𝑞𝜉
2𝑟 √𝑞2 𝜉2 − 12𝑟
= 𝜑𝑘± (see (51)) .
(82)
Hereto, we have completed the derivations for our main
results.
18
Abstract and Applied Analysis
𝑝
𝑦
−𝜑2
𝑦
𝑦
𝜑2
−𝜑2
𝜑
𝜑2
−𝜑2
𝜑
𝑦
𝜑2
𝜑
𝑦
𝐴8
𝐴7
𝑂
𝜑
𝐴9
−𝜑2
𝐴 12
𝐴 10
𝜑2
𝑙5
𝜑
𝑦
𝐴 11
𝑦
𝜑1
−𝜑2 −𝜑1
𝑦
𝜑
−𝜑2
𝜑
𝑦
𝜑2 𝜑
𝑦
𝜑2
𝜑1
−𝜑2 −𝜑1
𝜑2
𝜑
𝜑
𝑦
𝑦
𝑙6
−𝜑2 −𝜑1
𝜑
𝜑1 𝜑2
𝜑
𝑙7
Figure 16: The bifurcation phase portraits of system (54) for 𝑟 < 0.
4. Conclusions
In this paper, we have investigated the explicit expressions
of the nonlinear waves and bifurcation phenomena in (1).
Firstly, we obtained three types of explicit nonlinear wave
solutions. The first type is the fractional expressions 𝜑𝑎± and
𝜑𝑘± (see (29) and (51)). The second type is the trigonometric
expressions 𝜑𝑖± and 𝜑𝑗± (see (47) and (48)). The third type is
the exp-function expressions 𝜑𝑐± , 𝜑𝑑± , 𝜑𝑒± , and 𝜑𝑓± (see (33) and
(38)).
Secondly, we revealed five kinds of interesting bifurcation
phenomena in (1). The first phenomena is that the 1-blowup waves can be bifurcated from the periodic-blow-up (see
Figure 2 or Figure 13) and 2-blow-up waves (see Figure 8).
The second phenomena is that the 2-blow-up waves can be
bifurcated from the periodic-blow-up waves (see Figure 3).
The third kind is that the symmetric solitary waves can be
bifurcated from the symmetric periodic waves (see Figure 4).
The fourth kind is that the low-kink waves can be bifurcated
from the symmetric solitary waves (see Figure 6), the 1-blowup waves (see Figure 7), the tall-kink waves (see Figure 9),
and the antisymmetric solitary waves (see Figure 10). The fifth
kind is that the tall-kink waves can be bifurcated from the
symmetric periodic waves (see Figure 14).
Thirdly, we showed that some previous results are our
±
±
±
, 𝜑𝑐0
, and 𝜑𝑒0
are included in
special cases. For instants, 𝜑𝑏0
±
±
±
±
±
are
𝜑𝑐 , 𝜑𝑑 , 𝜑𝑒 , and 𝜑𝑓 (see (4), (5), (7), (33), and (38)). 𝜑𝑑0
±
±
included in 𝜑𝑐 and 𝜑𝑑 (see (6) and (33)).
Finally, we employed the software Mathematica to check
the correctness of these solutions. For example, the commands for 𝑢𝑎 and 𝐸𝑎 are as follows:
𝑤 = 𝛼𝛾2 ,
𝑝 = (𝛼𝛾2 − 𝑤)/𝛼,
𝑞 = 𝛽𝛿1 /𝛼(𝑐2 − 𝑐𝑠2 ) − 𝛿2 /𝛼,
𝑟 = −𝛿3 /𝛼,
𝑐 = 2𝛼𝛾,
𝜉 = 𝑥 − 𝑐𝑡,
𝜑𝑎 = √6𝑞/(3𝑞2 𝜉2 − 4𝑟),
Abstract and Applied Analysis
19
𝑢𝑎 = (𝛽/(𝑐2 − 𝑐𝑠2 ))𝜑𝑎2 ,
𝐸𝑎 = 𝜑𝑎 Exp[𝑖(𝛾𝑥 − 𝑤𝑡)],
Simplify[D[𝑢𝑎 ,{𝑡, 2}] − 𝑐𝑠2 D[𝑢𝑎 ,{𝑥, 2}] − 𝛽D[𝜑𝑎2 ,{𝑥, 2}]]
Simplify[𝑖D[𝐸𝑎 , 𝑡] + 𝛼D[𝐸𝑎 , {𝑥, 2}] − 𝛿1 𝑢𝑎 𝐸𝑎 +
𝛿2 𝜑𝑎2 𝐸𝑎 + 𝛿3 𝜑𝑎4 𝐸𝑎 ]
[13]
[14]
0
0
[15]
Acknowledgment
This paper is supported by the National Natural Science
Foundation (no. 11171115), Natural Science Foundation of
Guangdong Province (no. S2012040007959) and the Fundamental Research Funds for the Central Universities (no.
2012ZM0057).
References
[1] G.-T. Liu and T.-Y. Fan, “New applications of developed Jacobi
elliptic function expansion methods,” Physics Letters A, vol. 345,
no. 1–3, pp. 161–166, 2005.
[2] S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function
expansion method and periodic wave solutions of nonlinear
wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74,
2001.
[3] M. A. Abdou, “The extended F-expansion method and its
application for a class of nonlinear evolution equations,” Chaos,
Solitons & Fractals, vol. 31, no. 1, pp. 95–104, 2007.
[4] M. Wang and X. Li, “Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations,”
Physics Letters. A, vol. 343, no. 1–3, pp. 48–54, 2005.
[5] M. Wang, X. Li, and J. Zhang, “The 𝐺󸀠 /𝐺-expansion method
and travelling wave solutions of nonlinear evolution equations
in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp.
417–423, 2008.
[6] M. Song and Y. Ge, “Application of the 𝐺󸀠 /𝐺-expansion method
to (3+1)-dimensional nonlinear evolution equations,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1220–
1227, 2010.
[7] J. B. Li and Z. R. Liu, “Smooth and non-smooth traveling waves
in a nonlinearly dispersive equation,” Applied Mathematical
Modelling, vol. 25, pp. 41–56, 2000.
[8] Z. Liu and J. Li, “Bifurcations of solitary waves and domain wall
waves for KdV-like equation with higher order nonlinearity,”
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 2, pp. 397–407, 2002.
[9] Z. R. Liu and C. X. Yang, “The application of bifurcation method
to a higher-order KdV equation,” Journal of Mathematical
Analysis and Applications, vol. 275, no. 1, pp. 1–12, 2002.
[10] Z. R. Liu and Y. Liang, “The explicit nonlinear wave solutions
and their bifurcations of the generalized Camassa-Holm equation,” International Journal of Bifurcation and Chaos in Applied
Sciences and Engineering, vol. 21, no. 11, pp. 3119–3136, 2011.
[11] R. Liu, “Some new results on explicit traveling wave solutions of
𝐾(𝑚, 𝑛) equation,” Discrete and Continuous Dynamical Systems
B, vol. 13, no. 3, pp. 633–646, 2010.
[12] R. Liu, “Coexistence of multifarious exact nonlinear wave
solutions for generalized b-equation,” International Journal of
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
Bifurcation and Chaos in Applied Sciences and Engineering, vol.
20, no. 10, pp. 3193–3208, 2010.
M. Song, S. Li, and J. Cao, “New exact solutions for the (2+1)dimensional Broer-Kaup-Kupershmidt equations,” Abstract
and Applied Analysis, vol. 2010, Article ID 652649, 9 pages, 2010.
M. Song, X. Hou, and J. Cao, “Solitary wave solutions and
kink wave solutions for a generalized KDV-mKDV equation,”
Applied Mathematics and Computation, vol. 217, no. 12, pp.
5942–5948, 2011.
M. Song, J. Cao, and X. Guan, “Application of the bifurcation
method to the Whitham-Broer-Kaup-like equations,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 688–696,
2012.
D. Feng and J. Li, “Exact explicit travelling wave solutions for
the (𝑛 + 1) -dimensional 𝜙6 field model,” Physics Letters A, vol.
369, no. 4, pp. 255–261, 2007.
D.-J. Huang and H.-Q. Zhang, “New exact travelling waves solutions to the combined KdV-MKdV and generalized Zakharov
equations,” Reports on Mathematical Physics, vol. 57, no. 2, pp.
257–269, 2006.
S. A. El-Wakil, M. A. Abdou, and A. Elhanbaly, “New solitons
and periodic wave solutions for nonlinear evolution equations,”
Physics Letters A, vol. 353, pp. 40–47, 2006.
J. Wang, “Multisymplectic numerical method for the Zakharov
system,” Computer Physics Communications, vol. 180, no. 7, pp.
1063–1071, 2009.
B. J. Hong, D. C. Lu, and F. S. Sun, “The extended Jacobi
Elliptic Functions expansion method and new exact solutions
for the Zakharov equations,” World Journal of Modelling and
Simulation, vol. 5, pp. 216–224, 2009.
O. P. Layeni, “A new rational auxiliary equation method and
exact solutions of a generalized Zakharov system,” Applied
Mathematics and Computation, vol. 215, no. 8, pp. 2901–2907,
2009.
Y. Xia, Y. Xu, and C.-W. Shu, “Local discontinuous Galerkin
methods for the generalized Zakharov system,” Journal of
Computational Physics, vol. 229, no. 4, pp. 1238–1259, 2010.
G. Betchewe, B. B. Thomas, K. K. Victor, and K. T. Crepin,
“Dynamical survey of a generalized-Zakharov equation and
its exact travelling wave solutions,” Applied Mathematics and
Computation, vol. 217, no. 1, pp. 203–211, 2010.
A. Bekir and A. C. Cevikel, “New solitons and periodic solutions for nonlinear physical models in mathematical physics,”
Nonlinear Analysis. Real World Applications, vol. 11, no. 4, pp.
3275–3285, 2010.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014