Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 810217, 23 pages
doi:10.1155/2011/810217
Research Article
Trigonometric Function Periodic Wave Solutions
and Their Limit Forms for the KdV-Like and the
PC-Like Equations
Liu Zhengrong, Jiang Tianpei, Qin Peng, and Xu Qinfeng
Department of Mathematics, South China University of Technology, Guangzhou 510640, China
Correspondence should be addressed to Liu Zhengrong, liuzhr@scut.edu.cn
Received 15 February 2011; Accepted 26 May 2011
Academic Editor: Ming Li
Copyright q 2011 Liu Zhengrong et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We use the bifurcation method of dynamical systems to study the periodic wave solutions and
their limit forms for the KdV-like equation ut a1 buuux uxxx 0, and PC-like equation
vtt −vttxx −a1 v a2 v2 a3 v3 xx 0, respectively. Via some special phase orbits, we obtain some
new explicit periodic wave solutions which are called trigonometric function periodic wave
solutions because they are expressed in terms of trigonometric functions. We also show that the
trigonometric function periodic wave solutions can be obtained from the limits of elliptic function
periodic wave solutions. It is very interesting that the two equations have similar periodic wave
solutions. Our work extend previous some results.
1. Introduction
Many authors have investigated the KdV-like equation
ut a1 buuux uxxx 0,
1.1
and the PC-like equation
vtt − vttxx − a1 v a2 v2 a3 v3
xx
0.
1.2
For example, Dey 1, 2 studied the exact Himiltonian density and the conservation laws,
and gave two kink solutions for 1.1. Zhang et al. 3, 4 gave some solitary wave solutions
and singular wave solutions for 1.1 by using two different methods. Yu 5 got an exact
kink soliton for 1.1 by using homogeneous balance method. Grimshaw et al. 6 studied the
2
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large-amplitude solitons for 1.1. Fan 7, 8 gave some bell-shaped soliton solutions, kinkshaped soliton, and Jacobi periodic solutions for 1.1 by using algebraic method. Tang et
al. 9 investigated solitary waves and their bifurcations for 1.1 by employing bifurcation
method of dynamical systems. Peng 10 used the modified mapping method to get some
solitary wave solutions composed of hyperbolic functions, periodic wave solutions composed
of Jacobi elliptic functions, and singular wave solution composed of triangle functions for
1.1. Chow et al. 11 described the interaction between a soliton and a breather for 1.1
by using the Hirota bilinear method. Kaya and Inan 12 studied solitary wave solutions for
1.1 by using Adomian decomposition method. Yomba 13 used Fan’s subequation method
to construct exact traveling wave solutions composed of hyperbolic functions or Jacobi elliptic
functions for 1.1.
Zhang and Ma 14 gave some explicit solitary wave solutions composed of hyperbolic
functions by using solving algebraic equations for 1.2. Li and Zhang 15 used bifurcation
method of dynamical system to study the bifurcation of traveling wave solutions and
construct solitary wave solutions for 1.2. Kaya 16 discussed the exact and numerical
solitary wave solutions by using a decomposition method for 1.2. Rafei et al. 17 gave
numerical solutions by using He’s method for 1.2.
Recently, many authors have presented some useful methods to deal with the
problems in equations, for instance 18–30.
In this paper, we use the bifurcation method mentioned above to study the periodic
wave solutions for 1.1 and 1.2. Through some special phase orbits, we obtain new
expressions of periodic wave solutions which are composed of trigonometric functions sin ξ
or cos ξ. These solutions are called trigonometric function periodic wave solutions. We also
check the correctness by using the software Mathematica.
In Section 2, we will state our results for 1.1. In Section 3, we will state our results
for 1.2. In Sections 4, and 5, we will give derivations for our main results. Some discussions
and the orders for testing the correctness of the solutions will be given in Section 6.
2. Trigonometric Function Periodic Wave Solutions for 1.1
In this section, we state our main results for 1.1. In order to state these results conveniently,
we give some preparations. For given constant c / 0, on a − b plane we define some lines and
regions as follows.
1 When c < 0, we define lines
l1 : b 0,
l2 : b −
a
,
6c
l3 : b −
3a
,
16c
l4 : a 0,
and regions Ai i 1–8, as Figure 1a.
2.1
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3a
b = − 16c
b
l3
l4
A3
A4
l2
A2
A1
O
l1
A5
A6
l2
3a
b = − 16c
a
b = − 6c
l1
b=
B2
B3
k4
B4
k1
a
b
k3
a
− 6c
k2
B1
O
B8
B5
A8
A7
l3
3
B6
l4
B7
k4
k3
k2
a c < 0
k1 a
b c > 0
Figure 1: The locations of the lines li , ki i 1, 2, 3, 4 and the regions Aj , Bj j 1, 2, . . . , 8 for given
constant c /
0.
−1.4996
−20
−1.4998
−20
−10
−1.5002
10
20
−1.5004
a for b 0.66
−10
10
−2
20
−20
−10
−20
−4
−40
−6
−60
−8
−80
b for b 0.2
10
20
c for b 0.00001
Figure 2: The limiting precess of u1 ξ when c < 0, a, b ∈ A1 , and a, b tends to the line l1 , where a 4
and c −1.
2 When c > 0, we define lines
k1 : b 0,
k2 : a 0,
3a
,
16c
a
k4 : b − ,
6c
k3 : b −
2.2
and regions Bi i 1–8, as Figure 1b.
Using the lines and regions in Figure 1, we narrate our results as follows.
Proposition 2.1. For arbitrary given constant c /
0, let
ξ x − ct.
Then, 1.1 has the following periodic wave solutions.
2.3
4
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60
50
40
30
20
10
8
1.5004
6
1.5002
4
−20
−10
1.4998
10
20
2
−20
1.4996
−10
a for b −0.66
10
−20
20
b for b −0.2
−10
10
20
c for b −0.00001
Figure 3: The limiting precess of u1 ξ when c < 0, a, b ∈ A5 , and a, b tends to the line l1 , where a −4
and c −1.
(1) When c < 0 and a, b ∈ A1 or A5 , the expression of the periodic wave solution is
u1 ξ a
6c
aa 6bc cos
√
−cξ
,
2.4
which has the following limit forms.
i When c < 0, a, b ∈ A1 and a, b tends to the line l1 , u1 ξ tends to the periodic blow-up
solution
6c
u◦1 ξ √
a 1 cos −cξ
2.5
(see Figure 2).
ii When c < 0, a, b ∈ A5 and a, b tends to the line l1 , u1 ξ tends to the periodic blow-up
solution
6c
u∗1 ξ √
a 1 − cos −cξ
2.6
(see Figure 3).
iii When c < 0, a, b ∈ A1 or A5 , and a, b tends to l2 , u1 ξ tends to the trivial solution
uξ 6c/a.
(2) When c < 0 and a, b ∈ A2 , or when c > 0 and a, b ∈ B5 , the expression of the periodic
wave solution is
u2 ξ α0 cosw0 ξ β0
,
p0 cosw0 ξ q0
2.7
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5
where
α0 −
Δ 3a3a 16bc,
√ √ 3a Δ
a a− Δ
4a2 b2
√
3a 24bc Δ
β0 −
,
2ab2
√ a a− Δ
p0 ,
ab
√ 1 q0 −
a Δ ,
ab
√
3a 16bc Δ
.
w0 8b
2.8
,
2.9
2.10
2.11
2.12
2.13
The solution u2 ξ has the following limit forms.
i When c < 0, a, b ∈ A2 and a, b tends to l3 , the u2 ξ tends to the peak-shaped solitary
wave solution
u◦2 ξ
4c 3 2cξ2
a−9 2cξ2 2.14
(see Figure 4).
ii When c < 0, a, b ∈ A2 and a, b tends to l2 , u2 ξ tends to the trivial solution uξ 0.
iii When c > 0, a, b ∈ B5 and a, b tends to k1 , the u2 ξ tends to the periodic blow-up
solution
√
c 2 sin2 cξ/2 − 3
u∗2 ξ √
a sin2 cξ/2
2.15
(see Figure 5).
(3) When c < 0 and a, b ∈ A6 , or when c > 0 and a, b ∈ B1 , the expressions of the solution
is
u3 ξ α1 cosw1 ξ β1
,
p1 cosw1 ξ q1
2.16
6
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0.2
0.2
0.1
−40
−20
−0.1
−0.2
−0.3
−0.4
−40
20
−20
40
a for b 0.7
0.2
20
−0.2
−0.4
−0.6
−0.8
−1
−40
40
b for b 0.7499
−20
−0.2
−0.4
−0.6
−0.8
−1
40
20
c for b 0.7499999
Figure 4: The limiting precess of u2 ξ when c < 0, a, b ∈ A2 , and a, b tends to the line l3 , where a 4
and c −1.
3
40
2.5
2
30
1.5
1
20
10
0.5
−10
−5
350
300
250
200
150
100
50
5
10
−10
−5
a for b −1
5
10
−10
b for b −0.05
−5
5
10
c for b −0.0001
Figure 5: The limiting precess of u2 ξ when c > 0, a, b ∈ B5 , and a, b tends to the line k1 , where a −2
and c 1.
where
α1 −3a √ √ Δ
a a Δ
4a2 b2
√
3a 24bc − Δ
,
2ab2
√ a a Δ
p1 ,
ab
√
a− Δ
q1 ,
ab
√
3a 16bc − Δ
w1 .
8b
,
β1 2.17
The solution u3 ξ has the following limit forms.
i When c < 0, a, b ∈ A6 and a, b tends to l3 , the u3 ξ tends to the canyon-shaped
solitary wave (see Figure 6) solution u◦2 ξ.
ii When c < 0, a, b ∈ A6 and a, b tends to l2 , u3 ξ tends to the trivial solution uξ 0.
iii When c > 0, a, b ∈ B1 and a, b tends to k1 , the u3 ξ tends to the periodic blow-up
wave solution u∗1 ξ (see Figure 3).
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7
0.2
0.15
0.1
0.05
−40
−20
−0.05
20
−0.1
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
40
−40
a for b −1.5875
−20
−0.1
20
−40
40
b for b −1.6865
−20
−0.1
20
40
c for b −1.687499
Figure 6: The limiting precess of u3 ξ when c < 0, a, b ∈ A6 , and a, b tends to the line l3 , where a −9
and c −1.
Remark 2.2. Note that if u ϕξ is a solution of 1.1, then u ϕξ r also is solution of 1.1,
where r is a arbitrary constant. According to this fact and the results listed in Proposition 2.1,
the following nine functions also are periodic wave solutions of 1.1.
1 When c < 0 and a, b ∈ A1 or A5 , the functions are
u11 ξ u12 ξ u13 ξ a−
a
a−
6c
aa 6bc cos
√
−cξ
,
6c
√
,
aa 6bc sin −cξ
2.18
6c
√
.
aa 6bc sin −cξ
2 When c < 0 and a, b ∈ A2 or when c > 0 and a, b ∈ B5 , the functions are
u21 ξ −α0 cosw0 ξ β0
,
−p0 cosw0 ξ q0
u22 ξ α0 sinw0 ξ β0
,
p0 sinw0 ξ q0
u23 ξ −α0 sinw0 ξ β0
.
−p0 sinw0 ξ q0
2.19
3 When c < 0 and a, b ∈ A6 , or when c > 0 and a, b ∈ B1 , the functions are
u31 ξ −α1 cosw1 ξ β1
,
−p1 cosw1 ξ q1
u32 ξ α1 sinw1 ξ β1
,
p1 sinw1 ξ q1
u33 ξ −α1 sinw1 ξ β1
.
−p1 sinw1 ξ q1
2.20
8
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Γ5
Γ6
a3 =
a3 =
a3
a22
4(a1 −c2 )
a3 =
2a22
9(a1 −c2 )
Γ7
W7
a22
4(a1 −c2 )
a3 =
W5 W4
W6
a3
Γ3
Γ4
O
W3 W
2
L3
Γ2
L2
Ω1
a3 =
W1
a2
L4
L1
a2
Ω8
Ω4
9(a1 −c2 )
W8
O
Ω3
2a22
Γ1
Γ8
Ω2
Ω5
2a22
9(a1 −c2 )
a3 =
Ω7
Ω6
a3 =
a22
4(a1 −c2 )
L5
a c < a1
a3 =
L6
a22
4(a1 −c2 )
2a22
9(a1 −c2 )
L8
L7
b c > a1
2
2
Figure 7: The locations of the rays Γi , Li and the regions Wi , Ωi i 1, 2, . . . , 8 for given a1 and c.
Remark 2.3. In the given parametric regions, the solutions ui ξ, u1i ξ, u2i ξ, u3i ξ i 1, 2, 3,
and u◦2 ξ are nonsingular. The solutions u◦1 ξ, u∗1 ξ, and u∗2 ξ are singular. The relationships
of singular solutions and nonsingular solutions are displayed in the Proposition 2.1.
3. Trigonometric Function Periodic Wave Solutions for 1.2
In this section, we state our main results for 1.2. For given a1 and c a1 /
c2 , on a2 −a3 plane
we define some rays and regions as follows.
1 When c2 < a1 , we define curves
Γ1 : a2 > 0,
a3 0,
Γ2 : a2 > 0,
a3 2a22
,
9a1 − c2 Γ3 : a2 > 0,
a3 a22
,
4a1 − c2 Γ4 : a2 0,
a3 > 0,
Γ5 : a2 < 0,
a22
,
a3 4a1 − c2 Γ6 : a2 < 0,
a3 Γ7 : a2 < 0,
a3 0,
Γ8 : a2 0,
a3 < 0,
3.1
2a22
,
9a1 − c2 and region Wi as the domain surrounded by Γi and Γi1 i 1–7, W8 as the domain
surrounded by Γ8 and Γ1 see Figure 7a.
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2 When c
2
9
> a1 , we define curves
L1 : a2 > 0,
a3 0,
L2 : a2 0,
a3 > 0,
L3 : a2 < 0,
a3 0,
L4 : a2 < 0,
a3 2a22
,
9a1 − c2 L5 : a2 < 0,
a3 a21
,
4a1 − c2 L6 : a2 0,
a3 < 0,
L7 : a2 > 0,
a3 a22
,
4a1 − c2 L8 : a2 > 0,
a3 2a22
,
9a1 − c2 3.2
and region Ωi as the domain surrounded by Li and Li1 i 1–7, Ω8 as the domain
surrounded by L8 and L1 see Figure 7b.
Using the rays and regions above, we state our results as follows.
Proposition 3.1. For given parameter a1 and constant c satisfying c2 /
a1 , let ξ x − ct. Then, 1.2
has the following periodic wave solutions.
(1) When c 2 < a1 and a2 , a3 ∈ W1 or W6 , the expression of the periodic wave solution is
v1 ξ R0
,
R1 R2 cosR3 ξ
3.3
where
R0 2 c2 − a1 ,
2a2
,
3
1
R2 18a3 c2 − a1 4a22 ,
3
a1 − c2
R3 .
c2
R1 For a2 /
0, the periodic wave solution v1 ξ has the following limit forms.
3.4
10
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i When c2 < a1 , a2 , a3 ∈ W1 and a2 , a3 tends to the ray Γ1 , v1 ξ tends to the periodic
blow-up solution
v1◦ ξ
3 c2 − a1
.
a2 1 cos
a1 − c2 /|c| ξ
3.5
The limiting process is similar to that in Figure 2.
ii When c2 < a1 , a2 , a3 ∈ W6 and a2 , a3 tends to the ray Γ7 , v1 ξ tends to the periodic
blow-up solution
v1∗ ξ
3 c2 − a1
.
a2 1 − cos
a1 − c2 /|c| ξ
3.6
The limiting process is similar to that in Figure 3.
iii When c2 < a1 , a2 , a3 ∈ W1 and a2 , a3 tends to the curve Γ2 , or a2 , a3 ∈ W6 and
a2 , a3 tends to the curve Γ6 , v1 ξ tends to the trivial solution vξ 3c2 − a1 /a2 .
(2) When c2 < a1 and a2 , a3 ∈ W5 , or when c2 > a1 and a2 , a3 ∈ Ω1 , the expression of the
periodic wave solution is
v2 ξ S0 −
2S1
,
−S2 S3 cosS4 ξ
3.7
where
S0 S1 √
−a2 ω
,
2a3
√
−a22 4a3 a1 − c2 a2 ω
a23
√ 2 −a2 3 ω ,
3a3
√ 2
S3 a2 a2 3 ω ,
3a3
S1 a3
S4 − 2 ,
2c
2
ω a2 − 4a3 a1 − c2 .
S2 ,
3.8
3.9
The periodic wave solution v2 ξ has the following limit forms.
i When c2 < a1 , a2 , a3 ∈ W5 , and a2 , a3 tends to the curve Γ6 , v2 ξ tends to the trivial
solution vξ 0.
Mathematical Problems in Engineering
11
ii When c2 < a1 , a2 , a3 ∈ W5 , and a2 , a3 tends to the curve Γ5 , the v2 ξ tends to the
canyon-shaped solitary wave solution
v2◦ ξ
2 a1 − c2 12c2 − 9c2 − 2 a1 − c2 ξ2
.
a2 9c2 2a1 − c2 ξ2 3.10
The limiting process is similar to that in Figure 6.
iii When c2 > a1 , a2 , a3 ∈ Ω1 , and a2 , a3 tends to the ray L1 , v2 ξ tends to the periodic
blow-up wave solution
v2∗ ξ
⎞⎤
⎡
⎛
c2 − a1 ⎠⎦
a1 − c2 ⎣
2⎝
ξ
.
1 3 tan
2a2
4c2
3.11
The limiting process is similar to that in Figure 2.
(3) When c2 < a1 and a2 , a3 ∈ W2 , or when c2 > a1 and a2 , a3 ∈ Ω2 , the expression of the
periodic wave solution is
v3 ξ T0 2T1
,
−T2 T3 cosT4 ξ
3.12
where
T0 T1 √
−a2 − ω
,
2a3
√
−a22 4a3 a1 − c2 − a2 ω
a23
√ 2 a2 3 ω ,
3a3
√ 2
T3 a2 a2 − 3 ω ,
3a3
T1 a3
T4 − 2 .
2c
T2 ,
3.13
The periodic wave solution v3 ξ has the following limit forms:
i When c2 < a1 , a2 , a3 ∈ W2 , and a2 , a3 tends to the curve Γ2 , v3 ξ tends to the trivial
solution vξ 0.
ii When c2 < a1 , a2 , a3 ∈ W2 , and a2 , a3 tends to the curve Γ3 , the v3 ξ tends to the
peak-shaped solitary wave solution v2◦ ξ. The limiting process is similar to that in Figure 4.
iii When c2 > a1 , a2 , a3 ∈ Ω2 , and a2 , a3 tends to the ray L3 , the v3 ξ tends to the
periodic blow-up wave solution v2∗ ξ. The limiting process is similar to that in Figure 5.
12
Mathematical Problems in Engineering
Remark 3.2. Similar to the reason in Remark 2.2, the following nine functions also are periodic
wave solutions of 1.2.
1 When c2 < a1 and a2 , a3 ∈ W1 or W6 , the functions are
v11 ξ R0
,
R1 − R2 cosR3 ξ
v21 ξ R0
,
R1 R2 sinR3 ξ
v31 ξ R0
.
R1 − R2 sinR3 ξ
3.14
2 When c2 < a1 and a2 , a3 ∈ W5 or when c2 > a1 and a2 , a3 ∈ Ω1 , the functions are
v12 ξ S0 2S1
,
S2 S3 cosS4 ξ
v22 ξ S0 −
2S1
,
−S2 S3 sinS4 ξ
v32 ξ S0 2S1
.
S2 S3 sinS4 ξ
3.15
3 When c2 < a1 and a2 , a3 ∈ W2 or when c2 > a1 and a2 , a3 ∈ Ω2 , the functions are
v13 ξ T0 −
2T1
,
T2 T3 cosT4 ξ
v23 ξ T0 2T1
,
−T2 T3 sinT4 ξ
v33 ξ T0 −
2T1
.
T2 T3 sinT4 ξ
3.16
Remark 3.3. In the given regions, the solutions vi ξ, vi1 ξ, vi2 ξ, vi3 ξ i 1, 2, 3, and
v2◦ ξ are nonsingular. The solutions v1◦ ξ, v1∗ ξ, and v2∗ ξ are singular. The relationships
of nonsingular solutions and singular solutions are displayed in Proposition 3.1.
4. The Derivation on Proposition 2.1
In order to derive the Proposition 2.1, letting c be a constant and substituting u ϕξ with
ξ x − ct into 1.1, we have
−cϕ aϕϕ abϕ2 ϕ ϕ 0.
4.1
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13
Integrating 4.1 once and letting the integral constant be zero, it follows that
−cϕ a 2 ab 3
ϕ ϕ ϕ 0.
2
3
4.2
Letting ϕ y, yields the following planar system:
ϕ y,
y cϕ −
a ab 3
−
ϕ.
2
3
4.3
Obviously, system 4.3 has the first integral
6y2 − 6cϕ2 2aϕ3 abϕ4 h.
4.4
√
−3a − Δ
,
ϕ1 4ab
√
−3a Δ
ϕ2 ,
4ab
4.5
Let
where Δ is defined in 2.8. Then, it is easy to see that system 4.3 has three singular points
ϕ1 , 0, 0, 0 and ϕ2 , 0 when Δ > 0, two singular points −3/4b, 0 and 0, 0 when Δ 0,
unique singular point 0, 0 when Δ < 0.
Let ei and fi i 1, 2, 3 be, respectively,
√
a2 6abc
,
ab
√
−a a2 6abc
f1 ,
ab
√
√ 1
e2 −a Δ − 2 a a − Δ
,
4ab
e1 −a −
14
Mathematical Problems in Engineering
ϕ
O
y
y
e1
O
y
f1
ϕ
e3
y
O
A1
ϕ
ϕ
a
A8
ϕ
y
A7
y
ϕ
O
l1
O
O
A6
f3
f1 y
y
y
ϕ
O
ϕ
O
e1
A5
l2
y
ϕ
O
ϕ
O
A2
y
O
ϕ
O
ϕ
O
l4
A3
A4
l3 e y f2
2
y
b
y
O
ϕ
ϕ
O
Figure 8: When c < 0, the bifurcation phase portraits of system 4.3 and the locations of ei and fi i 1, 2, 3.
1
f2 4ab
−a 1
e3 −
4ab
1
f3 −
4ab
√
a
a
√ Δ2 a a− Δ
,
√
√ Δ2 a a Δ
,
√
√ Δ−2 a a Δ
.
4.6
Using the qualitative analysis of dynamical systems, we obtain the bifurcation phase
portraits of system 4.3 and the locations of ei and fi i 1, 2, 3 as Figures 8 and 9.
It is easy to test that the closed orbit passing ei , 0 passes fi , 0 i 1, 2, 3. Thus,
using the phase portraits in Figures 8 and 9, we derive ui ξ i 1, 2, 3 as follows.
1 When c < 0 and a, b ∈ A1 or A5 , the closed orbit passing the points e1 , 0 and
f1 , 0 has expression
y±
ab ϕ −e1 f1 e1 f1 ϕ − ϕ2 ,
6
where e1 ≤ ϕ ≤ f1 .
4.7
Substituting 4.7 into dϕ/y dξ, we have
dϕ
−e1 f1 e1 f1 ϕ − ϕ2
ab
dξ.
6
4.8
Mathematical Problems in Engineering
O
ϕ
O
y
e3
O
k4
B4
f3
y
ϕ
O
k2
y
ϕ
ϕ
b
ϕ
B2
B3
y
O
y
k3
y
O
15
y
ϕ
B1
O
B8
y
O
y e2
O
ϕ
O
f2
ϕ
ϕ
B5
y
k1
O
ϕ
a
y
ϕ
O
B7
B6
y
O
y
ϕ
ϕ
O
y
ϕ
O
Figure 9: When c > 0, the bifurcation phase portraits of system 4.3 and the locations of ei and fi i 1, 2, 3.
Integrating 4.8 along the closed orbit and noting that u ϕξ, we obtain the solution u1 ξ
as 2.4.
2 When c < 0 and a, b ∈ A2 or when c > 0 and a, b ∈ B5 , the closed orbit passing
the points e2 , 0 and f2 , 0 has expression
y±
ab ϕ − ϕ1 −e2 f2 e2 f2 ϕ − ϕ2 ,
6
where e2 ≤ ϕ ≤ f2 .
4.9
Substituting 4.9 into dϕ/y dξ, we get
dϕ
ϕ − ϕ1 −e2 f2 e2 f2 ϕ − ϕ2
ab
dξ.
6
4.10
Along the closed orbit integrating 4.10 and noting that u ϕξ, we get the solution u2 ξ
as 2.7.
3 When c < 0 and a, b ∈ A6 or when c > 0 and a, b ∈ B1 , the closed orbit passing
the points e3 , 0 and f3 , 0 has expression
y±
ab ϕ2 − ϕ −e3 f3 e3 f3 ϕ − ϕ2 ,
6
where e3 ≤ ϕ ≤ f3 .
4.11
16
Mathematical Problems in Engineering
Substituting 4.11 into dϕ/y dξ, it follows that
dϕ
ϕ2 − ϕ −e3 f3 e3 f3 ϕ − ϕ2
ab
dξ.
6
4.12
Similarly, along the closed orbit integrating 4.12, we obtain u3 ξ as 2.16. From the
expressions of these solutions, we get their limit forms. This completes the derivation on
Proposition 2.1.
5. The Derivation on Proposition 3.1
In this section, we give derivation on Proposition 3.1. Let v ψξ with ξ x − ct, where c is
a constant. Thus, 1.2 becomes
c2 ψ − c2 ψ − a1 ψ a2 ψ 2 a3 ψ 3 0.
5.1
Integrating 5.1 twice and letting integral constant be zero, we get
c2 ψ − ψ a1 ψ a2 ψ 2 a3 ψ 3 .
5.2
Letting ψ y, we have the planar system
ψ y,
c2 y c2 − a1 ψ − a2 ψ 2 − a3 ψ 3 .
5.3
It is easy to see that system 5.3 has the first integral
c2 y 2 ψ 2
a3 2 2a2
ψ ψ a1 − c2
2
3
h,
5.4
and three singular points 0, 0, ψ1 , 0, and ψ2 , 0, where
√
−a2 − ω
ψ1 ,
2a3
√
−a2 ω
ψ2 2a3
and ω is defined in 3.9.
5.5
Mathematical Problems in Engineering
Γ5
y
m2
Γ6
a3
y
n2
ϕ
O
17
ϕ
O
ϕ
y
W3
O
n3
ϕ
m1
n1
n1 y
O
W6
ϕ
W1
Γ7
Γ1
O
y
O
ϕ
ϕ
ϕ
O
Γ2
y
O
y
ϕ
y
O
W2
y
O
m1
ϕ
O
W4
O
m3
Γ4
W5
y
Γ3
y
a2
y
ϕ
W7
W8
Γ8
O
ϕ
Figure 10: When c2 < a1 , the bifurcation phase portraits of system 5.3 and the locations of mi and ni i 1, 2, 3.
Let mi and ni i 1, 2, 3 be, respectively,
m1 n1 m2 n2 −2a2 −
2 a22 − 9a1 a3 9a3 c2
3a3
−2a2 2 a22 − 9a1 a3 9a3 c2
3a3
√
√ −a2 − 3 ω − 2 a2 a2 3 ω
6a3
√
√ −a2 − 3 ω 2 a2 a2 3 ω
,
,
,
5.6
,
6a3
√
√ −a2 3 ω − 2 a2 a2 − 3 ω
m3 ,
6a3
√
√ −a2 3 ω 2 a2 a2 − 3 ω
n3 .
6a3
Similarly, using the qualitative analysis of dynamical systems, we get the bifurcation
phase portraits of system 5.3 and the locations of mi and ni i 1, 2, 3 as Figures 10 and 11.
It is easy to test that the closed orbit passing mi , 0 passes ni , 0 i 1, 2, 3. Thus,
using the phase portraits in Figures 10 and 11, we derive vi ξ i 1, 2, 3 as follows.
18
Mathematical Problems in Engineering
a3
y m3
n3
ϕ
O
m2
L2
Ω2
n2
y
O
L3
y
y
ϕ
O
y
y
O
y
ϕ
y
O
ϕ
Ω8
ϕ
O
y
O
L6
Ω4
L4
ϕ
y
L8
y
y
ϕ
ϕ
O
L5
ϕ
Ω7
Ω6
Ω5
a2
L1
O
Ω3
O
ϕ
O
Ω1
O
ϕ
O
ϕ
L7
Figure 11: When c2 > a1 , the bifurcation phase portraits of system 5.3 and the locations of mi and ni i 1, 2, 3.
1 When c 2 < a1 and a2 , a3 ∈ W1 or W6 , the closed orbit passing the points m1 , 0
and n1 , 0 has expression
y±
a3 ψ −m1 n1 m1 n1 ψ − ψ 2 ,
2c2
where m1 ≤ ψ ≤ n1 .
5.7
Substituting 5.7 into dψ/y dξ, we have
ψ
dψ
−m1 n1 m1 n1 ψ − ψ 2
a3
dξ.
2c2
5.8
Integrating 5.8 along the closed orbit and noting that v ψξ, we get the solution v1 ξ as
3.3.
2 When c2 < a1 and a2 , a3 ∈ W5 , or when c2 > a1 and a2 , a3 ∈ Ω1 , the closed orbit
passing the points m2 , 0 and n2 , 0 has expression
y±
a3 −
ψ
−m2 n2 m2 n2 ψ − ψ 2 ,
ψ
2
2c2
where m2 ≤ ψ ≤ n2 .
5.9
Mathematical Problems in Engineering
19
From dψ/y dξ and 5.9, it follows that
dψ
ψ2 − ψ −m2 n2 m2 n2 ψ − ψ 2
a3
dξ.
2c2
5.10
Integrating 5.10 along the closed orbit, we get v2 ξ as 3.7.
3 When c2 < a1 and a2 , a3 ∈ W2 , or when c2 > a1 and a2 , a3 ∈ Ω2 , the closed orbit
passing the points m3 , 0 and n3 , 0 has expression
y±
a3 −m3 n3 m3 n3 ψ − ψ 2 ,
ψ
−
ψ
1
2c2
where m3 ≤ ψ ≤ n3 .
5.11
Substituting 5.11 into dψ/y dξ, we have
dψ
ψ − ψ1 −m3 n3 m3 n3 ψ − ψ 2
a3
dξ.
2c2
5.12
Integrating 5.12 along the closed orbit, we obtain v3 ξ as 3.12. From the expressions
of these solutions, we get their limiting properties. This completes the derivation on
Proposition 3.1.
6. Discussions and Testing Orders
In this paper, Using the special closed orbits, we have obtained trigonometric function
periodic wave solutions for 1.1 and 1.2, respectively. Their limit forms have been given.
From these expressions, an interesting phenomena has been seen, that is, 1.1 and 1.2 have
similar periodic wave solutions. Our work has extended previous results on periodic wave
solutions.
Now, we point out that the trigonometric function periodic wave solutions can be
obtained from the limits of the elliplic function periodic wave solution. For given real number
μ, let
⎛
⎞
√ 3
aF
4
1
i
02
√
1 ⎜
⎟
2i i 3 F ⎠,
μ1 ⎝−4a 2 bμ 12ab
F
⎛
μ2 1 ⎜
⎝−4a 2 bμ 12ab
√ 4 1 − i 3 aF02
F
4aF02
1
μ3 −2a 2 bμ −
2F ,
6ab
F
⎞
√ ⎟
− 2i i 3 F ⎠,
6.1
20
Mathematical Problems in Engineering
y
lμ1
ϕ 1 μ e1
f1 μ1
lμ2
μ2 O
μ3 ϕ4
ϕ
Figure 12: The locations of lμ1 and lμ2 when c < 0 and a, b ∈ A1 .
where
F01 8 − 6bμ 15b2 μ2 10b3 μ3 ,
F02 −9bc a −2 bμ b2 μ2 ,
F03
2 3
,
a3 8F02
a −54bc −1 bμ aF01
6.2
1/3
F 54a2 bc −1 bμ − a3 F01 F03
.
Assume that c < 0, a, b ∈ A1 , and ϕ1 < μ < e1 . It is easy to check that μi i 1, 2, 3
are real and satisfy
μ < e1 < ϕ2 < f1 < μ1 < ϕ3 < μ2 < 0 < μ3 < ϕ4 .
6.3
There are two closed orbits lμ1 and lμ2 see Figure 12. The closed orbit lμ1 passes the
points μ, 0 and μ1 , 0. The closed orbit lμ2 passes the points μ2 , 0 and μ3 , 0.
On ϕ − y plane, the expression of lμ1 is
y2 ab μ3 − ϕ μ2 − ϕ μ1 − ϕ ϕ − μ ,
6
where μ ≤ ϕ ≤ μ1 .
6.4
Substituting 6.4 into dϕ/y dξ and integrating it along lμ1 , we have
−1
g sn sin z, k ab
|ξ|,
6
6.5
Mathematical Problems in Engineering
21
where
2
,
μ3 − μ1 μ2 − μ
g μ3 − μ2 μ1 − μ
k ,
μ3 − μ1 μ2 − μ
6.6
μ3 − μ1 ϕ − μ
sin z .
μ1 − μ μ3 − ϕ
Solving 6.5 for ϕ and noting that u ϕξ, we obtain an elliptic function periodic
wave solution
μ μ3 − μ1 μ3 μ1 − μ sn2 ηξ, k
uξ ,
μ3 − μ1 μ1 − μ sn2 ηξ, k
6.7
where
η
ab μ3 − μ1 μ2 − μ
.
24
6.8
Letting μ → e1 − 0, it follows
that μ1 → f1 , μ2 →0, μ3 → 0, k → 0, η →
abe1 f1 /24 and sn2 ηξ, k → sn2 abe1 f1 /24ξ, 0 sin2 abe1 f1 /24ξ.
Therefore, in 6.7 letting μ → e1 − 0, we obtain the trigonometric function periodic
wave solution
uξ e1 f1
2 abe1 f1 /24 ξ
f1 e1 − f1 sin
−6c
−a aa 6bc − 2 aa 6bcsin2
|c|/2 ξ
a−
6c
aa 6bc cos
|c|ξ
6.9
u11 ξ.
Via Remark 2.2 and u11 ξ, further we get u12 ξ, u13 ξ and u1 ξ. Similarly, we can derive
others trigonometric function periodic wave solutions.
We also have tested the correctness of each solution by using the software
Mathematica. Here, we list two testing orders. Others testing orders are similar.
22
Mathematical Problems in Engineering
1 The orders for testing u1 ξ
u
6c
√
a aa 6bc cos −cx − ct
6.10
Simplify Du, t a1 buDu, xu Du, {x, 3}.
2 The orders for testing v1 ξ
R0 2 −a1 c2 ,
R1 R2 2a2
,
3
2a3 −a1 c2 R3 v
4a22
,
9
a1 − c2 ,
c2
6.11
R0
,
R1 R2 cosR3 x − ct
vtt Dv, {t, 2} ,
vttxx Dvtt, {x, 2}
Simplify vtt − vttxx − Da1 v a2 v2 a3 v3 , {x, 2}.
Acknowledgment
Research is supported by the National Natural Science Foundation of China no. 10871073
and the Research Expences of Central Universities for students.
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