Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 284865, 7 pages
http://dx.doi.org/10.1155/2013/284865
Research Article
Abundant Exact Solition-Like Solutions to the Generalized
Bretherton Equation with Arbitrary Constants
Xiuqing Yu, Fengsheng Xu, and Lihua Zhang
Department of Mathematics Sciences, Dezhou University, Dezhou 253023, China
Correspondence should be addressed to Xiuqing Yu; sddzyxq@163.com
Received 20 January 2013; Accepted 26 February 2013
Academic Editor: Abdel-Maksoud A. Soliman
Copyright © 2013 Xiuqing Yu 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.
The Riccati equation is employed to construct exact travelling wave solutions to the generalized Bretherton equation. Taking full
advantage of the Riccati equation which has more new solutions, abundant new multiple solition-like solutions are obtained for the
generalized Bretherton equation.
1. Introduction
Nonlinear partial differential equations (PDE) are widely
chosen to describe complex phenomena in physics sciences.
Searching for exact solutions to nonlinear differential equations plays more and more important role in nonlinear science. Recently, various direct methods have been proposed,
such as the tanh-function method [1, 2], the Jacobi elliptic
function expansion method [3, 4], the F-expansion [5–8],
sine-cosine method [9, 10], and the homogeneous balance
method [11–13]. Among them, the tanh-function method is
improved continuously [14–17] as one of the most effectively
straightforward methods for constructing exact solutions to
PDEs. In the paper an extended tanh-function method is used
to solve the generalized Bretherton equation with arbitrary
constants.
In [18], Bretherton introduced the partial differential
equation
𝑢𝑡𝑡 + 𝑢𝑥𝑥 + 𝑢𝑥𝑥𝑥𝑥 + 𝑢 − 𝑢2 = 0
(1a)
in time and one spatial dimension as a model of a dispersive
wave system to study the resonant nonlinear interaction
between three liner models. The modified Bretherton equation
𝑢𝑡𝑡 + 𝑢𝑥𝑥 + 𝑢𝑥𝑥𝑥𝑥 + 𝑢 − 𝑢3 = 0
(1b)
was studied by Kudryashov [19], Kudryashov et al. [20], and
Berloff and Howard [21], and its travelling wave solutions
were obtained.
Our aim in this paper is to investigate multiple solitonlike solutions to the generalized Bretherton equation in [22]
by using the solutions to the Riccati equation:
𝑢𝑡𝑡 + 𝛼𝑢𝑥𝑥 + 𝛽𝑢𝑥𝑥𝑥𝑥 + 𝛿𝑢 + 𝛾𝑢3 = 0.
(1c)
2. Multiple Soliton-Like Solutions to the
Generalized Bretherton Equation
We assume the travelling wave variable
𝑢 (𝑥, 𝑡) = 𝑢 (𝜉) ,
𝜉 = 𝑥 − 𝑉𝑡,
(2)
where 𝑉 is the speed of the travelling wave.
Making use of the travelling wave transformation (2), (1c)
is converted into an ordinary differential equation (ODE) for
𝑢 = 𝑢(𝜉) as follows:
(𝑉2 + 𝛼) 𝑢󸀠󸀠 + 𝛽𝑢(4) + 𝛿𝑢 + 𝛾𝑢3 = 0.
(3)
We assume that the solutions to (3) can be expressed in the
form
𝑀
𝑢 (𝜉) = ∑ 𝑎𝑖 𝜙𝑖 ,
𝑖 = −𝑀
(4)
2
Abstract and Applied Analysis
+ 14𝛽𝑎−2 𝐶2 𝐵2 + 6𝛾𝑎2 𝑎0 𝑎−2 + 6𝛾𝑎1 𝑎−1 𝑎0
where 𝜙 is a solution of the Riccati equation,
𝜙󸀠 = 𝐴 + 𝐵𝜙 + 𝐶𝜙2 ,
(5)
where 𝑎𝑖 (𝑖 = 0, ±1, ±2, . . . , ±𝑀) 𝐴, 𝐵, and 𝐶 are constants to
be determined later, and either 𝑎−𝑀 or 𝑎𝑀 can be zero, but
they cannot be zero together.
Substituting (4) into (3) together with (5) and considering
the homogeneous balance between the highest-order derivative 𝑢(4) and the nonlinear term 𝑢3 , we obtain 𝑀 = 2. Thus
the solution to (3) takes the following form:
𝑢 (𝜉) = 𝑎2 𝜙2 + 𝑎1 𝜙 + 𝑎0 + 𝑎−1 𝜙−1 + 𝑎−2 𝜙−2 .
(6)
Substituting (6) with (5) into (3) and collecting all the terms of
the same power of 𝜙, the left-hand side of (3) is converted into
another polynomial of 𝜙. Setting the coefficients of 𝜙𝑖 (𝑖 =
0, ±1, ±2) to zero yields a set of algebraic equations
3
− 120𝛽𝑎−2 𝐴4 = 0,
𝛾𝑎−2
+ 8𝛽𝛼1 𝐶𝐵𝐴2 + 𝛾𝑎03 = 0,
6𝛾𝑎2 𝑎−1 𝑎0 + 𝛼𝑎−1 𝐵2 + 6𝑎2 𝑉2 𝐵𝐴 + 𝑎1 𝑉2 𝐵2
+ 3𝛾𝑎1 𝑎02 + 2𝑎1 𝑉2 𝐶𝐴 + 2𝛼𝑎1 𝐶𝐴 + 3𝛾𝑎12 𝑎−1 + 𝛿𝑎1
+ 6𝛼𝑎2 𝐵𝐴 + 22𝛽𝑎1 𝐶𝐵2 𝐴 + 30𝛽𝑎2 𝐵3 𝐴
+ 16𝛽𝑎1 𝐶2 𝐴2 + 𝛽𝑎1 𝐵4
+ 6𝛾𝑎2 𝑎1 𝑎−2 + 120𝛽𝑎2 𝐶𝐵𝐴2 = 0,
4𝛼𝑎2 𝐵2 + 8𝛼𝑎2 𝐶𝐴 + 136𝛽𝑎2 𝐶2 𝐴2 + 16𝛽𝑎2 𝐵4
+ 3𝛾𝑎2 𝑎02 + 𝛿𝑎2 + 3𝑉2 𝐵𝐶 + 3𝛼𝑎1 𝐶𝐵
+ 15𝛽𝑎1 𝐶𝐵3 + 4𝑎2 𝑉2 𝐵2 + 60𝛽𝑎1 𝐴𝐵𝐶2
+ 8𝑎2 𝑉2 𝐶𝐴 + 3𝛾𝑎12 𝑎0 + 23𝛽𝑎2 𝐶𝐵2 𝐴
+ 3𝛾𝑎22 𝑎−2 + 6𝛾𝑎2 𝑎1 𝑎−1 = 0,
2
3𝛾𝑎−1 𝑎−2
+ 24𝛽𝑎−1 𝐴4 + 96𝛽𝑎−2 𝐵𝐴3 = 0,
40𝛽𝑎1 𝐶3 𝐴 + 2𝛼𝑎1 𝐶2 + 50𝛽𝑎1 𝐶2 𝐵2 + 10𝛼𝑎2 𝐶𝐵
2
330𝛽𝑎−2 𝐵2 𝐴2 + 60𝛽𝑎−1 𝐵𝐴3 + 3𝛾𝑎−1
𝑎−2 + 6𝛼𝑎−2 𝐴2
2
+ 3𝛾𝑎0 𝑎−2
+ 6𝑉2 𝑎−2 𝐴2 = 0,
+ 6𝛾𝑎2 𝑎1 𝑎0 + 𝛾𝑎13 + 10𝑎2 𝑉2 𝐵𝐶
+ 3𝛾𝑎22 𝑎−1 + 130𝛽𝑎2 𝐶𝐵3 + 440𝛽𝑎2 𝐶2 𝐵𝐴
2
40𝛽𝑎−1 𝐴3 𝐶 + 3𝛾𝑎1 𝑎−2
+ 2𝑉2 𝑎−1 𝐴2 + 10𝑉2 𝑎−2 𝐵𝐴 + 2𝛼𝑎−1 𝐴2
3
+ 50𝛽𝑎−1 𝐵2 𝐴2 + 6𝛾𝑎−1 𝑎0 𝑎−2 + 130𝛽𝐵3 𝐴 + 𝛾𝑎−1
+ 2𝑎1 𝑉2 𝐶2 = 0,
3𝛾𝑎22 𝑎0 + 3𝛾𝑎12 𝑎2 + 6𝑎2 𝑉2 𝐶2 + 60𝛽𝑎1 𝐶3 𝐵
+ 440𝛽𝑎−2 𝐶𝐵𝐴2 + 10𝛼𝑎−2 𝐵𝐴 = 0,
+ 240𝛽𝑎2 𝐶3 𝐴 + 330𝛽𝑎2 𝐶2 𝐵2 + 6𝛼𝑎2 𝐶2 = 0,
2
6𝛾𝑎1 𝑎−1 𝑎−2 + 60𝛽𝑎−1 𝐴2 𝐶𝐵 + 3𝛾𝑎2 𝑎−2
+ 𝛿𝑎−2
3𝛾𝑎22 𝑎1 + 24𝛽𝑎1 𝐶4 + 336𝛽𝑎2 𝐶3 𝐵 = 0,
2
+ 3𝛾𝑎−1
𝑎0 + 3𝛼𝑎−1 𝐵𝐴 + 8𝑉2 𝑎−2 𝐶𝐴
𝛾𝑎23 + 120𝛽𝑎2 𝐶4 = 0.
(7)
+ 8𝛼𝑎−2 𝐶𝐴 + 136𝛽𝑎−2 𝐶2 𝐴2 + 3𝛾𝑎02 𝑎−2
+ 4𝛼𝑎−2 𝐵2 + 4𝑉2 𝑎−2 𝐵2 + 16𝛽𝑎−2 𝐵4
Solving (7) with the help of the symbolic computation
software Maple, we obtain the following.
+ 3𝑉2 𝑎−1 𝐵𝐴 + 15𝛽𝑎−1 𝐵3 𝐴 + 232𝛽𝑎−2 𝐶𝐵2 𝐴 = 0,
Case 1. One has
16𝛽𝑎−1 𝐴2 𝐶2 + 6𝑉2 𝑎−2 𝐶𝐵 + 2𝛼𝑎−1 𝐴𝐶 + 𝛽𝑎−1 𝐵4
𝑎−1 = 0,
+ 𝛿𝑎−1 + 6𝛾𝑎2 𝑎−1 𝑎−2 + 30𝛽𝑎−2 𝐶𝐵3 + 3𝛾𝑎−1 𝑎02
𝑎1 = 2𝐵𝐶√
+ 2𝑉2 𝑎−1 𝐴𝐶 + 6𝛾𝑎1 𝑎0 𝑎−2 + 𝛼𝑎−1 𝐵2
+ 120𝛽𝑎−2 𝐶2 𝐴𝐵 + 6𝛼𝑎−2 𝐶𝐵 + 22𝛽𝑎−1 𝐴𝐶𝐵2
−30𝛽
,
𝛾
Case 2. One has
2
+ 3𝛾𝑎2 𝑎−1
+ 8𝛽𝑎−1 𝐴𝐶2 𝐵 + 𝛿𝑎0 + 𝑎1 𝑉2 𝐵𝐴
𝑎−1 = 0,
+ 𝑉2 𝑎−1 𝐵𝐶 + 𝛼𝑎1 𝐵𝐴 + 𝛼𝑎−1 𝐵𝐶 + 16𝛽𝑎2 𝐶𝐴3
2
𝑎0 = 2𝐴𝐶√
−30𝛽
,
𝛾
𝛿 = 4𝛽 (−8𝐶𝐴𝐵2 + 𝐵4 + 16𝐴2 𝐶2 ) .
2𝛼𝑎−2 𝐶2 + 2𝛼𝑎2 𝐴2 + 2𝑎2 𝑉2 𝐴2 + 2𝑎−2 𝐶2 + 3𝛾𝑎12 𝑎−2
2
−30𝛽
,
𝛾
𝑎2 = 2𝐶2 √
𝑉 = ±√−𝛼 − 5𝐵2 𝛽 + 20𝐶𝐴𝛽,
2
+ 𝑎−1 𝑉2 𝐵2 + 3𝛾𝑎1 𝑎−1
= 0,
3
𝑎−2 = 0,
3
3
+ 𝛽𝑎1 𝐵 𝐴 + 14𝛽𝑎2 𝐵 𝐴 + 16𝛽𝑎−2 𝐶 𝐴 + 𝛽𝑎−1 𝐵 𝐶
𝑎−2 = 0,
𝑎1 = −2𝐵𝐶√
−30𝛽
,
𝛾
𝑎2 = −2𝐶2 √
𝑎0 = −2𝐴𝐶√
−30𝛽
,
𝛾
−30𝛽
,
𝛾
(8a)
Abstract and Applied Analysis
3
𝑉 = ±√−𝛼 − 5𝐵2 𝛽 + 20𝐶𝐴𝛽,
𝛿 = 4𝛽 (−8𝐶𝐴𝐵2 + 𝐵4 + 16𝐴2 𝐶2 ) ,
(8b)
where 𝐴, 𝐵, and 𝐶 are arbitrary constants, but 𝐶 cannot be
zero.
Case 3. One has
𝑎−1 = 0,
𝑎2 = 0,
𝑎−2 = 2𝐴2 √
𝑎1 = 0,
30𝛽
𝑎0 = 2𝐴𝐶√
,
𝛾
𝛿 = −16𝛽𝐴 𝐶 .
Case 4. One has
𝑎2 = 0,
𝑎0 = −
−15𝛽
(coth (𝑥 + 𝜀√−𝛼 − 5𝛽𝑡)
2𝛾
2
± csch (𝑥 + 𝜀√−𝛼 − 5𝛽𝑡))
− 𝜃√
2
(8c)
𝑎−1 = 0,
𝑢1𝑎 = 𝜃√
𝑎−2 = 2𝐴2 √
𝑎1 = 0,
30𝛽
−15 ± √165
𝐶𝐴√
,
15
𝛾
30𝛽
,
𝛾
2
− 𝜃√
𝑢1𝑐 = 𝜃√
2
+ 𝜃√
Case 5. One has
𝑎1 = 0,
30𝛽
𝑎0 = −2𝐴𝐶√
,
𝛾
𝑎−2
30𝛽
= −2𝐴 √
,
𝛾
2
𝑢1𝑑 = 𝜃√
2
𝐵 = 0,
+ 𝜃√
Case 6. One has
𝑎2 = 0,
𝑎0 =
𝑎1 = 0,
(−15 ± √165) 𝐶𝐴
15
𝑎−2 = −2𝐴2 √
30𝛽
√
,
𝛾
−15𝛽
,
2𝛾
−15𝛽
(csc (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡)
2𝛾
(8e)
𝑎−1 = 0,
(9c)
− cot (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡))
𝛿 = −16𝐵𝐴2 𝐶2 .
𝑉 = ±√−𝛼 − 60𝐶𝐴𝛽,
−15𝛽
,
2𝛾
± sec (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡))
(8d)
where 𝐴, 𝐵, and 𝐶 are arbitrary constants, but 𝐴 cannot be
zero.
30𝛽
,
𝛾
𝑢1𝑓 = 𝜃√
𝑉 = 𝜀√−𝛼 + (−30 ± 2√165) 𝐶𝐴𝛽,
𝛿 = −4𝐴2 𝐶2 𝛽 (13 ± √165) ,
(9d)
−15𝛽
,
2𝛾
tan (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡)
−15𝛽
)
𝑢1𝑒 = 𝜃√
(
2𝛾
(1 ± sec (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡))
+ 𝜃√
𝐵 = 0,
2
(9e)
−15𝛽
,
2𝛾
−15𝛽
(cot (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡)
2𝛾
2
± csc (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡))
(8f)
where 𝐴, 𝐵, and 𝐶 are arbitrary constants, while 𝐶 cannot be
zero in Cases 1 and 2 and 𝐴 cannot be zero in Cases 3–6. 𝜀 is
an arbitrary element of {−1, 1}.
(9b)
−15𝛽
(tan (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡)
2𝛾
𝛿 = −4𝐴2 𝐶2 𝛽 (13 ± √165) ,
𝑎2 = 0,
−15𝛽
,
2𝛾
tanh (𝑥 + 𝜀√−𝛼 − 5𝛽𝑡)
−15𝛽
)
𝑢1𝑏 = 𝜃√
(
2𝛾
(1 ± sech (𝑥 + 𝜀√−𝛼 − 5𝛽𝑡))
𝐵 = 0,
𝑉 = 𝜀√−𝛼 + (−30 ± 2√165) 𝐶𝐴𝛽,
𝑎−1 = 0,
(9a)
𝐵 = 0,
2
𝑉 = ±√−𝛼 − 60𝐶𝐴𝛽,
30𝛽
,
𝛾
Substituting ((8a), (8b), (8c), (8d), (8e), (8f)) into (6)
respectively and taking advantage of solitions to (5), we can
find the following solutions which contain multiple solitionlike and triangular periodic solutions for the generalized
Bretherton equation.
When 𝛿 = 4𝛽,
+ 𝜃√
−15𝛽
,
2𝛾
(9f)
4
Abstract and Applied Analysis
𝑢1𝑔 = 𝜃√
when 𝛿 = 1024𝛽,
−15𝛽
(sec (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡)
2𝛾
2
− tan (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡))
+ 𝜃√
𝑢1ℎ
(9g)
−15𝛽
,
2𝛾
− 8𝜃√
cot (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡)
−15𝛽
)
= 𝜃√
(
2𝛾
(1 ± csc (𝑥 + 𝜀√−𝛼 + 5𝛽𝑡))
tan (𝑥 ± √−𝛼 + 80𝛽𝑡)
−30𝛽
)
(
𝛾
(1 − tan2 (𝑥 ± √−𝛼 + 80𝛽𝑡))
+ 8𝜃√
−30𝛽
−30𝛽
coth2 (𝑥 ± √−𝛼 − 20𝛽𝑡) − 2𝜃√
,
𝛾
𝛾
(10b)
−30𝛽 2
−30𝛽
tan (𝑥 ± √−𝛼 + 20𝛽𝑡) + 2𝜃√
,
𝛾
𝛾
(10c)
𝑢2𝑑 = 2𝜃√
2
𝑢3𝑏 = 32𝜃√
−30𝛽
−30𝛽
= 2𝜃√
tanh2 (𝑥 ± √−𝛼 − 20𝛽𝑡) − 2𝜃√
,
𝛾
𝛾
(10a)
𝑢2𝑐 = 2𝜃√
−30𝛽 2
−30𝛽
cot (𝑥 ± √−𝛼 + 20𝛽𝑡) + 2𝜃√
,
𝛾
𝛾
(10d)
+ 4𝜃√
𝑢2𝑓 = 8𝜃√
𝑢4 =
∓ 8𝜃√
−30𝛽 cot (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡)
𝛾 (1 ± cot (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡))
+ 4𝜃√
−30𝛽
,
𝛾
±2√−30𝛽/𝛾
2
((𝑥 + 𝜀√−𝛼𝑡) + 𝑐0 )
2
.
(12)
When 𝛿 = −𝛽,
𝑢5𝑎 = 𝜃√
15𝛽
(coth (𝑥 + 𝜀√−𝛼 + 15𝛽𝑡)
2𝛾
± csch (𝑥 + 𝜀√−𝛼 + 15𝛽𝑡))
(10e)
−2
(13a)
15𝛽
,
2𝛾
−2
tanh (𝑥 + 𝜀√−𝛼 + 15𝛽𝑡)
15𝛽
)
𝑢5𝑏 = 𝜃√
(
2𝛾 (1 ± sech (𝑥 + 𝜀√−𝛼 + 15𝛽𝑡))
−30𝛽
,
𝛾
cot (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡)
−30𝛽
)
(
𝛾
(1 ± cot (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡))
(11c)
when 𝛿 = 0,
− 𝜃√
−30𝛽
𝛾 (1 ± tan (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡))
2
−30𝛽
+ 8𝜃√
,
𝛾
2
tan (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡)
(11b)
−30𝛽
,
𝛾
cot (𝑥 ± √−𝛼 + 80𝛽𝑡)
−30𝛽
)
𝑢3𝑐 = 32𝜃√
(
𝛾
(1 − cot2 (𝑥 ± √−𝛼 + 80𝛽𝑡))
tan (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡)
−30𝛽
)
𝑢2𝑒 = 8𝜃√
(
𝛾
(1 ± tan (𝑥 + 𝜀√−𝛼 + 20𝛽𝑡))
∓ 8𝜃√
−30𝛽
,
𝛾
(11a)
(9h)
when 𝛿 = 64𝛽,
𝑢2𝑏 = 2𝜃√
2
2
−15𝛽
+ 𝜃√
,
2𝛾
𝑢2𝑎
𝑢3𝑎
tanh (𝑥 ± √−𝛼 − 80𝛽𝑡)
−30𝛽
)
= 32𝜃√
(
𝛾
(1 + tanh2 (𝑥 ± √−𝛼 − 80𝛽𝑡))
− 𝜃√
2
𝑢5𝑐 = 𝜃√
(13b)
15𝛽
,
2𝛾
15𝛽
(tan (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡)
2𝛾
(10f)
± sec (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡))
+ 𝜃√
15𝛽
,
2𝛾
−2
(13c)
Abstract and Applied Analysis
5
−2
𝑢5𝑑
tan (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡)
15𝛽
)
= 𝜃√
(
2𝛾 (1 ± sec (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡))
+ 𝜃√
𝑢5𝑒 = 𝜃√
15𝛽
,
2𝛾
− 8𝜃√
15𝛽
(cot (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡)
2𝛾
+ 𝜃√
𝑢7𝑐 = 2𝜃√
(13e)
−2
(13f)
𝑢6𝑑 = 2𝜃√
(15c)
15𝛽
2𝛾
1
(−30 + 2𝜎√165) 𝛽𝑡)
4
± csch (𝑥 + 𝜀√ −𝛼 −
−2
(13g)
30𝛽
30𝛽
tanh−2 (𝑥 ± √−𝛼 + 60𝛽𝑡) − 2𝜃√
,
𝛾
𝛾
(14a)
30𝛽
30𝛽
coth−2 (𝑥 ± √−𝛼 + 60𝛽𝑡) − 2𝜃√
,
𝛾
𝛾
(14b)
30𝛽 −2
30𝛽
tan (𝑥 ± √−𝛼 − 60𝛽𝑡) + 2𝜃√
, (14c)
𝛾
𝛾
30𝛽 −2
30𝛽
cot (𝑥 ± √−𝛼 − 60𝛽𝑡) + 2𝜃√
, (14d)
𝛾
𝛾
when 𝛿 = −64𝛽,
−2
1
(−30 + 2𝜎√165) 𝛽𝑡))
4
√165
30𝛽
1
+ 𝜃√
(−1 + 𝜎
),
4
𝛾
15
(16a)
𝑢8𝑏 = 𝜃√
when 𝛿 = −16𝛽,
𝑢6𝑐 = 2𝜃√
−2
30𝛽
,
𝛾
× (coth (𝑥 + 𝜀√ −𝛼 −
15𝛽
,
2𝛾
−15𝛽
+ 𝜃√
,
2𝛾
𝑢6𝑏 = 2𝜃√
cot (𝑥 ± √−𝛼 − 240𝛽𝑡)
30𝛽
)
(
𝛾 (1 − cot2 (𝑥 ± √−𝛼 − 240𝛽𝑡))
+ 8𝜃√
𝑢8𝑎 = 𝜃√
cot (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡)
15𝛽
)
= 𝜃√
(
2𝛾 (1 ± csc (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡))
𝑢6𝑎 = 2𝜃√
30𝛽
,
𝛾
when 𝛿 = −(13 ± √165)𝛽,
15𝛽
(sec (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡)
2𝛾
+ 𝜃√
𝑢5𝑔
−2
15𝛽
,
2𝛾
− tan (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡))
−2
(15b)
± csc (𝑥 + 𝜀√−𝛼 − 15𝛽𝑡))
𝑢5𝑓 = 𝜃√
(13d)
tanh (𝑥 ± √−𝛼 + 240𝛽𝑡)
30𝛽
)
(
𝑢7𝑏 = 2𝜃√
𝛾 (1 + tanh2 (𝑥 ± √−𝛼 + 240𝛽𝑡))
15𝛽
2𝛾
−2
tanh (𝑥 + 𝜀√−𝛼 − (1/4) (−30 + 2𝜎√165) 𝛽𝑡)
×(
)
(1± sech(𝑥 +𝜀√−𝛼 − (1/4) (−30 + 2𝜎√165) 𝛽𝑡))
√165
30𝛽
1
+ 𝜃√
(−1 + 𝜎
),
4
𝛾
15
(16b)
𝑢8𝑐 = 𝜃√
15𝛽
1
(tan (𝑥 + 𝜀√ −𝛼 + (−30 + 2𝜎√165) 𝛽𝑡)
2𝛾
4
1
± sec (𝑥+𝜀√ −𝛼+ (−30+2𝜎√165) 𝛽𝑡))
4
−2
√165
30𝛽
1
− 𝜃√
(−1 + 𝜎
),
4
𝛾
15
(16c)
𝑢7𝑎 = 2𝜃√
tanh (𝑥 ± √−𝛼 − 240𝛽𝑡)
30𝛽
)
(
𝛾 (1 − tanh2 (𝑥 ± √−𝛼 − 240𝛽𝑡))
+ 8𝜃√
30𝛽
,
𝛾
−2
𝑢8𝑑 = 𝜃√
(15a)
15𝛽
2𝛾
× (csc (𝑥 + 𝜀√ −𝛼 +
1
(−30 + 2𝜎√165) 𝛽𝑡)
4
6
Abstract and Applied Analysis
1
− cot (𝑥+ 𝜀√ −𝛼+ (−30 +2𝜎√165) 𝛽𝑡))
4
−2
when 𝛿 = −4(13 ± √165)𝛽,
√165
30𝛽
1
− 𝜃√
(−1 + 𝜎
),
4
𝛾
15
𝑢9𝑎 = 2𝜃√
(16d)
30𝛽
tanh−2 (𝑥 + 𝜀√−𝛼 − (−30 + 2𝜎√165) 𝛽𝑡)
𝛾
+ 𝜃√
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(17a)
𝑢8𝑒
= 𝜃√
15𝛽
2𝛾
𝑢9𝑏 = 2𝜃√
tan (𝑥 + 𝜀√−𝛼 + (1/4) (−30+ 2𝜎√165) 𝛽𝑡)
−2
+ 𝜃√
)
×(
(1± sec (𝑥 + 𝜀√−𝛼 + (1/4) (−30+ 2𝜎√165) 𝛽𝑡))
𝑢9𝑐 = 2𝜃√
(16e)
30𝛽 −2
tan (𝑥 + 𝜀√−𝛼 + (−30 + 2𝜎√165) 𝛽𝑡)
𝛾
− 𝜃√
15𝛽
2𝛾
× (cot (𝑥 + 𝜀√ −𝛼 +
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(17b)
√165
30𝛽
1
− 𝜃√
(−1 + 𝜎
),
4
𝛾
15
𝑢8𝑓 = 𝜃√
30𝛽
coth−2 (𝑥 + 𝜀√−𝛼 − (−30 + 2𝜎√165) 𝛽𝑡)
𝛾
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(17c)
1
(−30 + 2𝜎√165) 𝛽𝑡)
4
± csc (𝑥 + 𝜀√ −𝛼 +
𝑢9𝑑 = 2𝜃√
30𝛽 −2
cot (𝑥 + 𝜀√−𝛼 + (−30 + 2𝜎√165) 𝛽𝑡)
𝛾
−2
1
(−30 + 2𝜎√165) 𝛽𝑡))
4
− 𝜃√
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(17d)
√165
30𝛽
1
− 𝜃√
(−1 + 𝜎
),
4
𝛾
15
when 𝛿 = −64(13 ± √165)𝛽,
(16f)
𝑢8𝑔 = 𝜃√
𝑢10𝑎
15𝛽
2𝛾
× (sec (𝑥 + 𝜀√ −𝛼 +
= 2𝜃√
1
(−30 + 2𝜎√165) 𝛽𝑡)
4
− tan (𝑥 + 𝜀√ −𝛼 +
1
(−30 + 2𝜎√165) 𝛽𝑡))
4
−2
√165
30𝛽
1
− 𝜃√
(−1 + 𝜎
),
4
𝛾
15
30𝛽
𝛾
−2
tanh (𝑥 + 𝜀√−𝛼 − 4 (−30 + 2𝜎√165) 𝛽𝑡)
×(
)
(1 + tanh2 (𝑥 + 𝜀√−𝛼 − 4 (−30 + 2𝜎√165) 𝛽𝑡))
+ 4𝜃√
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(16g)
(18a)
𝑢10𝑏
𝑢8ℎ
= 𝜃√
15𝛽
2𝛾
= 2𝜃√
cot (𝑥 + 𝜀√−𝛼 + (1/4) (−30 + 2𝜎√165) 𝛽𝑡)
30𝛽
𝛾
−2
tanh (𝑥 + 𝜀√−𝛼 + 4 (−30 + 2𝜎√165) 𝛽𝑡)
−2
)
×(
(1± csc (𝑥 + 𝜀√−𝛼 + (1/4) (−30+ 2𝜎√165) 𝛽𝑡))
×(
)
(1 − tanh2 (𝑥 + 𝜀√−𝛼 + 4 (−30 + 2𝜎√165) 𝛽𝑡))
√165
30𝛽
1
− 𝜃√
(−1 + 𝜎
),
4
𝛾
15
− 4𝜃√
(16h)
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(18b)
Abstract and Applied Analysis
7
𝑢10𝑐
= 2𝜃√
30𝛽
𝛾
cot (𝑥 + 𝜀√−𝛼 + 4 (−30 + 2𝜎√165) 𝛽𝑡)
−2
×(
)
(1 − cot2 (𝑥+ 𝜀√−𝛼 + 4 (−30 + 2𝜎√165) 𝛽𝑡))
− 4𝜃√
√165
30𝛽
(−1 + 𝜎
),
𝛾
15
(18c)
where 𝜀, 𝜃, and 𝜎 are arbitrary elements of {−1, 1}.
3. Conclusion
In this paper, we have used solutions to the Riccati equation
to solve the generalized Bretherton equation with arbitrary
constants and obtained abundant new multiple solition-like
and triangular periodic solutions. It is significant to observe
practical denotation of the obtained solutions, so the obtained
solutions involving arbitrary constants in this paper have
potential applications in dispersive wave systems to research
for resonant nonlinear interactions.
Acknowledgment
This work is supported by the Natural Science Foundation of
Shandong Province of China (ZR2010AL019).
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