# Abundant Exact Solition-Like Solutions to the Generalized Bretherton Equation with Arbitrary Constants

Authors: Xiuqing Yu; Fengsheng Xu; Lihua Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 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(1a)utt+uxx+uxxxx+u-u2=0
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
(1b)utt+uxx+uxxxx+u-u3=0
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 soliton-like solutions to the generalized Bretherton equation in [22] by using the solutions to the Riccati equation:
(1c)utt+αuxx+βuxxxx+δu+γu3=0.


## 2. Multiple Soliton-Like Solutions to the Generalized Bretherton Equation

We assume the travelling wave variable(2)u(x,t)=u(ξ),ξ=x-Vt,
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:
(3)(V2+α)u′′+βu(4)+δu+γu3=0.
We assume that the solutions to (3) can be expressed in the form
(4)u(ξ)=∑i=-MMaiϕi,
where ϕ is a solution of the Riccati equation,
(5)ϕ′=A+Bϕ+Cϕ2,
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:
(6)u(ξ)=a2ϕ2+a1ϕ+a0+a-1ϕ-1+a-2ϕ-2.
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
(7)γa-23-120βa-2A4=0,3γa-1a-22+24βa-1A4+96βa-2BA3=0,330βa-2B2A2+60βa-1BA3+3γa-12a-2+6αa-2A2+3γa0a-22+6V2a-2A2=0,40βa-1A3C+3γa1a-22+2V2a-1A2+10V2a-2BA+2αa-1A2+50βa-1B2A2+6γa-1a0a-2+130βB3A+γa-13+440βa-2CBA2+10αa-2BA=0,6γa1a-1a-2+60βa-1A2CB+3γa2a-22+δa-2+3γa-12a0+3αa-1BA+8V2a-2CA+8αa-2CA+136βa-2C2A2+3γa02a-2+4αa-2B2+4V2a-2B2+16βa-2B4+3V2a-1BA+15βa-1B3A+232βa-2CB2A=0,16βa-1A2C2+6V2a-2CB+2αa-1AC+βa-1B4+δa-1+6γa2a-1a-2+30βa-2CB3+3γa-1a02+2V2a-1AC+6γa1a0a-2+αa-1B2+120βa-2C2AB+6αa-2CB+22βa-1ACB2+a-1V2B2+3γa1a-12=0,2αa-2C2+2αa2A2+2a2V2A2+2a-2C2+3γa12a-2+3γa2a-12+8βa-1AC2B+δa0+a1V2BA+V2a-1BC+αa1BA+αa-1BC+16βa2CA3+βa1B3A+14βa2B2A2+16βa-2C3A+βa-1B3C+14βa-2C2B2+6γa2a0a-2+6γa1a-1a0+8βα1CBA2+γa03=0,6γa2a-1a0+αa-1B2+6a2V2BA+a1V2B2+3γa1a02+2a1V2CA+2αa1CA+3γa12a-1+δa1+6αa2BA+22βa1CB2A+30βa2B3A+16βa1C2A2+βa1B4+6γa2a1a-2+120βa2CBA2=0,4αa2B2+8αa2CA+136βa2C2A2+16βa2B4+3γa2a02+δa2+3V2BC+3αa1CB+15βa1CB3+4a2V2B2+60βa1ABC2+8a2V2CA+3γa12a0+23βa2CB2A+3γa22a-2+6γa2a1a-1=0,40βa1C3A+2αa1C2+50βa1C2B2+10αa2CB+6γa2a1a0+γa13+10a2V2BC+3γa22a-1+130βa2CB3+440βa2C2BA+2a1V2C2=0,3γa22a0+3γa12a2+6a2V2C2+60βa1C3B+240βa2C3A+330βa2C2B2+6αa2C2=0,3γa22a1+24βa1C4+336βa2C3B=0,γa23+120βa2C4=0.
Solving (7) with the help of the symbolic computation software Maple, we obtain the following.Case 1.
One has(8a)a-1=0,a-2=0,a2=2C2-30βγ,a1=2BC-30βγ,a0=2AC-30βγ,V=±-α-5B2β+20CAβ,δ=4β(-8CAB2+B4+16A2C2).Case 2.
One has(8b)a-1=0,a-2=0,a2=-2C2-30βγ,a1=-2BC-30βγ,a0=-2AC-30βγ,V=±-α-5B2β+20CAβ,δ=4β(-8CAB2+B4+16A2C2),
where A, B, and C are arbitrary constants, but C cannot be zero.Case 3.
One has(8c)a-1=0,a2=0,a1=0,a-2=2A230βγ,a0=2AC30βγ,B=0,V=±-α-60CAβ,δ=-16βA2C2.Case 4.
One has(8d)a-1=0,a2=0,a1=0,a-2=2A230βγ,a0=--15±16515CA30βγ,B=0,V=ε-α+(-30±2165)CAβ,δ=-4A2C2β(13±165),
where A,B, and C are arbitrary constants, but A cannot be zero.Case 5.
One has(8e)a-1=0,a2=0,a1=0,a-2=-2A230βγ,a0=-2AC30βγ,B=0,V=±-α-60CAβ,δ=-16BA2C2.Case 6.
One has(8f)a-1=0,a2=0,a1=0,a-2=-2A230βγ,a0=(-15±165)CA1530βγ,B=0,V=ε-α+(-30±2165)CAβ,δ=-4A2C2β(13±165),
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}.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 solition-like and triangular periodic solutions for the generalized Bretherton equation.Whenδ=4β,(9a)u1a=θ-15β2γ(coth(x+ε-α-5βt)θ-15β2γc±csch(x+ε-α-5βt))2-θ-15β2γ,(9b)u1b=θ-15β2γ(tanh(x+ε-α-5βt)(1±sech(x+ε-α-5βt)))2-θ-15β2γ,(9c)u1c=θ-15β2γ(tan(x+ε-α+5βt)θ-15β2γcc±sec(x+ε-α+5βt))2+θ-15β2γ,(9d)u1d=θ-15β2γ(csc(x+ε-α+5βt)θ-15β2γc-cot(x+ε-α+5βt))2+θ-15β2γ,(9e)u1e=θ-15β2γ(tan(x+ε-α+5βt)(1±sec(x+ε-α+5βt)))2+θ-15β2γ,(9f)u1f=θ-15β2γ(cot(x+ε-α+5βt)θ-15β2γcc±csc(x+ε-α+5βt))2+θ-15β2γ,(9g)u1g=θ-15β2γ(sec(x+ε-α+5βt)θ-15β2γc-tan(x+ε-α+5βt))2+θ-15β2γ,(9h)u1h=θ-15β2γ(cot(x+ε-α+5βt)(1±csc(x+ε-α+5βt)))2+θ-15β2γ,
when δ=64β,(10a)
u
2
a
=
2
θ
-
30
β
γ
tanh
2
(
x
±
-
α
-
20
β
t
)
-
2
θ
-
30
β
γ
,
(10b)
u
2
b
=
2
θ
-
30
β
γ
cot
h
2
(
x
±
-
α
-
20
β
t
)
-
2
θ
-
30
β
γ
,
(10c)
u
2
c
=
2
θ
-
30
β
γ
ta
n
2
(
x
±
-
α
+
20
β
t
)
+
2
θ
-
30
β
γ
,
(10d)
u
2
d
=
2
θ
-
30
β
γ
cot
2
(
x
±
-
α
+
20
β
t
)
+
2
θ
-
30
β
γ
,
(10e)
u
2
e
=
8
θ
-
30
β
γ
(
tan
(
x
+
ε
-
α
+
20
β
t
)
(
1
±
tan
(
x
+
ε
-
α
+
20
β
t
)
)
)
2
∓
8
θ
-
30
β
γ
tan
(
x
+
ε
-
α
+
20
β
t
)
(
1
±
tan
(
x
+
ε
-
α
+
20
β
t
)
)
+
4
θ
-
30
β
γ
,
(10f)
u
2
f
=
8
θ
-
30
β
γ
(
cot
(
x
+
ε
-
α
+
20
β
t
)
(
1
±
cot
(
x
+
ε
-
α
+
20
β
t
)
)
)
2
∓
8
θ
-
30
β
γ
cot
(
x
+
ε
-
α
+
20
β
t
)
(
1
±
cot
(
x
+
ε
-
α
+
20
β
t
)
)
+
4
θ
-
30
β
γ
,
when δ=1024β,(11a)
u
3
a
=
32
θ
-
30
β
γ
(
tanh
(
x
±
-
α
-
80
β
t
)
(
1
+
tanh
2
(
x
±
-
α
-
80
β
t
)
)
)
2
-
8
θ
-
30
β
γ
,
(11b)
u
3
b
=
32
θ
-
30
β
γ
(
tan
(
x
±
-
α
+
80
β
t
)
(
1
-
ta
n
2
(
x
±
-
α
+
80
β
t
)
)
)
2
+
8
θ
-
30
β
γ
,
(11c)
u
3
c
=
32
θ
-
30
β
γ
(
cot
(
x
±
-
α
+
80
β
t
)
(
1
-
cot
2
(
x
±
-
α
+
80
β
t
)
)
)
2
+
8
θ
-
30
β
γ
,
when δ=0,(12)u4=±2-30β/γ((x+ε-αt)2+c0)2.
When δ=-β,(13a)u5a=θ15β2γ(coth(x+ε-α+15βt)θ15β2γcc±csch(x+ε-α+15βt))-2-θ15β2γ,(13b)u5b=θ15β2γ(tanh(x+ε-α+15βt)(1±sech(x+ε-α+15βt)))-2-θ15β2γ,(13c)u5c=θ15β2γ(tan(x+ε-α-15βt)θ15β2γcc±sec(x+ε-α-15βt))-2+θ15β2γ,(13d)u5d=θ15β2γ(tan(x+ε-α-15βt)(1±sec(x+ε-α-15βt)))-2+θ15β2γ,(13e)u5e=θ15β2γ(cot(x+ε-α-15βt)θ15β2γcc±csc(x+ε-α-15βt))-2+θ15β2γ,(13f)u5f=θ15β2γ(sec(x+ε-α-15βt)θ15β2γcc-tan(x+ε-α-15βt))-2+θ15β2γ,(13g)u5g=θ15β2γ(cot(x+ε-α-15βt)(1±csc(x+ε-α-15βt)))-2+θ-15β2γ,
when δ=-16β,(14a)
u
6
a
=
2
θ
30
β
γ
tanh
-
2
(
x
±
-
α
+
60
β
t
)
-
2
θ
30
β
γ
,
(14b)
u
6
b
=
2
θ
30
β
γ
cot
h
-
2
(
x
±
-
α
+
60
β
t
)
-
2
θ
30
β
γ
,
(14c)
u
6
c
=
2
θ
30
β
γ
ta
n
-
2
(
x
±
-
α
-
60
β
t
)
+
2
θ
30
β
γ
,
(14d)
u
6
d
=
2
θ
30
β
γ
cot
-
2
(
x
±
-
α
-
60
β
t
)
+
2
θ
30
β
γ
,
when δ=-64β,(15a)
u
7
a
=
2
θ
30
β
γ
(
tanh
(
x
±
-
α
-
240
β
t
)
(
1
-
tanh
2
(
x
±
-
α
-
240
β
t
)
)
)
-
2
+
8
θ
30
β
γ
,
(15b)
u
7
b
=
2
θ
30
β
γ
(
tanh
(
x
±
-
α
+
240
β
t
)
(
1
+
tanh
2
(
x
±
-
α
+
240
β
t
)
)
)
-
2
-
8
θ
30
β
γ
,
(15c)
u
7
c
=
2
θ
30
β
γ
(
cot
(
x
±
-
α
-
240
β
t
)
(
1
-
cot
2
(
x
±
-
α
-
240
β
t
)
)
)
-
2
+
8
θ
30
β
γ
,
when δ=-(13±165)β,(16a)
u
8
a
=
θ
15
β
2
γ
×
(
coth
(
x
+
ε
-
α
-
1
4
(
-
30
+
2
σ
165
)
β
t
)
×
c
c
±
csch
(
x
+
ε
-
α
-
1
4
(
-
30
+
2
σ
165
)
β
t
)

)
-
2
+
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16b)
u
8
b
=
θ
15
β
2
γ
×
(

tanh
(
x
+
ε
-
α
-
(
1
/
4
)
(
-
30
+
2
σ
165
)
β
t
)
(
1

±
sech
(
x
+

ε
-
α
-
(
1
/
4
)
(
-
30
+
2
σ
165
)
β
t
)
)

)
-
2
+
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16c)
u
8
c
=
θ
15
β
2
γ
(
tan
(
x
+
ε
-
α
+
1
4
(
-
30
+
2
σ
165
)
β
t
)
θ
15
β
2
γ
c
c
±
sec
(

x

+

ε
-
α

+

1
4
(
-
30

+

2
σ
165
)
β
t

)

)
-
2
-
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16d)
u
8
d
=
θ
15
β
2
γ
×
(
csc
(
x
+
ε
-
α
+
1
4
(
-
30
+
2
σ
165
)
β
t
)
×
c
c
-
cot
(

x

+
ε
-
α

+

1
4
(
-
30
+

2
σ
165
)
β
t

)

)
-
2
-
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16e)
u
8
e
=
θ
15
β
2
γ
×
(

tan
(
x
+
ε
-
α
+
(
1
/
4
)
(
-
30

+
2
σ
165
)
β
t
)
(
1

±
sec
(
x
+
ε
-
α
+
(
1
/
4
)
(
-
30

+
2
σ
165
)
β
t
)
)

)
-
2
-
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16f)
u
8
f
=
θ
15
β
2
γ
×
(
cot
(
x
+
ε
-
α
+
1
4
(
-
30
+
2
σ
165
)
β
t
)
cccc
±
csc
(
x
+
ε
-
α
+
1
4
(
-
30
+
2
σ
165
)
β
t
)
)
-
2
-
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16g)
u
8
g
=
θ
15
β
2
γ
×
(
sec
(
x
+
ε
-
α
+
1
4
(
-
30
+
2
σ
165
)
β
t
)
×
c
c
-
tan
(
x
+
ε
-
α
+
1
4
(
-
30
+
2
σ
165
)
β
t
)

)
-
2
-
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(16h)
u
8
h
=
θ
15
β
2
γ
×
(

cot
(
x
+
ε
-
α
+
(
1
/
4
)
(
-
30
+
2
σ
165
)
β
t
)
(
1

±
csc
(
x
+
ε
-
α
+
(
1
/
4
)
(
-
30

+
2
σ
165
)
β
t
)
)

)
-
2
-
1
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
when δ=-4(13±165)β,(17a)
u
9
a
=
2
θ
30
β
γ
tanh
-
2
(
x
+
ε
-
α
-
(
-
30
+
2
σ
165
)
β
t
)
+
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(17b)
u
9
b
=
2
θ
30
β
γ
cot
h
-
2
(
x
+
ε
-
α
-
(
-
30
+
2
σ
165
)
β
t
)
+
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(17c)
u
9
c
=
2
θ
30
β
γ
ta
n
-
2
(
x
+
ε
-
α
+
(
-
30
+
2
σ
165
)
β
t
)
-
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(17d)
u
9
d
=
2
θ
30
β
γ
cot
-
2
(
x
+
ε
-
α
+
(
-
30
+
2
σ
165
)
β
t
)
-
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
when δ=-64(13±165)β,(18a)
u
10
a
=
2
θ
30
β
γ
×
(

tanh
(
x
+
ε
-
α
-
4
(
-
30
+
2
σ
165
)
β
t
)
(
1
+
tanh
2
(

x
+
ε
-
α
-
4
(
-
30
+
2
σ
165
)
β
t
)

)

)
-
2
+
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(18b)
u
10
b
=
2
θ
30
β
γ
×
(

tanh
(
x
+
ε
-
α
+
4
(
-
30
+
2
σ
165
)
β
t
)
(
1
-
tanh
2
(

x
+
ε
-
α
+
4
(
-
30
+
2
σ
165
)
β
t
)

)

)
-
2
-
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
(18c)
u
10
c
=
2
θ
30
β
γ
×
(

cot
(
x
+
ε
-
α
+
4
(
-
30
+
2
σ
165
)
β
t
)
(
1
-
cot
2
(

x

+
ε
-
α
+
4
(
-
30
+
2
σ
165
)
β
t
)
)

)
-
2
-
4
θ
30
β
γ
(
-
1
+
σ
165
15
)
,
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.



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*Source: 284865-2013-04-08.xml*