# Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients

Authors: Gang-Wei Wang; Tian-Zhou Xu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The simplified modified Kawahara equation with variable coefficients is studied by using Lie symmetry method. Then we obtain the corresponding Lie algebra, optimal system, and the similarity reductions. At last, we also give some new explicit solutions for some special forms of the equations.


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## Body

## 1. Introduction

Lie’s classical theory of symmetries of differential equations is an inspiring source for various generalizations aiming to find the ways for obtaining explicit solutions. Lie’s theory provides a standard method [1–5] for finding the Lie point symmetry group of a nonlinear system. And above all, Lie’s method of infinitesimal transformation groups which essentially reduces the number of independent variables in partial differential equation (PDE) and reduces the order of ordinary differential equation (ODE) has been widely used in equations of mathematical physics. Lie method is an effective and the simplest method among group theoretic techniques and a large number of equations [6–11] are solved with the aid of this method.In this paper, by using the Lie symmetry group method, we will consider the following simplified modified Kawahara equation:(1)ut+u2ux+α(t)u+β(t)uxxxxx=0.Here in (1) the first term represents the evolution term while the second term represents the nonlinear term. The third term represents the linear damping [6, 7, 12] while the fourth term is the dispersion term. The time dependent coefficients of damping and dispersion are, respectively, α(t) and β(t) which are arbitrary smooth functions of the variable t. If α(t)=0 and β(t)=constant, (1) becomes the standard simplified modified Kawahara equation (see [13, 14] and references therein).These KdV types of equations have been derived to model many physical phenomena, such as gravity-capillary waves on a shallow layer and magneto-sound propagation in plasmas, (see [14] and references therein). Many studies have been conducted with these types of equations [6–8, 12–18]. In [14] similarity solutions for some classes of (1) were considered. Abundant solitons solutions are obtained by using the tanh method in [17]. The paper [18] is mainly concerned with the local well-posedness of the initial-value problems for the Kawahara and the modified Kawahara equations in Sobolev spaces.Our aim in the present work is to perform the variable coefficients version of the simplified modified Kawahara equation with the help of Lie’s method. Then we get symmetry reductions and group-invariant solutions.


## 2. Lie Group Classification

### 2.1. Lie Symmetry Analysis of (1)

In this section, we will perform Lie group method for (1).If (1) is invariant under a one-parameter Lie group of point transformations
(2)t*=t+ϵτ(x,t,u)+O(ϵ2),x*=x+ϵξ(x,t,u)+O(ϵ2),u*=u+ϵη(x,t,u)+O(ϵ2),
with infinitesimal generator
(3)V=τ(x,t,u)∂∂t+ξ(x,t,u)∂∂x+η(x,t,u)∂∂u,
then the invariant condition reads as
(4)ηt+u2ηx+2ηuux+α′(t)τu+β′(t)τuxxxxx+α(t)η+β(t)ηxxxxx=0,
where
(5)ηt=Dt(η)-uxDt(ξ)-utDt(τ),ηx=Dx(η)-uxDx(ξ)-utDx(τ),ηxx=Dx(ηx)-uxtDx(τ)-uxxDx(ξ),ηxxx=Dx(ηxx)-uxxtDx(τ)-uxxxDx(ξ),ηxxxx=Dx(ηxxx)-uxxxtDx(τ)-uxxxxDx(ξ),ηxxxxx=Dx(ηxxxx)-uxxxxtDx(τ)-uxxxxxDx(ξ).
Here, Di denotes the total derivative operator and is defined by
(6)Di=∂∂xi+ui∂∂u+uij∂∂uj+⋯,i=1,2,
and (x1,x2)=(t,x).Solving (4) with the help of (5) we obtain
(7)τ=τ(t),ξu=0,ηuu=0,(8)β(t)τt+β′(t)τ-5β(t)ξx=0,(9)ηxu-2ξxx=0,(10)τtu2-ξxu2+2ηu-ξt+β(t)(5ηxxxxu-ξxxxxx)=0,(11)ηt+α(t)η+α(t)uτt-α(t)uηu+uα′(t)τ+u2ηx+β(t)ηxxxxx=0.The structure of the determining equations (10) and (11) may give the selection of the following three forms for the coefficient α(t).Case 1 (α(t)=0).
In this case, solving (7)–(11) we get
(12)ξ=2c1x+c2,τ=2c3t+c4,η=(c1-c3)u,
and (8) becomes
(13)β′(t)(2c3t+c4)+β(t)(2c3-10c1)=0,
where c1,…,c4 are constants. The analysis of (13) leads to the following three possibilities for β(t).(1.1)
β(t) is arbitrary.For this case, we obtain the vector field(14)V1=∂∂x.(1.2)
β(t)=b0(≠0) is a constant.Equation (1) admits a three-dimensional Lie algebra spanned by
(15)V1=∂∂x,V2=∂∂t,V3=x∂∂x+5t∂∂t-2u∂∂u.(1.3)
β(t)=b0emt, b0,m(≠0) are constant.We obtain the corresponding two Lie point symmetry generators(16)V1=∂∂x,V2=∂∂t+m5x∂∂x+m10u∂∂u.Case 2 (α(t) is a nonzero constant as a).
Similarly, in this case, solving (7)–(11) for the infinitesimals, we obtain
(17)ξ=2c1x+c3,τ=-c2ae2at+c4,η=(c1+c2e2at)u,
and (8) becomes(18)β′(t)(-c2ae2at+c4)+β(t)(-2c2e2at-10c1)=0,
where c1,…,c4 are constants. The analysis of (18) gives rise to the following four possibilities for β(t).(2.1)
β(t) is arbitrary.For this case, we obtain the vector field(19)V1=∂∂x.(2.2)
β(t)=b0(≠0) is a constant.We have the corresponding two-dimensional Lie algebra(20)V1=∂∂x,V2=∂∂t.(2.3)
β(t)=b0e-2at, b0 is a constant.Equation (1) admits a three-dimensional Lie algebra spanned by
(21)V1=∂∂x,V2=e2at∂∂t-ae2atu∂∂u,V3=5∂∂t-2ax∂∂x-au∂∂u.(2.4)
β(t)=b0emt(b0≠0) is a constant.Substitutingβ(t) into (18), one can get
(22)c2e2at(ma+2)=10c1-c4m,
where   m≠-2a. We obtain the corresponding two-dimensional Lie algebra
(23)V1=∂∂x,V2=∂∂t+m5x∂∂x+m10u∂∂u.Case 3 (α(t)=1/t).
In this case, solving (7)–(11), we get
(24)ξ=c3x+c4,τ=-c1t2-2c2t+c3t,η=(c1t+c2)u,
and (8) becomes
(25)β′(t)(-c1t2-2c2t+c3t)+β(t)(-4c3-2c2-2c1t)=0,
where c1,…,c4 are constants. Similarly, the analysis of (25) gives rise to the following three possibilities for β(t).(3.1)
β(t) is arbitrary.For this case, we obtain the vector field(26)V1=∂∂x.(3.2)
β(t)=b0(≠0) is a constant.We obtain the corresponding two-dimensional Lie algebra(27)V1=∂∂x,V2=x∂∂x+5t∂∂t-2u∂∂u.(3.3)
β(t)=b0/t2, b0(≠0) is a constant.The Lie algebra is extended by the symmetry generators(28)V1=∂∂x,V2=t2∂∂t-tu∂∂u,V3=x∂∂x-5t∂∂t+3u∂∂u.


### 2.2. Optimal System of One-Dimensional Lie Algebras

First of all, we briefly review the main definitions [1] which will be used in the following sections.Definition 1 (see [1]).
LetG  be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subgroups with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: ħ=Adg(h), g∈G.To obtain the optimal system, we apply the formula [1]
(29)Ad(exp(ϵVi))Vj=Vj-ϵ[Vi,Vj]+12ϵ2[Vi,[Vi,Vj]]-⋯,
where ϵ is a real constant. Here [Vi,Vj] is the commutator for the Lie algebra given by
(30)[Vi,Vj]=ViVj-VjVi.The commutator table of the Lie point symmetries of (1.2) and the adjoint representations of the symmetry group of (1.2) on its Lie algebra are given in Tables1 and 2, respectively. In the same way, the commutator table and the adjoint representations of (2.3), (2.4), and (3.3) are given in Tables 3, 4, 5, 6, 7, and 8, respectively. We give in Table 9 optimal system of subalgebras for (1.2),  (2.3),  (2.4),  and  (3.3), respectively.Table 1
Commutator table of the Lie algebra of  (1.2).
[
V
i
,
V
j
]
V
1
V
2
V
3
V
1
0
0
V
1
V
2
0
0
5
V
2
V
3
-
V
1
-
5
V
2
0Table 2
Adjoint table of the Lie algebra of  (1.2).
Ad
V
1
V
2
V
3
V
1
V
1
V
2
V
3
-
ϵ
V
1
V
2
V
1
V
2
V
3
-
5
ϵ
V
2
V
3
e
ϵ
V
1
e
5
ϵ
V
2
V
3Table 3
Commutator table of the Lie algebra of  (2.3).
[
V
i
,
V
j
]
V
1
V
2
V
3
V
1
0
0
-
2
a
V
1
V
2
0
0
-
10
a
V
2
V
3
2
a
V
1
10
a
V
2
0Table 4
Adjoint table of the Lie algebra of(2.3).
Ad
V
1
V
2
V
3
V
1
V
1
V
2
V
3
+
2
a
ϵ
V
1
V
2
V
1
V
2
V
3
+
10
a
ϵ
V
2
V
3
e
-
2
a
ϵ
V
1
e
-
10
a
ϵ
V
2
V
3Table 5
Commutator table of the Lie algebra of  (2.4).
[
V
i
,
V
j
]
V
1
V
2
V
1
0
0
V
2
0
0Table 6
Adjoint table of the Lie algebra of(2.4).
Ad
V
1
V
2
V
1
V
1
V
2
V
2
V
1
V
2Table 7
Commutator table of the Lie algebra of  (3.3).
[
V
i
,
V
j
]
V
1
V
2
V
3
V
1
0
0
V
1
V
2
0
0
5
V
2
V
3
-
V
1
-
5
V
2
0Table 8
Adjoint table of the Lie algebra of(3.3).
Ad
V
1
V
2
V
3
V
1
V
1
V
2
V
3
-
ϵ
V
1
V
2
V
1
V
2
V
3
-
5
ϵ
V
2
V
3
e
ϵ
V
1
e
5
ϵ
V
2
V
3Table 9
Optimal system of subalgebras of some special cases of (1.1).
Equation number
Equations
Optimal system
(1.2)
u
t
+
u
2
u
x
+
b
0
u
x
x
x
x
x
=
0
V
1
,
V
2
+
λ
V
1
,
V
3
(
2.3
)
u
t
+
u
2
u
x
+
a
u
+
(
b
0
e
-
2
a
t
)
u
x
x
x
x
x
=
0
V
1
,
V
2
+
μ
V
1
,
V
3
(
2.4
)
u
t
+
u
2
u
x
+
a
u
+
(
b
0
e
m
t
)
u
x
x
x
x
x
=
0
V
1
,
V
2
(
3.3
)
u
t
+
u
2
u
x
+
(
1
/
t
)
u
+
(
b
0
/
t
2
)
u
x
x
x
x
x
=
0
V
1
,
V
2
+
μ
V
1
,
V
3
Hereμ=0,μ=±1, and λ is a nonzero constant.Remark 2.
For simplicity, we will refer to the equation corresponding to the case (1.3) as (1.3), and so on.Remark 3.
For brevity we only consider the optimal system for the equation concerned in case (1.2) in detail and the rest will be listed in Table9 as they can be derived in a similar manner.


## 2.1. Lie Symmetry Analysis of (1)

In this section, we will perform Lie group method for (1).If (1) is invariant under a one-parameter Lie group of point transformations
(2)t*=t+ϵτ(x,t,u)+O(ϵ2),x*=x+ϵξ(x,t,u)+O(ϵ2),u*=u+ϵη(x,t,u)+O(ϵ2),
with infinitesimal generator
(3)V=τ(x,t,u)∂∂t+ξ(x,t,u)∂∂x+η(x,t,u)∂∂u,
then the invariant condition reads as
(4)ηt+u2ηx+2ηuux+α′(t)τu+β′(t)τuxxxxx+α(t)η+β(t)ηxxxxx=0,
where
(5)ηt=Dt(η)-uxDt(ξ)-utDt(τ),ηx=Dx(η)-uxDx(ξ)-utDx(τ),ηxx=Dx(ηx)-uxtDx(τ)-uxxDx(ξ),ηxxx=Dx(ηxx)-uxxtDx(τ)-uxxxDx(ξ),ηxxxx=Dx(ηxxx)-uxxxtDx(τ)-uxxxxDx(ξ),ηxxxxx=Dx(ηxxxx)-uxxxxtDx(τ)-uxxxxxDx(ξ).
Here, Di denotes the total derivative operator and is defined by
(6)Di=∂∂xi+ui∂∂u+uij∂∂uj+⋯,i=1,2,
and (x1,x2)=(t,x).Solving (4) with the help of (5) we obtain
(7)τ=τ(t),ξu=0,ηuu=0,(8)β(t)τt+β′(t)τ-5β(t)ξx=0,(9)ηxu-2ξxx=0,(10)τtu2-ξxu2+2ηu-ξt+β(t)(5ηxxxxu-ξxxxxx)=0,(11)ηt+α(t)η+α(t)uτt-α(t)uηu+uα′(t)τ+u2ηx+β(t)ηxxxxx=0.The structure of the determining equations (10) and (11) may give the selection of the following three forms for the coefficient α(t).Case 1 (α(t)=0).
In this case, solving (7)–(11) we get
(12)ξ=2c1x+c2,τ=2c3t+c4,η=(c1-c3)u,
and (8) becomes
(13)β′(t)(2c3t+c4)+β(t)(2c3-10c1)=0,
where c1,…,c4 are constants. The analysis of (13) leads to the following three possibilities for β(t).(1.1)
β(t) is arbitrary.For this case, we obtain the vector field(14)V1=∂∂x.(1.2)
β(t)=b0(≠0) is a constant.Equation (1) admits a three-dimensional Lie algebra spanned by
(15)V1=∂∂x,V2=∂∂t,V3=x∂∂x+5t∂∂t-2u∂∂u.(1.3)
β(t)=b0emt, b0,m(≠0) are constant.We obtain the corresponding two Lie point symmetry generators(16)V1=∂∂x,V2=∂∂t+m5x∂∂x+m10u∂∂u.Case 2 (α(t) is a nonzero constant as a).
Similarly, in this case, solving (7)–(11) for the infinitesimals, we obtain
(17)ξ=2c1x+c3,τ=-c2ae2at+c4,η=(c1+c2e2at)u,
and (8) becomes(18)β′(t)(-c2ae2at+c4)+β(t)(-2c2e2at-10c1)=0,
where c1,…,c4 are constants. The analysis of (18) gives rise to the following four possibilities for β(t).(2.1)
β(t) is arbitrary.For this case, we obtain the vector field(19)V1=∂∂x.(2.2)
β(t)=b0(≠0) is a constant.We have the corresponding two-dimensional Lie algebra(20)V1=∂∂x,V2=∂∂t.(2.3)
β(t)=b0e-2at, b0 is a constant.Equation (1) admits a three-dimensional Lie algebra spanned by
(21)V1=∂∂x,V2=e2at∂∂t-ae2atu∂∂u,V3=5∂∂t-2ax∂∂x-au∂∂u.(2.4)
β(t)=b0emt(b0≠0) is a constant.Substitutingβ(t) into (18), one can get
(22)c2e2at(ma+2)=10c1-c4m,
where   m≠-2a. We obtain the corresponding two-dimensional Lie algebra
(23)V1=∂∂x,V2=∂∂t+m5x∂∂x+m10u∂∂u.Case 3 (α(t)=1/t).
In this case, solving (7)–(11), we get
(24)ξ=c3x+c4,τ=-c1t2-2c2t+c3t,η=(c1t+c2)u,
and (8) becomes
(25)β′(t)(-c1t2-2c2t+c3t)+β(t)(-4c3-2c2-2c1t)=0,
where c1,…,c4 are constants. Similarly, the analysis of (25) gives rise to the following three possibilities for β(t).(3.1)
β(t) is arbitrary.For this case, we obtain the vector field(26)V1=∂∂x.(3.2)
β(t)=b0(≠0) is a constant.We obtain the corresponding two-dimensional Lie algebra(27)V1=∂∂x,V2=x∂∂x+5t∂∂t-2u∂∂u.(3.3)
β(t)=b0/t2, b0(≠0) is a constant.The Lie algebra is extended by the symmetry generators(28)V1=∂∂x,V2=t2∂∂t-tu∂∂u,V3=x∂∂x-5t∂∂t+3u∂∂u.


## 2.2. Optimal System of One-Dimensional Lie Algebras

First of all, we briefly review the main definitions [1] which will be used in the following sections.Definition 1 (see [1]).
LetG  be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subgroups with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: ħ=Adg(h), g∈G.To obtain the optimal system, we apply the formula [1]
(29)Ad(exp(ϵVi))Vj=Vj-ϵ[Vi,Vj]+12ϵ2[Vi,[Vi,Vj]]-⋯,
where ϵ is a real constant. Here [Vi,Vj] is the commutator for the Lie algebra given by
(30)[Vi,Vj]=ViVj-VjVi.The commutator table of the Lie point symmetries of (1.2) and the adjoint representations of the symmetry group of (1.2) on its Lie algebra are given in Tables1 and 2, respectively. In the same way, the commutator table and the adjoint representations of (2.3), (2.4), and (3.3) are given in Tables 3, 4, 5, 6, 7, and 8, respectively. We give in Table 9 optimal system of subalgebras for (1.2),  (2.3),  (2.4),  and  (3.3), respectively.Table 1
Commutator table of the Lie algebra of  (1.2).
[
V
i
,
V
j
]
V
1
V
2
V
3
V
1
0
0
V
1
V
2
0
0
5
V
2
V
3
-
V
1
-
5
V
2
0Table 2
Adjoint table of the Lie algebra of  (1.2).
Ad
V
1
V
2
V
3
V
1
V
1
V
2
V
3
-
ϵ
V
1
V
2
V
1
V
2
V
3
-
5
ϵ
V
2
V
3
e
ϵ
V
1
e
5
ϵ
V
2
V
3Table 3
Commutator table of the Lie algebra of  (2.3).
[
V
i
,
V
j
]
V
1
V
2
V
3
V
1
0
0
-
2
a
V
1
V
2
0
0
-
10
a
V
2
V
3
2
a
V
1
10
a
V
2
0Table 4
Adjoint table of the Lie algebra of(2.3).
Ad
V
1
V
2
V
3
V
1
V
1
V
2
V
3
+
2
a
ϵ
V
1
V
2
V
1
V
2
V
3
+
10
a
ϵ
V
2
V
3
e
-
2
a
ϵ
V
1
e
-
10
a
ϵ
V
2
V
3Table 5
Commutator table of the Lie algebra of  (2.4).
[
V
i
,
V
j
]
V
1
V
2
V
1
0
0
V
2
0
0Table 6
Adjoint table of the Lie algebra of(2.4).
Ad
V
1
V
2
V
1
V
1
V
2
V
2
V
1
V
2Table 7
Commutator table of the Lie algebra of  (3.3).
[
V
i
,
V
j
]
V
1
V
2
V
3
V
1
0
0
V
1
V
2
0
0
5
V
2
V
3
-
V
1
-
5
V
2
0Table 8
Adjoint table of the Lie algebra of(3.3).
Ad
V
1
V
2
V
3
V
1
V
1
V
2
V
3
-
ϵ
V
1
V
2
V
1
V
2
V
3
-
5
ϵ
V
2
V
3
e
ϵ
V
1
e
5
ϵ
V
2
V
3Table 9
Optimal system of subalgebras of some special cases of (1.1).
Equation number
Equations
Optimal system
(1.2)
u
t
+
u
2
u
x
+
b
0
u
x
x
x
x
x
=
0
V
1
,
V
2
+
λ
V
1
,
V
3
(
2.3
)
u
t
+
u
2
u
x
+
a
u
+
(
b
0
e
-
2
a
t
)
u
x
x
x
x
x
=
0
V
1
,
V
2
+
μ
V
1
,
V
3
(
2.4
)
u
t
+
u
2
u
x
+
a
u
+
(
b
0
e
m
t
)
u
x
x
x
x
x
=
0
V
1
,
V
2
(
3.3
)
u
t
+
u
2
u
x
+
(
1
/
t
)
u
+
(
b
0
/
t
2
)
u
x
x
x
x
x
=
0
V
1
,
V
2
+
μ
V
1
,
V
3
Hereμ=0,μ=±1, and λ is a nonzero constant.Remark 2.
For simplicity, we will refer to the equation corresponding to the case (1.3) as (1.3), and so on.Remark 3.
For brevity we only consider the optimal system for the equation concerned in case (1.2) in detail and the rest will be listed in Table9 as they can be derived in a similar manner.


## 3. Symmetry Reductions and ExactGroup-Invariant Solutions

In this section, we will use Table9 to obtain symmetry reductions and exact group-invariant solutions for  (1.2),  (2.3),  (2.4),  and  (3.3).


### 3.1. Symmetry Reductions and Exact Solutions to (1.2)

#### 3.1.1.V1

For the generatorV1, the group-invariant solution is u=f(ξ), where ξ=t   is the group-invariant, the substitution of this solution into (1.2) gives the trivial solution u(x,t)=C, and C is a constant.


#### 3.1.2.V2+λV1

For the linear combinationV2+λV1, we have
(31)u=f(ξ),
where ξ=x-λt is the group-invariant. Substituting (31) into (1.2), we reduce it to the following ODE:
(32)b0f(5)+f2f′-λf′=0.


#### 3.1.3.V3

For the generatorV3, we have
(33)u=t-(2/5)f(ξ),
where ξ=xt-(1/5) is the group-invariant. Substituting (33) into (1.2), we reduce it to the following ODE:
(34)5b0f(5)+5f2f′-2f-ξf′=0,
where  f′=df/dξ.


### 3.2. Symmetry Reductions and Exact Solutions to(2.3)

#### 3.2.1.V1

For the generatorV1, we get that the group-invariant solution of (2.3) is u(x,t)=ce-at, where c is an arbitrary constant.


#### 3.2.2.V3

For the generatorV3, we have
(35)u=e-at/5f(ξ),
where  ξ=xe2at/5. Substituting (35) into (2.3), one can get
(36)5b0f(5)+5f2f′+4af+2aξf′=0,
where f′=df/dξ.


#### 3.2.3.V2+μV1

(i) Forμ≠0, we have
(37)u=f(ξ)e-at,
where   ξ=x+(μe-2at/2a). Substituting (37) into (2.3) yields
(38)b0f(5)+f2f′-μf′=0.(ii) Forμ=0, we get
(39)b0f(5)+f2f′=0.


### 3.3. Symmetry Reductions and Exact Solutions to(2.4)

#### 3.3.1.V1

For the generatorV1, we get that the group-invariant solution of (2.4) is
(40)u(x,t)=ce-at,
where c is an arbitrary constant.


#### 3.3.2.V2

For this case, we have(41)ξ=xe-(mt/5),u=emt/10f(ξ);
then substituting (41) into (2.4) gives rise to
(42)10b0f(5)+10f2f′+10af-2mξf′+mf=0.


### 3.4. Symmetry Reductions and Exact Solutions to(3.3)

#### 3.4.1.V1

For the generatorV1, we get that the group-invariant solution of (3.3) is
(43)u(x,t)=ct,
where c is an arbitrary constant.


#### 3.4.2.V3

For the generatorV3, we have
(44)u=t-(3/5)f(ξ),
where ξ=xt1/5. Substituting (44) into (3.3), one can get
(45)5b0f(5)+5f2f′+2f+ξf′=0,
where f′=df/dξ.


#### 3.4.3.V2+μV1

(i) Forμ≠0, we have
(46)u=1tf(ξ),
where ξ=x+μ/t. Substituting (46) into (3.3) yields
(47)b0f(5)+f2f′-μf′=0.In order to search for other explicit solutions, by using the Jacobi elliptic function expansion method [19]. By virtue of the technique of solution we introduce the ansatz
(48)f=a0+a1sn(ξ)+a2sn2(ξ).
Substituting (48) into (47), one can get
(49)b=-1360a22m4,m=m,μ=115a22(-m2+1+m4)m4,a0=-13a2(1+m2)m2,a1=0,a2=a2.Thus, we obtain Jacobi elliptic function solutions of(3.3) as follows:
(50)u(x,t)=-13a2(1+m2)m2+a2sn2(ξ).
In particular, when m→1, we can obtain hyperbolic function solutions
(51)u(x,t)=-13a2(1+m2)m2+a2tanh2(ξ).
When m→0, we can obtain trigonometric function solutions
(52)u(x,t)=-13a2(1+m2)m2+a2sin2(ξ),
where ξ=x+μ/t.(ii) Forμ=0, we get
(53)b0f(5)+f2f′=0.Remark 4.
It is not difficult to find out that the reduced ODEs may be classified into four classes:(54)f(5)+Af2f′=0,f(5)+Af2f′+Bf′=0,f(5)+Af2f′+Bf′+Cf=0,f(5)+Af2f′+Bf′+Cξf′=0.


## 3.1. Symmetry Reductions and Exact Solutions to (1.2)

### 3.1.1.V1

For the generatorV1, the group-invariant solution is u=f(ξ), where ξ=t   is the group-invariant, the substitution of this solution into (1.2) gives the trivial solution u(x,t)=C, and C is a constant.


### 3.1.2.V2+λV1

For the linear combinationV2+λV1, we have
(31)u=f(ξ),
where ξ=x-λt is the group-invariant. Substituting (31) into (1.2), we reduce it to the following ODE:
(32)b0f(5)+f2f′-λf′=0.


### 3.1.3.V3

For the generatorV3, we have
(33)u=t-(2/5)f(ξ),
where ξ=xt-(1/5) is the group-invariant. Substituting (33) into (1.2), we reduce it to the following ODE:
(34)5b0f(5)+5f2f′-2f-ξf′=0,
where  f′=df/dξ.


## 3.1.1.V1

For the generatorV1, the group-invariant solution is u=f(ξ), where ξ=t   is the group-invariant, the substitution of this solution into (1.2) gives the trivial solution u(x,t)=C, and C is a constant.


## 3.1.2.V2+λV1

For the linear combinationV2+λV1, we have
(31)u=f(ξ),
where ξ=x-λt is the group-invariant. Substituting (31) into (1.2), we reduce it to the following ODE:
(32)b0f(5)+f2f′-λf′=0.


## 3.1.3.V3

For the generatorV3, we have
(33)u=t-(2/5)f(ξ),
where ξ=xt-(1/5) is the group-invariant. Substituting (33) into (1.2), we reduce it to the following ODE:
(34)5b0f(5)+5f2f′-2f-ξf′=0,
where  f′=df/dξ.


## 3.2. Symmetry Reductions and Exact Solutions to(2.3)

### 3.2.1.V1

For the generatorV1, we get that the group-invariant solution of (2.3) is u(x,t)=ce-at, where c is an arbitrary constant.


### 3.2.2.V3

For the generatorV3, we have
(35)u=e-at/5f(ξ),
where  ξ=xe2at/5. Substituting (35) into (2.3), one can get
(36)5b0f(5)+5f2f′+4af+2aξf′=0,
where f′=df/dξ.


### 3.2.3.V2+μV1

(i) Forμ≠0, we have
(37)u=f(ξ)e-at,
where   ξ=x+(μe-2at/2a). Substituting (37) into (2.3) yields
(38)b0f(5)+f2f′-μf′=0.(ii) Forμ=0, we get
(39)b0f(5)+f2f′=0.


## 3.2.1.V1

For the generatorV1, we get that the group-invariant solution of (2.3) is u(x,t)=ce-at, where c is an arbitrary constant.


## 3.2.2.V3

For the generatorV3, we have
(35)u=e-at/5f(ξ),
where  ξ=xe2at/5. Substituting (35) into (2.3), one can get
(36)5b0f(5)+5f2f′+4af+2aξf′=0,
where f′=df/dξ.


## 3.2.3.V2+μV1

(i) Forμ≠0, we have
(37)u=f(ξ)e-at,
where   ξ=x+(μe-2at/2a). Substituting (37) into (2.3) yields
(38)b0f(5)+f2f′-μf′=0.(ii) Forμ=0, we get
(39)b0f(5)+f2f′=0.


## 3.3. Symmetry Reductions and Exact Solutions to(2.4)

### 3.3.1.V1

For the generatorV1, we get that the group-invariant solution of (2.4) is
(40)u(x,t)=ce-at,
where c is an arbitrary constant.


### 3.3.2.V2

For this case, we have(41)ξ=xe-(mt/5),u=emt/10f(ξ);
then substituting (41) into (2.4) gives rise to
(42)10b0f(5)+10f2f′+10af-2mξf′+mf=0.


## 3.3.1.V1

For the generatorV1, we get that the group-invariant solution of (2.4) is
(40)u(x,t)=ce-at,
where c is an arbitrary constant.


## 3.3.2.V2

For this case, we have(41)ξ=xe-(mt/5),u=emt/10f(ξ);
then substituting (41) into (2.4) gives rise to
(42)10b0f(5)+10f2f′+10af-2mξf′+mf=0.


## 3.4. Symmetry Reductions and Exact Solutions to(3.3)

### 3.4.1.V1

For the generatorV1, we get that the group-invariant solution of (3.3) is
(43)u(x,t)=ct,
where c is an arbitrary constant.


### 3.4.2.V3

For the generatorV3, we have
(44)u=t-(3/5)f(ξ),
where ξ=xt1/5. Substituting (44) into (3.3), one can get
(45)5b0f(5)+5f2f′+2f+ξf′=0,
where f′=df/dξ.


### 3.4.3.V2+μV1

(i) Forμ≠0, we have
(46)u=1tf(ξ),
where ξ=x+μ/t. Substituting (46) into (3.3) yields
(47)b0f(5)+f2f′-μf′=0.In order to search for other explicit solutions, by using the Jacobi elliptic function expansion method [19]. By virtue of the technique of solution we introduce the ansatz
(48)f=a0+a1sn(ξ)+a2sn2(ξ).
Substituting (48) into (47), one can get
(49)b=-1360a22m4,m=m,μ=115a22(-m2+1+m4)m4,a0=-13a2(1+m2)m2,a1=0,a2=a2.Thus, we obtain Jacobi elliptic function solutions of(3.3) as follows:
(50)u(x,t)=-13a2(1+m2)m2+a2sn2(ξ).
In particular, when m→1, we can obtain hyperbolic function solutions
(51)u(x,t)=-13a2(1+m2)m2+a2tanh2(ξ).
When m→0, we can obtain trigonometric function solutions
(52)u(x,t)=-13a2(1+m2)m2+a2sin2(ξ),
where ξ=x+μ/t.(ii) Forμ=0, we get
(53)b0f(5)+f2f′=0.Remark 4.
It is not difficult to find out that the reduced ODEs may be classified into four classes:(54)f(5)+Af2f′=0,f(5)+Af2f′+Bf′=0,f(5)+Af2f′+Bf′+Cf=0,f(5)+Af2f′+Bf′+Cξf′=0.


## 3.4.1.V1

For the generatorV1, we get that the group-invariant solution of (3.3) is
(43)u(x,t)=ct,
where c is an arbitrary constant.


## 3.4.2.V3

For the generatorV3, we have
(44)u=t-(3/5)f(ξ),
where ξ=xt1/5. Substituting (44) into (3.3), one can get
(45)5b0f(5)+5f2f′+2f+ξf′=0,
where f′=df/dξ.


## 3.4.3.V2+μV1

(i) Forμ≠0, we have
(46)u=1tf(ξ),
where ξ=x+μ/t. Substituting (46) into (3.3) yields
(47)b0f(5)+f2f′-μf′=0.In order to search for other explicit solutions, by using the Jacobi elliptic function expansion method [19]. By virtue of the technique of solution we introduce the ansatz
(48)f=a0+a1sn(ξ)+a2sn2(ξ).
Substituting (48) into (47), one can get
(49)b=-1360a22m4,m=m,μ=115a22(-m2+1+m4)m4,a0=-13a2(1+m2)m2,a1=0,a2=a2.Thus, we obtain Jacobi elliptic function solutions of(3.3) as follows:
(50)u(x,t)=-13a2(1+m2)m2+a2sn2(ξ).
In particular, when m→1, we can obtain hyperbolic function solutions
(51)u(x,t)=-13a2(1+m2)m2+a2tanh2(ξ).
When m→0, we can obtain trigonometric function solutions
(52)u(x,t)=-13a2(1+m2)m2+a2sin2(ξ),
where ξ=x+μ/t.(ii) Forμ=0, we get
(53)b0f(5)+f2f′=0.Remark 4.
It is not difficult to find out that the reduced ODEs may be classified into four classes:(54)f(5)+Af2f′=0,f(5)+Af2f′+Bf′=0,f(5)+Af2f′+Bf′+Cf=0,f(5)+Af2f′+Bf′+Cξf′=0.


## 4. The Explicit Power Series Solutions

In this section, we will consider the explicit analytic solutions of some special forms of reduced equations by using the power series method.Now, we seek a solution of (45) in a power series of the following form:
(55)f(ξ)=∑n=0∞cnξn.Substituting (55) into (45), we get
(56)600bc5+b∑n=1∞(n+1)(n+2)(n+3)(n+4)(n+5)cn+5ξn+5c02c1+5∑n=1∞∑k=0n∑j=0k(n+1-k)cjck-jcn+1-kξn+2c0+2∑n=1∞cnξn+∑n=1∞ncnξn=0.Now from (56), comparing coefficients, for  n=0, one can get
(57)c5=-2c0+5c02c1600b.Generally, forn≥1, we obtain
(58)cn+5=-1b(n+1)(n+2)(n+3)(n+4)(n+5)×((n+2)cn+5∑k=0n∑j=0k(n+k-1)cjck-jcn+1-k).From (57) and (58), we can obtain all the coefficients cn(n≥5) of the power series (55). For arbitrary chosen constant numbers c0, c1, c2, c3, and c4, the other terms also can be determined successively from (57) and (58) in a unique way. In addition, it is easy to prove the convergence of the power series (55) with the coefficients given by (57) and (58) [20, 21]. The details are omitted here. In this connection, this power series solution is an explicit analytic solution.So, the power series solution of (45) can be written as follows:
(59)f(ξ)=c0+c1ξ+c2ξ2+c3ξ3+c4ξ4+c5ξ5+∑n=1∞cn+5ξn+5=c0+c1ξ+c2ξ2+c3ξ3+c4ξ4-2c0+5c02c1600bξ5-∑n=0∞1b(n+1)(n+2)(n+3)(n+4)(n+5)×(+5∑k=0n‍∑j=0k‍(n+k-1)cjck-jcn+1-k(n+2)cn+5∑k=0n‍∑j=0k‍(n+k-1)cjck-jcn+1-k)ξn+5.Thus, the exact power series solution of(3.3) is
(60)u(x,t)=[∑n=0∞cn+5(xt1/5)n+5c0+c1(xt1/5)+c2(xt1/5)2+c3(xt1/5)3+c4(xt1/5)4-2c0+5c02c1600b(xt1/5)5+∑n=0∞cn+5(xt1/5)n+5](t-(3/5))=[+5∑k=0n∑j=0k(n+k-1)cjck-jcn+1-k)c0+c1(xt1/5)+c2(xt1/5)2+c3(xt1/5)3+c4(xt1/5)4-2c0+5c02c1600b(xt1/5)5-∑n=0∞1b(n+1)(n+2)(n+3)(n+4)(n+5)×(∑j=0k(n+2)cn+5∑k=0n∑j=0k(n+k-1)cjck-jcn+1-k)×ξn+5+5∑k=0n∑j=0k(n+k-1)cjck-jcn+1-k)](t-(3/5)),
where ci(i=0,1,2,3,4) are arbitrary constants, and the other coefficients cn(n≥5) can be determined successively from (57) and (58).Of course, in physical applications, it will be convenient to write the solution of (45) in the approximate form
(61)u(x,t)=[2c0+5c02c1600b(xt1/5)5c0+c1(xt1/5)+c2(xt1/5)2+c3(xt1/5)3c0+c1(xt1/5)+c2(xt1/5)2+c3(xt1/5)3+c4(xt1/5)4-2c0+5c02c1600b(xt1/5)5+⋯]×(t-(3/5)).Remark 5.
The exact solution of the rest of equations can be derived in a similar manner. We have details omitted here.Remark 6.
It is easy to see that the reduced equations (54) are all higher-order nonlinear ODEs or with nonconstant coefficients. If we obtain a one-parameter symmetry group of an ODE, then we could reduce the order of the equation by one. However, we find out that such reduced ODEs are more complicated than the original equation. In general, we cannot obtain the exact explicit solutions for higher-order nonlinear ODEs or with nonconstant coefficients by using the elementary functions and integrals. However, the power series can be used to solve them. In view of this, we can find that the power series method [14, 15, 20–23] is an effective tool of solving such ODEs. Moreover, from our model, we could find that these power series solutions are important for computations in numerical analysis and physical applications. And above all, these power series play an important role in the investigation of physical phenomena and other natural phenomena.Remark 7.
Indeed, for all the rest of the cases presented in Section2, it is possible to get optimal systems and symmetry reductions, but for brevity we have omitted them here.


## 5. Conclusions

We have performed Lie symmetry analysis for the simplified modified Kawahara equation with variable coefficients. Then the similarity reductions and exact solutions are obtained based on the optimal system of the one-dimensional Lie algebras, for some special forms of the equations. Moreover, the power series solution of the reduced equation are given simultaneously. These are new solutions for the simplified modified Kawahara equation with variable coefficients. The symmetry analysis based on the Lie group method is a very powerful method and is worthy of being studyied further.



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*Source: 139160-2013-10-07.xml*