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

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.

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: 𝑢_𝑡+𝑢²𝑢_𝑥+𝛼(𝑡)𝑢+𝛽(𝑡)𝑢_{𝑥𝑥𝑥𝑥𝑥}=0. (1)

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 𝑡^∗=𝑡+𝜖𝜏(𝑥,𝑡,𝑢)+𝑂𝜖²,𝑥^∗=𝑥+𝜖𝜉(𝑥,𝑡,𝑢)+𝑂𝜖²,𝑢^∗=𝑢+𝜖𝜂(𝑥,𝑡,𝑢)+𝑂𝜖², (2) with infinitesimal generator ()𝜕𝑉=𝜏𝑥,𝑡,𝑢()𝜕𝜕𝑡+𝜉𝑥,𝑡,𝑢()𝜕𝜕𝑥+𝜂𝑥,𝑡,𝑢,𝜕𝑢 (3) then the invariant condition reads as 𝜂^𝑡+𝑢²𝜂^𝑥+2𝜂𝑢𝑢_𝑥+𝛼^′(𝑡)𝜏𝑢+𝛽^′(𝑡)𝜏𝑢_{𝑥𝑥𝑥𝑥𝑥}+𝛼(𝑡)𝜂+𝛽(𝑡)𝜂^{𝑥𝑥𝑥𝑥𝑥}=0, (4) where 𝜂^𝑡=𝐷_𝑡𝜂−𝑢_𝑥𝐷_𝑡(𝜉)−𝑢_𝑡𝐷_𝑡(𝜏),𝜂^𝑥=𝐷_𝑥𝜂−𝑢_𝑥𝐷_𝑥(𝜉)−𝑢_𝑡𝐷_𝑥(𝜏),𝜂^𝑥𝑥=𝐷_𝑥𝜂^𝑥−𝑢_𝑥𝑡𝐷_𝑥(𝜏)−𝑢_𝑥𝑥𝐷_𝑥(𝜉),𝜂^𝑥𝑥𝑥=𝐷_𝑥𝜂^𝑥𝑥−𝑢_𝑥𝑥𝑡𝐷_𝑥(𝜏)−𝑢_𝑥𝑥𝑥𝐷_𝑥(𝜉),𝜂^{𝑥𝑥𝑥𝑥}=𝐷_𝑥𝜂^𝑥𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑡}𝐷_𝑥(𝜏)−𝑢_{𝑥𝑥𝑥𝑥}𝐷_𝑥(𝜉),𝜂^{𝑥𝑥𝑥𝑥𝑥}=𝐷_𝑥𝜂^{𝑥𝑥𝑥𝑥}−𝑢_{𝑥𝑥𝑥𝑥𝑡}𝐷_𝑥(𝜏)−𝑢_{𝑥𝑥𝑥𝑥𝑥}𝐷_𝑥(𝜉). (5) Here, 𝐷_𝑖 denotes the total derivative operator and is defined by 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢_𝑖𝜕𝜕𝑢+𝑢_𝑖𝑗𝜕𝜕𝑢_𝑗+⋯,𝑖=1,2, (6) and (𝑥¹,𝑥²)=(𝑡,𝑥).

Solving (4) with the help of (5) we obtain 𝜏=𝜏(𝑡),𝜉_𝑢=0,𝜂_𝑢𝑢𝛽(𝑡)𝜏=0,_𝑡+𝛽^′(𝑡)𝜏−5𝛽(𝑡)𝜉_𝑥𝜂=0,_𝑥𝑢−2𝜉_𝑥𝑥𝜏=0,_𝑡𝑢²−𝜉_𝑥𝑢²+2𝜂𝑢−𝜉_𝑡+𝛽(𝑡)5𝜂_{𝑥𝑥𝑥𝑥𝑢}−𝜉_{𝑥𝑥𝑥𝑥𝑥}𝜂=0,_𝑡+𝛼(𝑡)𝜂+𝛼(𝑡)𝑢𝜏_𝑡−𝛼(𝑡)𝑢𝜂_𝑢+𝑢𝛼^′(𝑡)𝜏+𝑢²𝜂_𝑥+𝛽(𝑡)𝜂_{𝑥𝑥𝑥𝑥𝑥}=0. (7)(8)(9)(10)(11)

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 𝜉=2𝑐₁𝑥+𝑐₂,𝜏=2𝑐₃𝑡+𝑐₄,𝜂=𝑐₁−𝑐₃𝑢, (12)
and (8) becomes 𝛽^′(𝑡)2𝑐₃𝑡+𝑐₄+𝛽(𝑡)2𝑐₃−10𝑐₁=0, (13) 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 𝑉₁=𝜕.𝜕𝑥 (14)(1.2) 𝛽(𝑡)=𝑏₀(≠0) is a constant.

Equation (1) admits a three-dimensional Lie algebra spanned by 𝑉₁=𝜕𝜕𝑥,𝑉₂=𝜕,𝑉𝜕𝑡₃𝜕=𝑥𝜕𝜕𝑥+5𝑡𝜕𝜕𝑡−2𝑢.𝜕𝑢 (15)(1.3)𝛽(𝑡)=𝑏₀e^𝑚𝑡, 𝑏₀,𝑚(≠0) are constant.

We obtain the corresponding two Lie point symmetry generators 𝑉₁=𝜕𝜕𝑥,𝑉₂=𝜕+𝑚𝜕𝑡5𝑥𝜕+𝑚𝜕𝑥𝑢𝜕10.𝜕𝑢 (16)

Case 2 (𝛼(𝑡) is a nonzero constant as 𝑎). Similarly, in this case, solving (7)–(11) for the infinitesimals, we obtain 𝜉=2𝑐₁𝑥+𝑐₃𝑐,𝜏=−²𝑎e^2𝑎𝑡+𝑐₄,𝜂=𝑐₁+𝑐₂e^2𝑎𝑡𝑢, (17) and (8) becomes𝛽^′(𝑡)−𝑐₂𝑎e^2𝑎𝑡+𝑐₄+𝛽(𝑡)−2𝑐₂e^2𝑎𝑡−10𝑐₁=0, (18) 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 𝑉₁=𝜕.𝜕𝑥 (19)(2.2)𝛽(𝑡)=𝑏₀(≠0) is a constant.

We have the corresponding two-dimensional Lie algebra 𝑉₁=𝜕𝜕𝑥,𝑉₂=𝜕.𝜕𝑡 (20)(2.3)𝛽(𝑡)=𝑏₀e^−2𝑎𝑡, 𝑏₀ is a constant.

Equation (1) admits a three-dimensional Lie algebra spanned by 𝑉₁=𝜕𝜕𝑥,𝑉₂=e^2𝑎𝑡𝜕𝜕𝑡−𝑎e^2𝑎𝑡𝑢𝜕,𝑉𝜕𝑢₃𝜕=5𝜕𝜕𝑡−2𝑎𝑥𝜕𝜕𝑥−𝑎𝑢.𝜕𝑢 (21)(2.4)𝛽(𝑡)=𝑏₀e^𝑚𝑡(𝑏₀≠0) is a constant.

Substituting 𝛽(𝑡) into (18), one can get 𝑐₂e^2𝑎𝑡𝑚𝑎+2=10𝑐₁−𝑐₄𝑚, (22) where ??𝑚≠−2𝑎. We obtain the corresponding two-dimensional Lie algebra 𝑉₁=𝜕,𝑉𝜕𝑥₂=𝜕+𝑚𝜕𝑡5𝑥𝜕+𝑚𝜕𝑥𝑢𝜕10.𝜕𝑢 (23)

Case 3 (𝛼(𝑡)=1/𝑡). In this case, solving (7)–(11), we get 𝜉=𝑐₃𝑥+𝑐₄,𝜏=−𝑐₁𝑡²−2𝑐₂𝑡+𝑐₃𝑐𝑡,𝜂=₁𝑡+𝑐₂𝑢, (24)
and (8) becomes 𝛽^′(𝑡)−𝑐₁𝑡²−2𝑐₂𝑡+𝑐₃𝑡+𝛽(𝑡)−4𝑐₃−2𝑐₂−2𝑐₁𝑡=0, (25) 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 𝑉₁=𝜕.𝜕𝑥 (26)(3.2)𝛽(𝑡)=𝑏₀(≠0) is a constant.

We obtain the corresponding two-dimensional Lie algebra 𝑉₁=𝜕𝜕𝑥,𝑉₂𝜕=𝑥𝜕𝜕𝑥+5𝑡𝜕𝜕𝑡−2𝑢.𝜕𝑢 (27)(3.3)𝛽(𝑡)=𝑏₀/𝑡², 𝑏₀(≠0) is a constant.

The Lie algebra is extended by the symmetry generators 𝑉₁=𝜕𝜕𝑥,𝑉₂=𝑡²𝜕𝜕𝜕𝑡−𝑡𝑢,𝑉𝜕𝑢₃𝜕=𝑥𝜕𝜕𝑥−5𝑡𝜕𝜕𝑡+3𝑢.𝜕𝑢 (28)

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]). Let ?𝐺? 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] 𝐴𝑑exp𝜖𝑉_𝑖𝑉_𝑗=𝑉_𝑗−𝜖𝑉_𝑖,𝑉_𝑗+12𝜖²𝑉_𝑖,𝑉_𝑖,𝑉_𝑗−⋯, (29) where 𝜖 is a real constant. Here [𝑉_𝑖,𝑉_𝑗] is the commutator for the Lie algebra given by 𝑉_𝑖,𝑉_𝑗=𝑉_𝑖𝑉_𝑗−𝑉_𝑗𝑉_𝑖. (30)

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 Tables 1 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).

Table 2 

Adjoint table of the Lie algebra of (1.2).

Table 3 

Commutator table of the Lie algebra of (2.3).

Table 4 

Adjoint table of the Lie algebra of (2.3).

Table 5 

Commutator table of the Lie algebra of (2.4).

Table 6 

Adjoint table of the Lie algebra of (2.4).

Table 7 

Commutator table of the Lie algebra of (3.3).

Table 8 

Adjoint table of the Lie algebra of (3.3).

Table 9 

Optimal system of subalgebras of some special cases of (1.1).

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 Table 9 as they can be derived in a similar manner.

3. Symmetry Reductions and Exact Group-Invariant Solutions

In this section, we will use Table 9 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 generator 𝑉₁, 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 combination 𝑉₂+𝜆𝑉₁, we have 𝑢=𝑓(𝜉), (31) where 𝜉=𝑥−𝜆𝑡 is the group-invariant. Substituting (31) into (1.2), we reduce it to the following ODE: 𝑏₀𝑓⁽⁵⁾+𝑓²𝑓^′−𝜆𝑓^′=0. (32)

3.1.3. 𝑉₃

For the generator 𝑉₃, we have 𝑢=𝑡^−(2/5)𝑓(𝜉), (33) where 𝜉=𝑥𝑡^−(1/5) is the group-invariant. Substituting (33) into (1.2), we reduce it to the following ODE: 5𝑏₀𝑓⁽⁵⁾+5𝑓²𝑓^′−2𝑓−𝜉𝑓^′=0, (34) where??𝑓^′=𝑑𝑓/𝑑𝜉.

3.2. Symmetry Reductions and Exact Solutions to (2.3)

3.2.1. 𝑉₁

For the generator 𝑉₁, 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 generator 𝑉₃, we have 𝑢=𝑒^{−𝑎𝑡/5}𝑓(𝜉), (35) where ?𝜉=𝑥𝑒^2𝑎𝑡/5. Substituting (35) into (2.3), one can get 5𝑏₀𝑓⁽⁵⁾+5𝑓²𝑓^′+4𝑎𝑓+2𝑎𝜉𝑓^′=0, (36) where?𝑓^′=𝑑𝑓/𝑑𝜉.

3.2.3. 𝑉₂+𝜇𝑉₁

(i) For 𝜇≠0, we have 𝑢=𝑓(𝜉)e^−𝑎𝑡, (37) where ??𝜉=𝑥+(𝜇e^−2𝑎𝑡/2𝑎). Substituting (37) into (2.3) yields 𝑏₀𝑓⁽⁵⁾+𝑓²𝑓^′−𝜇𝑓^′=0. (38)

(ii) For 𝜇=0, we get 𝑏₀𝑓⁽⁵⁾+𝑓²𝑓^′=0. (39)

3.3. Symmetry Reductions and Exact Solutions to (2.4)

3.3.1. 𝑉₁

For the generator 𝑉₁, we get that the group-invariant solution of (2.4) is 𝑢(𝑥,𝑡)=𝑐𝑒^−𝑎𝑡, (40) where 𝑐 is an arbitrary constant.

3.3.2. 𝑉₂

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

3.4. Symmetry Reductions and Exact Solutions to (3.3)

3.4.1. 𝑉₁

For the generator 𝑉₁, we get that the group-invariant solution of (3.3) is 𝑢()=𝑐𝑥,𝑡𝑡, (43) where 𝑐 is an arbitrary constant.

3.4.2. 𝑉₃

For the generator 𝑉₃, we have 𝑢=𝑡^−(3/5)𝑓(𝜉), (44) where 𝜉=𝑥𝑡^1/5. Substituting (44) into (3.3), one can get 5𝑏₀𝑓⁽⁵⁾+5𝑓²𝑓^′+2𝑓+𝜉𝑓^′=0, (45) where?𝑓^′=𝑑𝑓/𝑑𝜉.

3.4.3. 𝑉₂+𝜇𝑉₁

(i) For 𝜇≠0, we have 1𝑢=𝑡𝑓(𝜉), (46) where 𝜉=𝑥+𝜇/𝑡. Substituting (46) into (3.3) yields 𝑏₀𝑓⁽⁵⁾+𝑓²𝑓^′−𝜇𝑓^′=0. (47)

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 𝑓=𝑎₀+𝑎₁sn(𝜉)+𝑎₂sn²(𝜉). (48) Substituting (48) into (47), one can get 1𝑏=−𝑎360𝑚²²⁴1,𝑚=𝑚,𝜇=𝑎15²²−𝑚²+1+𝑚⁴𝑚⁴,𝑎₀1=−3𝑎₂1+𝑚²𝑚²,𝑎₁=0,𝑎₂=𝑎₂. (49)

Thus, we obtain Jacobi elliptic function solutions of (3.3) as follows: 𝑢()1𝑥,𝑡=−3𝑎₂1+𝑚²𝑚²+𝑎₂sn²(𝜉). (50) In particular, when 𝑚→1, we can obtain hyperbolic function solutions 𝑢()1𝑥,𝑡=−3𝑎₂1+𝑚²𝑚²+𝑎₂tanh²(𝜉). (51) When 𝑚→0, we can obtain trigonometric function solutions 𝑢()1𝑥,𝑡=−3𝑎₂1+𝑚²𝑚²+𝑎₂sin²(𝜉), (52) where 𝜉=𝑥+𝜇/𝑡.

(ii) For 𝜇=0, we get 𝑏₀𝑓⁽⁵⁾+𝑓²𝑓^′=0. (53)

Remark 4. It is not difficult to find out that the reduced ODEs may be classified into four classes: 𝑓⁽⁵⁾+𝐴𝑓²𝑓^′𝑓=0,₍₅₎+𝐴𝑓²𝑓^′+𝐵𝑓^′𝑓=0,₍₅₎+𝐴𝑓²𝑓^′+𝐵𝑓^′𝑓+𝐶𝑓=0,₍₅₎+𝐴𝑓²𝑓^′+𝐵𝑓^′+𝐶𝜉𝑓^′=0. (54)

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: 𝑓(𝜉)=^∞_𝑛=0𝑐_𝑛𝜉^𝑛. (55)

Substituting (55) into (45), we get 600𝑏𝑐₅+𝑏^∞_𝑛=1(𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)(𝑛+5)𝑐_𝑛+5𝜉^𝑛+5𝑐²₀𝑐₁+5^∞𝑛_𝑛=1^𝑘_𝑘=0𝑗=0(𝑛+1−𝑘)𝑐_𝑗𝑐_𝑘−𝑗𝑐_{𝑛+1−𝑘}𝜉^𝑛+2𝑐₀+2^∞_𝑛=1𝑐_𝑛𝜉^𝑛+^∞_𝑛=1𝑛𝑐_𝑛𝜉^𝑛=0. (56)

Now from (56), comparing coefficients, for ?𝑛=0, one can get 𝑐₅=−2𝑐₀+5𝑐²₀𝑐₁.600𝑏 (57)

Generally, for 𝑛≥1, we obtain 𝑐_𝑛+51=−𝑏()×()𝑐𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)(𝑛+5𝑛+2_𝑛+5^𝑛𝑘_𝑘=0𝑗=0(𝑛+𝑘−1)𝑐_𝑗𝑐_𝑘−𝑗𝑐_{𝑛+1−𝑘}. (58)

From (57) and (58), we can obtain all the coefficients 𝑐_𝑛(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: 𝑓(𝜉)=𝑐₀+𝑐₁𝜉+𝑐₂𝜉²+𝑐₃𝜉³+𝑐₄𝜉⁴+𝑐₅𝜉⁵+^∞_𝑛=1𝑐_𝑛+5𝜉^𝑛+5=𝑐₀+𝑐₁𝜉+𝑐₂𝜉²+𝑐₃𝜉³+𝑐₄𝜉⁴−2𝑐₀+5𝑐²₀𝑐₁𝜉600𝑏⁵−^∞_𝑛=01𝑏()×()𝑐𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)(𝑛+5𝑛+2_𝑛+5^𝑛𝑘_𝑘=0𝑗=0(𝑛+𝑘−1)𝑐_𝑗𝑐_𝑘−𝑗𝑐_{𝑛+1−𝑘}𝜉^𝑛+5. (59)

Thus, the exact power series solution of (3.3) is 𝑢(𝑥,𝑡)=𝑐₀+𝑐₁𝑥𝑡^1/5+𝑐₂𝑥𝑡^1/52+𝑐₃𝑥𝑡^1/53+𝑐₄𝑥𝑡^1/54−2𝑐₀+5𝑐²₀𝑐₁600𝑏𝑥𝑡^1/55+^∞_𝑛=0𝑐_𝑛+5𝑥𝑡^1/5𝑛+5𝑡^−(3/5)=𝑐₀+𝑐₁𝑥𝑡^1/5+𝑐₂𝑥𝑡^1/52+𝑐₃𝑥𝑡^1/53+𝑐₄𝑥𝑡^1/54−2𝑐₀+5𝑐²₀𝑐₁600𝑏𝑥𝑡^1/55−^∞_𝑛=01𝑏()×()𝑐𝑛+1)(𝑛+2)(𝑛+3)(𝑛+4)(𝑛+5𝑛+2_𝑛+5^𝑛𝑘_𝑘=0𝑗=0(𝑛+𝑘−1)𝑐_𝑗𝑐_𝑘−𝑗𝑐_{𝑛+1−𝑘}×𝜉^𝑛+5𝑡^−(3/5), (60) where 𝑐_𝑖(𝑖=0,1,2,3,4) are arbitrary constants, and the other coefficients 𝑐_𝑛(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 𝑢(𝑥,𝑡)=𝑐₀+𝑐₁𝑥𝑡^1/5+𝑐₂𝑥𝑡^1/52+𝑐₃𝑥𝑡^1/53+𝑐₄𝑥𝑡^1/54−2𝑐₀+5𝑐²₀𝑐₁600𝑏𝑥𝑡^1/55×𝑡+⋯_−(3/5). (61)

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 Section 2, 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.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (NNSFC) (Grant no. 11171022). The authors express their sincere thanks to the referees for their careful review of this paper and their useful suggestions.