The Lie symmetry method is performed for the fifth-order nonlinear evolution Kudryashov-Sinelshchikov equation. We will find ones and two-dimensional optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions is investigated.

1. Introduction

The theory of Lie symmetry groups of differential equations was developed by Lie [1]. Such Lie groups are invertible point transformations of both the dependent and independent variables of the differential equations. The symmetry group methods provide an ultimate arsenal for analysis of differential equations and are of great importance to understand and to construct solutions of differential equations. Several applications of Lie groups in the theory of differential equations were discussed in the literature, and the most important ones are reduction of order of ordinary differential equations, construction of invariant solutions, mapping solutions to other solutions, and the detection of linearizing transformations (for many other applications of Lie symmetries, see [2–4]).

In the present paper, we study the following fifth-order nonlinear evolution equation: ut+auux+bux3+cux4+dux5=eux2,
where a, b, c, d, and e are positive constants. This equation was introduced recently by Kudryashov and Sinelshchikov [5], which is the generalization of the famous Kawahara equation. By using the reductive perturbation method, they obtained (1.1). Kudryashov and Demina found some exact solutions of the generalized nonlinear evolution equations such as (1.1); see [6]. The study of nonlinear wave processes in viscoelastic tube is the important problem in such tubes similar to large arteries (see [7–9]).

In this paper, by using the Lie point symmetry method, we will investigate (1.1), looking at the representation of the obtained symmetry group on its Lie algebra. We will find the preliminary classification of group-invariant solutions, and then we can reduce (1.1) to an ordinary differential equation.

This work is organized as follows. In Section 2, we recall some results needed to construct Lie point symmetries of a given system of differential equations. In Section 3, we give the general form of an infinitesimal generator admitted by (1.1) and find transformed solutions. Section 4 is devoted to the construction of the group-invariant solutions and its classification which provides in each case reduced forms of (1.1).

2. Method of Lie Symmetries

In this section, we recall the general procedure for determining symmetries for any system of partial differential equations (see [2, 3, 10]). To begin, let us consider that the general case of a nonlinear system of partial differential equations of order nth in p-independent and q-dependent variables is given as a system of equations Δν(x,u(n))=0,ν=1,…,l,
involving x=(x1,…,xp), u=(u1,…,uq) and the derivatives of u with respect to x up to n, where u(n) represents all the derivatives of u of all orders from 0 to n. We consider a one-parameter Lie group of infinitesimal transformations acting on the independent and dependent variables of the system (2.1), x̃i=xi+sξi(x,u)+O(s2),i=1,…,p,ũj=uj+sηj(x,u)+O(s2),j=1,…,q,
where s is the parameter of the transformation and ξi, ηj are the infinitesimals of the transformations for the independent and dependent variables, respectively. The infinitesimal generator v associated with the above group of transformations can be written as v=∑i=1pξi(x,u)∂xi+∑j=1qηj(x,u)∂uj.
A symmetry of a differential equation is a transformation, which maps solutions of the equation to other solutions. The invariance of the system (2.1) under the infinitesimal transformations leads to the invariance conditions (Theorem 2.36 of [2]) Pr(n)v[Δν(x,u(n))]=0,ν=1,…,l,wheneverΔν(x,u(n))=0,
where Pr(n) is called the nth-order prolongation of the infinitesimal generator given by Pr(n)v=v+∑α=1q∑JφαJ(x,u(n))∂uJα,
where J=(j1,…,jk), 1≤jk≤p, 1≤k≤n, and the sum is over all J^{'}s of order 0<#J≤n. If #J=k, the coefficient φJα of ∂uJα will only depend on kth and lower-order derivatives of u, φαJ(x,u(n))=DJ(φα-∑i=1pξiuiα)+∑i=1pξiuJ,iα,
where uiα:=∂uα/∂xi and uJ,iα:=∂uJα/∂xi.

One of the most important properties of these infinitesimal symmetries is that they form a Lie algebra under the usual Lie bracket.

3. Lie Symmetries of (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>)

We consider the one-parameter Lie group of infinitesimal transformations on(x1=x,x2=t,u1=u),
x̃=x+sξ(x,t,u)+O(s2),t̃=x+sη(x,t,u)+O(s2),ũ=x+sφ(x,t,u)+O(s2),
where s is the group parameter, and ξ1=ξ and ξ2=η, and φ1=φ are the infinitesimals of the transformations for the independent and dependent variables, respectively. The associated vector field is of the form v=ξ(x,t,u)∂x+η(x,t,u)∂t+φ(x,t,u)∂u,
By (2.5), its fifth prolongation is Pr(5)v=v+φx∂ux+φt∂ut+φx2∂ux2+φxt∂uxt+⋯+φt2∂ut2+φxt4∂uxt4+φt5∂ut5,
where, for instance, by (2.6) we have φx=Dx(φ-ξux-ηut)+ξux2+ηuxt,φt=Dt(φ-ξux-ηut)+ξuxt+ηut2,⋮φt5=Dx5(φ-ξux-ηut)+ξux5t+ηut5,
where Dx and Dt are the total derivatives with respect to x and t respectively. By (2.4), the vector field v generates a one-parameter symmetry group of (1.1) if and only if Pr(5)v[ut+auux+bux3+cux4+dux5-eux2]=0,wheneverut+auux+bux3+cux4+dux5-eux2=0.
The condition (3.5) is equivalent to auxφ+auφx+φt-eφx2+bφx3+cφx4+dφx5=0,wheneverut+auux+bux3+cux4+dux5-eux2=0.

Substituting (3.4) into (3.6), and equating the coefficients of the various monomials in partial derivatives with respect to x and various power of u, we can find the determining equations for the symmetry group of (1.1). Solving this equation, we get the following forms of the coefficient functions: ξ=c2at+c3,η=c1,φ=c2,
where c1, c2, and c3 are arbitrary constants. Thus, the Lie algebra of infinitesimal symmetry of (1.1) is spanned by the three vector fields: v1=∂x,v2=∂t,v3=t∂x+1a∂u.
The commutation relations between these vector fields are given in Table 1.

The commutator table.

[vi,vj]

v1

v2

v3

v1

0

0

0

v2

0

0

v1

v3

0

-v1

0

Theorem 3.1.

The Lie algebra £_{3} spanned by v1,v2,andv3 is second Bianchi class type and it is solvable and Nilpotent [11].

To obtain the group transformation which is generated by the infinitesimal generators vi for i=1,2,3, we need to solve the three systems of first-order ordinary differential equations dx̃(s)ds=ξi(x̃(s),t̃(s),ũ(s)),x̃(0)=x,dt̃(s)ds=ηi(x̃(s),t̃(s),ũ(s)),t̃(0)=t,dũ(s)ds=φi(x̃(s),t̃(s),ũ(s)),ũ(0)=u,i=1,2,3.
Exponentiating the infinitesimal symmetries of (1.1), we get the one-parameter groups Gi(s) generated by vi for i=1,2,3, G1:(t,x,u)⟼(x+s,t,u),G2:(t,x,u)⟼(x,t+s,u),G3:(t,x,u)⟼(x+ts,t,u+sa).
Consequently, we have the following.

Theorem 3.2.

If u=f(x,t) is a solution of (1.1), so are the functions
G1(s)⋅f(x,t)=f(x-s,t),G2(s)⋅f(x,t)=f(x,t-s),G3(s)⋅f(x,t)=f(x-ts,t)+sa.

4. Optimal System and Invariant Solution of (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>)

In this section, we obtain the optimal system and reduced forms of (1.1) by using symmetry group properties obtained in the previous section. Since the original partial differential equation has two independent variables, then this partial differential equation transforms into the ordinary differential equation after reduction.

Definition 4.1.

Let G be a Lie group with Lie algebra 𝔤. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subalgebras with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras form 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 𝔥¯=Ad(g(𝔥)) [2].

Theorem 4.2.

Let H and H¯ be connected s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras 𝔥 and 𝔥¯ of the Lie algebra 𝔤 of G, thenH¯=gHg-1 are conjugate subgroups if and only if 𝔥¯=Ad(g(𝔥)) are conjugate subalgebras [2].

By Theorem 4.2, the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For one-dimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation, since each one-dimensional subalgebra is determined by nonzero vector in 𝔤. This problem is attacked by the naïve approach of taking a general element V in 𝔤 and subjecting it to various adjoint transformation so as to “simplify” it as much as possible. Thus, we will deal with th construction of the optimal system of subalgebras of 𝔤.

To compute the adjoint representation, we use the Lie series Ad(exp(εvi)vj)=vj-ε[vi,vj]+ε22[vi,[vi,vj]]-⋯,
where [vi,vj] is the commutator for the Lie algebra, ε is a parameter, and i,j=1,2,3. Then, we have Table 2.

Adjoint representation table of the infinitesimal generators v_{i}.

Ad(exp(εvi))vj

v1

v2

v3

v1

v1

v2

v3

v2

v1

v2

v3-εv1

v3

v1

v2+εv1

v3

Theorem 4.3.

An optimal system of one-dimensional Lie algebras of (1.1) is provided by (1) v2, and (2) v3+αv2.

Proof.

Consider the symmetry algebra 𝔤 of (1.1) whose adjoint representation was determined in Table 2, and
V=a1v1+a2v2+a3v3
is a nonzero vector field in 𝔤. We will simplify as many of the coefficients ai as possible through judicious applications of adjoint maps to V. Suppose first that a3≠0. Scaling V if necessary, we can assume that a3=1. Referring to Table 2, if we act on such a V by Ad(exp(a1v2)), we can make the coefficient of v1 vanish, and the vector field V takes the form
V′=Ad(exp(a1v2))V=a2′v2+v3
for certain scalar a2′. So, depending on the sign of a2′, we can make the coefficient of v2 either +1, -1, or 0. In other words, every one-dimensional subalgebra generated by a V with a3≠0 is equivalent to one spanned by either v3+v2, v3-v2, or v3.

The remaining one-dimensional subalgebras are spanned by vectors of the above form with a3=0. If a2≠0, we scale to make a2=1, and then the vector field V takes the form
V′′=a1′′v1+v2,
for certain scalar a1′′. Similarly, we can vanish a1′′, so every one-dimensional subalgebra generated by a V with a3=0 is equivalent to the subalgebra spanned by v2.

Theorem 4.4.

An optimal system of two-dimensional Lie algebras of (1.1) is provided by
〈αv2+v3,βv1+γv3〉.

Symmetry group method will be applied to (1.1) to be connected directly to some order differential equations. To do this, a particular linear combinations of infinitesimals are considered and their corresponding invariants are determined.

Equation (1.1) is expressed in the coordinates (x,t,u), so to reduce this equation is to search for its form in specific coordinates. Those coordinates will be constructed by searching for independent invariants (χ,ζ) corresponding to the infinitesimal generator. So using the chain rule, the expression of the equation in the new coordinate allows us to get the reduced equation.

In what follows, we begin the reduction process of (1.1); note that the reduced form does not allow us to have the self-similar solutions.

4.1. Galilean-Invariant Solutions

First, consider v3=t∂x+(1/a)∂u. To determine independent invariants I, we need to solve the first partial differential equations vi(I)=0, that is, invariants ζ and χ can be found by integrating the corresponding characteristic system, which is dt0=dxt=adu1.
The obtained solution is given by χ=t,ζ=u-xat.
Therefore, a solution of our equation in this case is u=f(x,χ,ζ)=ζ+xat.
The derivatives of u are given in terms of ζ and χ as ux=1at,ux2=ux3=ux4=ux5=0,ut=ζχ-1at2x.
Substituting (4.9) into (1.1), we obtain the order ordinary differential equation ζχ+1χζ=0.
The solution of this equation is ζ=c1/χ. Consequently, we obtain that u(x,t)=x+ac1at.

4.2. Travelling Wave Solutions

The invariants of v2+c0v1=c0∂x+∂t are χ=x-c0t and ζ=u, so the reduced form of (1.1) is -c0ζχ+aζζχ+bζχ3+cζχ4+dζχ5-eζχ2=0.
The family of the periodic solution for (4.12) when a=1 takes the following form (see [5]): ζ=a0+Asn4{mχ,k}+Bsn{mχ,k}ddχsn{mχ,k},
where sn{mχ,k} is Jacobi elliptic function.

The invariants of v3+βv2=t∂x+β∂t+(1/a)∂u are χ=x-t2/2β and ζ=u-t/aβ, so the reduced form of (1.1) is 1aβ-tβζχ+aζζχ+bζχ3+cζχ4+dζχ5-eζχ2=0.

The invariants of v2=∂t are χ=x and ζ=u, then the reduced form of (1.1) is aζζχ+bζχ3+cζχ4+dζχ5-eζχ2=0.

The invariants of v1=∂x are χ=t and ζ=u then the reduced form of (1.1) is ζχ=0, then the solution of this equation is u(x,t)=cte.

Acknowledgments

This research was supported by Islamic Azad University of Karaj branch. The authors are grateful to the referee for careful reading and useful suggestions.

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