Lie Symmetry Analysis of Kudryashov-Sinelshchikov Equation

Abstract

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: 𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥3+𝑐𝑢_𝑥4+𝑑𝑢_𝑥5=𝑒𝑢_𝑥2, (1.1) 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 Δ_𝜈𝑥,𝑢^(𝑛)=0,𝜈=1,…,𝑙, (2.1) involving 𝑥=(𝑥¹,…,𝑥^𝑝), 𝑢=(𝑢¹,…,𝑢^𝑞) 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), 𝑥^𝑖=𝑥^𝑖+𝑠𝜉^𝑖(𝑥,𝑢)+𝑂𝑠²𝑢,𝑖=1,…,𝑝,_𝑗=𝑢^𝑗+𝑠𝜂^𝑗(𝑥,𝑢)+𝑂𝑠²,𝑗=1,…,𝑞, (2.2) where 𝑠 is the parameter of the transformation and 𝜉^𝑖, 𝜂^𝑗 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=^𝑝_𝑖=1𝜉^𝑖(𝑥,𝑢)𝜕_𝑥𝑖+^𝑞_𝑗=1𝜂^𝑗(𝑥,𝑢)𝜕_𝑢𝑗. (2.3) 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)𝐯Δ_𝜈𝑥,𝑢^(𝑛)=0,𝜈=1,…,𝑙,wheneverΔ_𝜈𝑥,𝑢^(𝑛)=0, (2.4) where Pr^(𝑛) is called the 𝑛th-order prolongation of the infinitesimal generator given by Pr^(𝑛)𝐯=𝐯+^𝑞_𝛼=1𝐽𝜑^𝐽_𝛼𝑥,𝑢^(𝑛)𝜕_𝑢^𝛼_𝐽, (2.5) where 𝐽=(𝑗₁,…,𝑗_𝑘), 1≤𝑗_𝑘≤𝑝, 1≤𝑘≤𝑛, and the sum is over all 𝐽^'s of order 0<#𝐽≤𝑛. If #𝐽=𝑘, the coefficient 𝜑^𝛼_𝐽 of 𝜕_𝑢^𝛼_𝐽 will only depend on 𝑘th and lower-order derivatives of 𝑢, 𝜑^𝐽_𝛼𝑥,𝑢^(𝑛)=𝐷_𝐽𝜑_𝛼−^𝑝_𝑖=1𝜉^𝑖𝑢^𝛼_𝑖+^𝑝_𝑖=1𝜉^𝑖𝑢^𝛼_𝐽,𝑖, (2.6) where 𝑢^𝛼_𝑖∶=𝜕𝑢^𝛼/𝜕𝑥^𝑖 and 𝑢^𝛼_𝐽,𝑖∶=𝜕𝑢^𝛼_𝐽/𝜕𝑥^𝑖.

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 (1.1)

We consider the one-parameter Lie group of infinitesimal transformations on(𝑥¹=𝑥,𝑥²=𝑡,𝑢¹=𝑢), 𝑥=𝑥+𝑠𝜉(𝑥,𝑡,𝑢)+𝑂𝑠²,𝑡̃=𝑥+𝑠𝜂(𝑥,𝑡,𝑢)+𝑂𝑠²,𝑢=𝑥+𝑠𝜑(𝑥,𝑡,𝑢)+𝑂𝑠², (3.1) 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, (3.2) By (2.5), its fifth prolongation is Pr⁽⁵⁾𝐯=𝐯+𝜑^𝑥𝜕_𝑢𝑥+𝜑^𝑡𝜕_𝑢𝑡+𝜑^𝑥2𝜕_𝑢𝑥2+𝜑^𝑥𝑡𝜕_𝑢𝑥𝑡+⋯+𝜑^𝑡2𝜕_𝑢𝑡2+𝜑^𝑥𝑡4𝜕_{𝑢4𝑥𝑡}+𝜑^𝑡5𝜕_𝑢𝑡5, (3.3) where, for instance, by (2.6) we have 𝜑^𝑥=𝐷_𝑥𝜑−𝜉𝑢_𝑥−𝜂𝑢_𝑡+𝜉𝑢_𝑥2+𝜂𝑢_𝑥𝑡,𝜑^𝑡=𝐷_𝑡𝜑−𝜉𝑢_𝑥−𝜂𝑢_𝑡+𝜉𝑢_𝑥𝑡+𝜂𝑢_𝑡2,⋮𝜑^𝑡5=𝐷⁵_𝑥𝜑−𝜉𝑢_𝑥−𝜂𝑢_𝑡+𝜉𝑢_𝑥5𝑡+𝜂𝑢_𝑡5, (3.4) where 𝐷_𝑥 and 𝐷_𝑡 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⁾𝐯𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥3+𝑐𝑢_𝑥4+𝑑𝑢_𝑥5−𝑒𝑢_𝑥2=0,whenever𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥3+𝑐𝑢_𝑥4+𝑑𝑢_𝑥5−𝑒𝑢_𝑥2=0. (3.5) The condition (3.5) is equivalent to 𝑎𝑢_𝑥𝜑+𝑎𝑢𝜑^𝑥+𝜑^𝑡−𝑒𝜑^𝑥2+𝑏𝜑^𝑥3+𝑐𝜑^𝑥4+𝑑𝜑^𝑥5=0,whenever𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥3+𝑐𝑢_𝑥4+𝑑𝑢_𝑥5−𝑒𝑢_𝑥2=0. (3.6)

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, (3.7) 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. (3.8) The commutation relations between these vector fields are given in Table 1.

Table 1 

The commutator table.

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 𝐯_𝑖 for 𝑖=1,2,3, we need to solve the three systems of first-order ordinary differential equations 𝑑𝑥(𝑠)𝑑𝑠=𝜉_𝑖𝑥(𝑠),𝑡̃(𝑠),𝑢(𝑠),𝑥(0)𝑑𝑡̃(𝑠)=𝑥,𝑑𝑠=𝜂_𝑖𝑥(𝑠),𝑡̃(𝑠),𝑢(𝑠),𝑡̃(0)𝑑𝑢(𝑠)=𝑡,𝑑𝑠=𝜑_𝑖𝑥(𝑠),𝑡̃(𝑠),𝑢(𝑠),𝑢(0)=𝑢,𝑖=1,2,3. (3.9) Exponentiating the infinitesimal symmetries of (1.1), we get the one-parameter groups 𝐺_𝑖(𝑠) generated by 𝐯_𝑖 for 𝑖=1,2,3, 𝐺₁∶()⟼(),𝐺𝑡,𝑥,𝑢𝑥+𝑠,𝑡,𝑢₂∶()⟼(),𝐺𝑡,𝑥,𝑢𝑥,𝑡+𝑠,𝑢₃∶()⟼𝑠𝑡,𝑥,𝑢𝑥+𝑡𝑠,𝑡,𝑢+𝑎. (3.10) 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)()(),𝐺⋅𝑓𝑥,𝑡=𝑓𝑥−𝑠,𝑡₂(𝑠)()(),𝐺⋅𝑓𝑥,𝑡=𝑓𝑥,𝑡−𝑠₃(𝑠)()()+𝑠⋅𝑓𝑥,𝑡=𝑓𝑥−𝑡𝑠,𝑡𝑎. (3.11)

4. Optimal System and Invariant Solution of (1.1)

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 Adexp𝜀𝐯_𝑖𝐯_𝑗=𝐯_𝑗−𝜀𝐯_𝑖,𝐯_𝑗+𝜀²2𝐯_𝑖,𝐯_𝑖,𝐯_𝑗−⋯, (4.1) where [𝐯_𝑖,𝐯_𝑗] is the commutator for the Lie algebra, 𝜀 is a parameter, and 𝑖,𝑗=1,2,3. Then, we have Table 2.

Table 2 

Adjoint representation table of the infinitesimal generators v_i.

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 (4.2) is a nonzero vector field in 𝔤. We will simplify as many of the coefficients 𝑎_𝑖 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^′=Adexp𝑎₁𝐯₂𝐕=𝑎^′₂𝐯₂+𝐯₃ (4.3) for certain scalar 𝑎^′₂. So, depending on the sign of 𝑎^′₂, 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, (4.4) 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. (4.5)

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 𝐯_𝑖(𝐼)=0, that is, invariants 𝜁 and 𝜒 can be found by integrating the corresponding characteristic system, which is 𝑑𝑡0=𝑑𝑥𝑡=𝑎𝑑𝑢1. (4.6) The obtained solution is given by 𝑥𝜒=𝑡,𝜁=𝑢−𝑎𝑡. (4.7) Therefore, a solution of our equation in this case is 𝑥𝑢=𝑓𝑥,𝜒,𝜁=𝜁+𝑎𝑡. (4.8) The derivatives of 𝑢 are given in terms of 𝜁 and 𝜒 as 𝑢_𝑥=1𝑎𝑡,𝑢_𝑥2=𝑢_𝑥3=𝑢_𝑥4=𝑢_𝑥5=0,𝑢_𝑡=𝜁_𝜒−1𝑎𝑡²𝑥. (4.9) Substituting (4.9) into (1.1), we obtain the order ordinary differential equation 𝜁_𝜒+1𝜒𝜁=0. (4.10) The solution of this equation is 𝜁=𝑐₁/𝜒. Consequently, we obtain that 𝑢(𝑥,𝑡)=𝑥+𝑎𝑐₁𝑎𝑡. (4.11)

4.2. Travelling Wave Solutions

The invariants of 𝐯₂+𝑐₀𝐯_𝟏=𝑐₀𝜕_𝑥+𝜕_𝑡 are 𝜒=𝑥−𝑐₀𝑡 and 𝜁=𝑢, so the reduced form of (1.1) is −𝑐₀𝜁_𝜒+𝑎𝜁𝜁_𝜒+𝑏𝜁_𝜒3+𝑐𝜁_𝜒4+𝑑𝜁_𝜒5−𝑒𝜁_𝜒2=0. (4.12) The family of the periodic solution for (4.12) when 𝑎=1 takes the following form (see [5]): 𝜁=𝑎₀+𝐴𝑠𝑛⁴𝑑𝑚𝜒,𝑘+𝐵𝑠𝑛𝑚𝜒,𝑘𝑑𝜒𝑠𝑛𝑚𝜒,𝑘, (4.13) 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 1−𝑡𝑎𝛽𝛽𝜁_𝜒+𝑎𝜁𝜁_𝜒+𝑏𝜁_𝜒3+𝑐𝜁_𝜒4+𝑑𝜁_𝜒5−𝑒𝜁_𝜒2=0. (4.14)

The invariants of 𝐯₂=𝜕_𝑡 are 𝜒=𝑥 and 𝜁=𝑢, then the reduced form of (1.1) is 𝑎𝜁𝜁_𝜒+𝑏𝜁_𝜒3+𝑐𝜁_𝜒4+𝑑𝜁_𝜒5−𝑒𝜁_𝜒2=0. (4.15)

The invariants of 𝐯₁=𝜕_𝑥 are 𝜒=𝑡 and 𝜁=𝑢 then the reduced form of (1.1) is 𝜁_𝜒=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.