# Lie Algebra Classification, Conservation Laws, and Invariant Solutions for a Generalization of the Levinson–Smith Equation

Authors: G. Loaiza; Y. Acevedo; O.M.L. Duque; Danilo A. García Hernández
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.


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## Body

## 1. Introduction

Lie group symmetry method is a powerful tool employed to study ODEs, PDEs, FPDEs, FODEs, and so on. This theory was introduced in the 19th century by Sophus Lie [1], following the idea of Galois theory in algebra. Lie group method applied to differential equations has received great interest among researchers in different fields of science such as mathematics, theoretical, and applied physics, due to the physical interpretations of the underlying equations that are studied. As a consequence, this method leads to construct, for example, conservation laws, using the well known Noether’s theorem [2], even more applying Ibragimov’s approach [3]. In the same way, it is possible to build similarity solutions which, in the traditional methods, are not possible.Furthermore, this method contributes to establish schemes and the usefulness of some numerical methods; here, many packages are being built in different environments of computations, e.g., [4, 5]. In general, taking into account the importance of the equations’ study (such as ODEs, PDEs, and others), this method can be interesting to different researchers. A vast reference in Lie group method can be found in the literature, e.g., [6–9]. Recently, the Lie group method approach has been applied to solve and analyze different problems in many scientific fields, e.g., in [10], the authors applied the Lie symmetry method to investigate a fourth-order 1 + 2 evolutionary partial differential equation which has been proposed for the image processing noise reduction. References in the latest progress in symmetry analysis can be found in [11–18] and therein.In [19], Kamke proposes the following differential equation:(1)yxx−fyyfyyx2+gxyx+hxfy=0,where f,g,h are the arbitrary functions, for this equation presents the transformation yx=ηξx which reduces this equation to a system of two first-order equations. Equation (1) can be written as(2)yxx+ϕx,y,yxyx=γx,y,where ϕx,y,yx=−fyy/fyyx+gx and γx,y=−hxfy. This means that the coefficient of friction, i.e., ϕ is a function that depends on x,y, and yx, and it will almost always be a nonlinear function, and on the other hand, function γ, which is known as function of disturbance, is also nonlinear. It is worth remembering that this type of equation (2) is known as generalizations of the Levinson–Smith equation. Also, equation (2), which is a particular case of the generalized Levinson–Simth equations, is related to the Liénard-type second-order nonlinear differential equation [20]. The underlying equations describe several phenomena in different areas such as electronics, biology, mechanics, seismology, chemistry, physics, and cosmology, for example, an important model in physical and biological sciences is the van der Pol equation, which describes a nonconservative oscillator with nonlinear damping. Levinson and Smith in [21] studied a general equation for relaxation oscillations.In [8], the equation,(3)yxx=2yx2y−1+x−1yx+x−1y2,is presented; note that equation (3) is a particular case of equation (1), with fy=y2,gx=−x−1 and hx=−x−1; in [8], Cantwell states that equation (3) has a group of symmetries 5−dimensional; but it does not exhibit the development of said statement; they affirm the Lie group of symmetries of (3), using a ODEtools Maple package. In fact, the goal of this work is (i) to calculate the 5−dimensional Lie symmetry group in detail, (ii) to present the optimal system (optimal algebra) for (3), (iii) to make use of all elements of the optimal algebra to propose invariant solutions for (3), then (iv) to construct the Lagrangian with which we could determine the variational symmetries using Noether’s theorem and thus to present conservation laws associated, and (iv) also to build some nontrivial conservation laws using Ibragimov’s method, and finally (v) to classify the Lie algebra associated to (3), corresponding to the symmetry group.


## 2. Continuous Group of Lie Symmetries

In this section, we study the Lie symmetry group for (3). The main result of this section can be presented as follows.Proposition 1.
The Lie symmetry group for equation (3) is generated by the following vector fields:(4)Π1=−x∂∂x+y∂∂y,Π2=y2∂∂y,Π3=−x∂∂x+xy2∂∂y,Π4=x2y2∂∂y,Π5=x3∂∂x−2x2y+x3y2∂∂y.Proof.
A general form of the one-parameter Lie group admitted by (3) is given by(5)x⟶x+ɛξx,y+Oɛ2,y⟶y+ɛηx,y+Oɛ2,where ϵ is the group parameter. The vector field associated with the group of transformations shown above can be written as Γ=ξx,y∂/∂x+ηx,y∂/∂y, where ξ,η are differentiable functions in ℝ2. Applying its second prolongation,(6)Γ2=Γ+ηx∂∂yx+ηxx∂∂yxx,to equation (3), we must find the infinitesimals ξ,η, satisfying the symmetry condition,(7)ξx−2y2+yxx−2+η2yx2y−2−2x−1y+ηx−4y−1yx−x−1+ηxx=0,associated with (3). Here, ηx,ηxx are the coefficients in Γ2 given by(8)ηx=Dxη−Dxξyx=ηx+ηy−ξxyx−ξyyx2,ηxx=Dxηx−Dxξyxx,=ηxx+2ηxy−ξxxyx+ηyy−2ξxyyx2−ξyyyx3+ηy−2ξxyxx−3ξyyxyxx.
BeingDx as the total derivative operator, Dx=∂x+yx∂y+yxx∂yx+⋯. Replacing (8) into (7) and using (3), we obtain(9)−ξyy−2y−1ξyyx3+ηyy−2ξxy−2y−1ηy−2ξyx−1+2y−2ηyx2+2ηxy−ξxx−x−1ξx+x−2ξ−3ξyx−1y2−4y−1ηxyx+ηxx−x−1ηx−2x−1yη+x−1y2ηy−2x−1y2ξx+x−2y2ξ=0.
From (9), canceling the coefficients of the monomial variables in derivatives 1,yx3,yx2, and yx, we obtain the determining equations for the symmetry group of (3). That is,(9a)yξyy+2ξy=0,(9b)y2xηyy−2y2xξxy−2yxηy−2y2ξy+2xη=0,(9c)2x2yηxy−x2yξxx−xyξx+yξ−3xy3ξy−4x2ηx=0,(9d)x2ηxx−xηx−2xyη+xy2ηy−2xy2ξx+y2ξ=0.
Solving the system of equations (9a)–(9d) for ξ and η, we get(10)ξ=−c1x−c3x+c5x3,η=c1y+c2y2+c3xy2+c4x2y2+c5−2x2y2+x3y2.
Thus, the infinitesimal generators of the group of symmetries of (3) are the operators Π1−Π5 described in the statement of Proposition 1, thus having the proposed result.


## 3. Optimal System

Taking into account [22–25], we present in this section the optimal system associated to the symmetry group of (3), which shows a systematic way to classify the invariant solutions. To obtain the optimal system, we should first calculate the corresponding commutator table, which can be obtained from the operator(11)Πα,Πβ=ΠαΠβ−ΠβΠα=∑i=1nΠαξβi−Πβξαi∂∂xi,where i=1,2, with α,β=1,…,5, and ξαi,ξβi are the corresponding coefficients of the infinitesimal operators Πα,Πβ. After applying the operator (11) to the symmetry group of (3), we obtain the operators that are shown in Table 1.Now, the next thing is to calculate the adjoint action representation of the symmetries of (3), and to do that, we use Table 1 and the operator(12)AdexpλΠH=∑n=0∞λnn!adΠnGfor the symmetriesΠandG.Table 1
Commutators table associated to the symmetry group of (3).
Π1Π2Π3Π4Π5Π10Π20−Π4−2Π5Π2−Π20002Π4Π3000−2Π4−2Π5Π4Π402Π400Π52Π5−2Π42Π500Making use of this operator, we can construct Table2, which shows the adjoint representation for each Πi.Table 2
Adjoint representation of the symmetry group of (3).
adj[, ]Π1Π2Π3Π4Π5Π1Π1e−λΠ2Π3eλΠ4e2λΠ5Π2Π1+λΠ2Π2Π3Π4Π5−2λΠ4Π3Π1Π2Π3e2λΠ4e2λΠ5Π4Π1−λΠ4Π2Π3−2λΠ4Π4Π5Π5Π1−2λΠ5Π2+2λΠ4Π3−2λΠ5Π4Π5Proposition 2.
The optimal system associated to equation (3) is given by the vector fields(13)Π5,a2Π2,Π4,a1Π1+a3Π3,b1Π4+Π5,−a3Π1+a3Π3,a3Π3−2a3b12Π5,Π2+b17Π4,−a3Π1+a3Π3+Π5,a2Π2+a3Π3+b9+a2a3Π4,a2Π2+b10Π4+Π5,a1Π1+a3Π3+b2Π5,−Π1+Π3+b3Π4,−2Π1+Π3+b14Π5,a1Π1+b5Π4+b6Π5,Π1+b7Π2+b8Π5,−2a3Π1+a3Π3+Π4+2a3b13Π5,a2Π2+a3Π3+b11Π4−a3b11a2Π5,a1Π1+Π3+b3Π4+b4Π5,−2Π1+Π3+b15Π4+b16Π5.Proof.
To calculate the optimal system, we start with the generators of symmetries (4) and a generic nonzero vector. Let(14)G=a1Π1+a2Π2+a3Π3+a4Π4+a5Π5.
The objective is to simplify as many coefficients,ai, as possible, through maps adjoint to G, using Table 2.(1)
Assuminga5=1 in (14), we have that G=a1Π1+a2Π2+a3Π3+a4Π4+Π5. Applying the adjoint operator to Π1,G and Π3,G, we do not have any reduction; on the other hand, applying the adjoint operator to Π2,G, we get(15)G1=Adexpλ1Π2G=a1Π1+Π2a2+a1λ1+a3Π3+Π4a4−2λ1+Π5.(1.1) Casea1≠0. Using λ1=−a2/a1a1≠0, in (15), Π2 is eliminated, therefore G1=a1Π1+a3Π3+b1Π4+Π5, where b1=a4+2a2/a1. Now, applying the adjoint operator to Π4,G1, we get G2=Adexpλ2Π4G1=a1Π1+a3Π3+b1−λ2a1+2a3Π4+Π5.1.1.ACasea1+2a3≠0. Using λ2=b1/a1+2a3, with a1+2a3≠0, eliminated is Π4, and then G2=a1Π1+a3Π3+Π5. Applying the adjoint operator to Π5,G2, we get(16)G3=Adexpλ3Π5G2=a1Π1+a3Π3+1−2λ3a1+a3Π5.1.1.A.A1Casea1+a3≠0. Using λ3=1/2a1+a3, with a1+a3≠0, in (16), Π5 is eliminated, therefore G3=a1Π1+a3Π3. Then, we have the first element of the optimal system.(17)G3=a1Π1+a3Π3,witha1≠0, a1+2a3≠0, and a1+a3≠0. This is how the first reduction of the generic element (14) ends.1.1.A.A2Casea1+a3=0. We get G3=−a3Π1+a3Π3+Π5. Then, we have the other element of the optimal system.(18)G3=−a3Π1+a3Π3+Π5,witha3≠0. This is how the other reduction of the generic element (14) ends.1.1.BCasea1+2a3=0. We get G2=−2a3Π1+a3Π3+b1Π4+Π5. Applying the adjoint operator to Π5,G2, we have(19)G19=Adexpλ19Π5G2=−2a3Π1+a3Π3+b1Π4+1+4a3λ19Π5.1.1.B.1Casea3≠0. Using λ19=−1/4a3, with a3≠0, in (19), Π5 is eliminated, therefore G19=−2a3Π1+a3Π3+b1Π4. Then, we have the other element of the optimal system.(20)G19=−2a3Π1+a3Π3+b1Π4,witha1≠0, a1+2a3≠0, and a1+a3≠0. This is how the other reduction of the generic element (14) ends.1.1.B.2Casea3=0. We get G19=b1Π4+Π5. Then, we have the other element of the optimal system.(21)G19=b1Π4+Π5.This is how the other reduction of the generic element (14) ends.(1.2) Casea1=0. We get G1=a2Π2+a3Π3+a4−2λ1Π4+Π5, using λ1=a4/2, then Π4 is eliminated, and then G1=a2Π2+a3Π3+Π5. Now, applying the adjoint operator to Π4,G1, we have G16=Adexpλ16Π4G1=a2Π2+a3Π3−2a3λ16Π4+Π5. It is clear that we do not have any reduction.(1.2.1) Casea3≠0. Then, using λ16=−b9/2a3, with a3≠0, we get G16=a2Π2+a3Π3+b9Π4+Π5. Applying the adjoint operator to Π5,G16, we have(22)G17=Adexpλ17Π5G16=a2Π2+a3Π3+b9+2a2λ17Π4+1−2a3λ17Π5.Usingλ17=1/2a3, with a3≠0, in (22), Π5 is eliminated, therefore G17=a2Π2+a3Π3+b9+a2/a3Π4. Then, we have the other element of the optimal system.(23)G17=a2Π2+a3Π3+b9+a2a3Π4,witha3≠0.(1.2.2) Casea3=0. We get G16=a2Π2+Π5. Applying the adjoint operator to Π5,G16, we have(24)G18=Adexpλ18Π5G16=a2Π2+2a2λ18Π4+Π5.It is clear that we do not have any reduction.1.2.2.ACasea2≠0. Then, using λ18=−b10/2a2, with a2≠0, in (24), we get G18=a2Π2+b10Π4+Π5. Then, we have the other element of the optimal system.(25)G18=a2Π2+b10Π4+Π5,witha2≠0.1.2.2.BCasea2=0. Then, we get G18=Π5, hence we have the other element of the optimal system.(26)G18=Π5.(2)
Assuminga5=0 and a4=1 in (14), we have that G=a1Π1+a2Π2+a3Π3+Π4. Applying the adjoint operator to Π1,G and Π3,G, we do not have any reduction; on the other hand, applying the adjoint operator to Π2,G, we get(27)G4=Adexpλ4Π2G=a1Π1+a2+a1λ4Π2+a3Π3+Π4.(2.1) Casea1≠0. Using λ4=−a2/a1 with a1≠0, in (27), Π2 is eliminated, therefore G4=a1Π1+a3Π3+Π4. Now, applying the adjoint operator to Π4,G4, we get G5=Adexpλ5Π4G4=a1Π1+a3Π3+1−λ5a1+2a3Π4.2.1.ACasea1+2a3≠0. Using λ5=1/a1+2a3, with a1+2a3≠0, eliminated is Π4, then G5=a1Π1+a3Π3. Applying the adjoint operator to Π5,G5, we get(28)G6=Adexpλ6Π5G5=a1Π1+a3Π3−2λ6a1+a3Π5.It is clear that we do not have any reduction.2.1.A.A1Casea1+a3≠0. Then, substituting λ6=−b2/2a1+a3 with a1+a3≠0, we have the other element of the optimal system(29)G6=a1Π1+a3Π3+b2Π5,witha1≠0, a1+2a3≠0, and a1+a3≠0. This is how the other reduction of the generic element (14) ends.2.1.A.A2Casea1+a3=0. We get G6=−a3Π1+a3Π3, and then we have the other element of the optimal system,(30)G6=−a3Π1+a3Π3,witha1,a3≠0 and a1+2a3≠0, a1=−a3. This is how the other reduction of the generic element (14) ends.2.1.BCasea1+2a3=0. We get G5=−2a3Π1+a3Π3+Π4. Applying the adjoint operator to Π5,G5, we have(31)G23=Adexpλ23Π5G5=−2a3Π1+a3Π3+Π4+2a3λ23Π5.It is clear that we do not have any reduction; it is also clear thata1≠0 and then a3≠0; then, substituting λ23=b13, we have the other element of the optimal system(32)G23=−2a3Π1+a3Π3+Π4+2a3b13Π5,witha3≠0ya1+2a3=0. This is how the other reduction of the generic element (14) ends.(2.2) Casea1=0. We get G4=a2Π2+a3Π3+Π4. Now, applying the adjoint operator to Π4,G4, we have G20=Adexpλ20Π4G4=a2Π2+a3Π3+1−2a3λ20Π4.2.2.ACasea3≠0. Using λ20=1/2a3, with a3≠0, Π4 is eliminated, then G20=a2Π2+a3Π3. Applying the adjoint operator to Π5,G20, we get(33)G21=Adexpλ21Π5G20=a2Π2+a3Π3+2a2λ21Π4−2a3λ21Π5.It is clear that we do not have any reduction.2.2.A.1Casea2≠0. Then, substituting λ21=+b11/2a2 with a2≠0, we have other element of the optimal system(34)G21=a2Π2+a3Π3+b11Π4−a3b11a2Π5,witha2,a3≠0. This is how the other reduction of the generic element (14) ends.2.2.A.2Casea2=0. We get G21=a3Π3−2a3λ21Π5; we do not have any reduction; then, using λ21=b12, we have the other element of the optimal system(35)G21=a3Π3−2a3b12Π5,witha2≠0.This is how the other reduction of the generic element (14) ends.2.2.BCasea3=0. We get G20=a2Π2+Π4. Applying the adjoint operator to Π5,G20, we get(36)G22=Adexpλ22Π5G20=a2Π2+1−2a2λ22Π4.2.2.B.1Casea2≠0. Using λ22=1/2a2 with a2≠0, Π4 is eliminated, then we have the other element of the optimal system(37)G22=a2Π2,witha2≠0.This is how the other reduction of the generic element (14) ends.2.2.B.2Casea2=0. We get G22=Π4, and then we have the other element of the optimal system(38)G22=Π4.This is how the other reduction of the generic element (14) ends.(3)
Following a procedure analogous to the previous one and analyzing the respective cases fora4=a5=0,a3=1 in (14); a3=a4=a5=0,a2=1 in (14) and a2=a3=a4=a5=0,a1=1 in (14); we can reduce and obtain all the elements presented for the optimal system.


## 4. Invariant Solutions by Some Generators of the Optimal System

In this section, we characterize invariant solutions taking into account all operators that generate the optimal system presented in Proposition2. For this purpose, we use the method of invariant curve condition [23] (presented in Section 4.3), which is given by the following equation:(39)Qx,y,yx=η−yxξ=0.Using the elementΠ5 from Proposition 2, under the condition (39), we obtain that Q=η5−yxξ5=0, which implies −2x2y+x3y2−yxx3=0; then, solving this ODE, we have yx=1/xc1x+1, where c is an arbitrary constant, which is an invariant solution for (3); using an analogous procedure with all of the elements of the optimal system (Proposition 2), we obtain both implicit and explicit invariant solutions that are shown in Table 3, with c being a constant.Table 3
Solutions for (3) using invariant curve condition.
ElementsQx,y,yx=0SolutionsType solution1Π5−2x2y+x3y2−yxx3=0yx=1/xcx+1Explicit2Π2y2−yx0=0yx=0Trivial3Π4x2y2−yx0=0yx=0Trivial4Π1+Π3y+xy2−yx2x=0−3x/c1+x3/2Explicit5Π4+Π5x3y2+x2y2−2x2y−yxx3=0yx=2/cx2+2x+1Explicit6−Π1+Π3xy2−y−yx0=0xy2−y=0.Implicit7Π3−2Π5−2x3y2+4x2y+xy2−yx−2x3−x=0yx=1/cx2+x+c.Explicit8Π2+Π4y21+x2−yx0=0yx=0Trivial9−Π1+Π3+Π5x3y2+xy2−2x2y−y−yxx3=0yx=e1/2x2/xcx+e1/2x2Explicit10Π2+Π3+2Π4xy2+2x2y2+y2−yx−x=0yx=1/c+x2+x+logxExplicit11Π2+Π4+Π5y2+x2y2+x3y2−2x2y−yxx3=0yx=4x2/cx4+4x3+2x2+1Explicit12Π1+Π3+Π5x3y2+xy2−2x2y+y−yx−2x+x3=0yx=1/c2−x23/4x+xExplicit13−Π1+Π3+Π4x2y2+xy2−y−yx0=0x2y+xy−1=0,yx≠0Implicit14−2Π1+Π3+Π5x3y2+xy2−2x2y−2y−yxx+x3=0yx=1/xcx+1Explicit15Π1+Π4+Π5x3y2+x2y2−2x2y+y−yxx3−x=0yx=1/cx1−x2+xx+1Explicit16Π1+Π2+Π5x3y2−2x2y+y2+y−yxx3−x=0yx=1/c1−x2x+2x2+x−1Explicit17−2Π1+Π3+Π4+Π5x3y2+x2y2+xy2−2x2y−y−yxx+2x3=0yx=−4/xc−log2x2+1+4logx+22tan−12xExplicit18Π2+Π3+Π4−Π5−x3y2+x2y2+xy2+y2+2x2y−yx−x−x3=0yx=2/2c1x2+2c1+2x2+1logx−x2+1logx2+1+2xExplicit19Π1+Π3+Π4+Π5x3y2+x2y2+xy2−2x2y+y−yxx3−2x=0yx=3/cx2−x23/4+xx+3Explicit20−2Π1+Π3+Π4+Π5x3y2+x2y2+xy2−2x2y−2y−yxx3+x=0yx=2/xcx+xlogx2+1−2xlogx+2Explicit


## 5. Variational Symmetries and Conserved Quantities

In this section, we present the variational symmetries of (3) and we are going to use them to define conservation laws via Noether’s theorem [26]. First of all, we are going to determine the Lagrangian using the Jacobi last multiplier method, presented by Nucci in [27], and for this reason, we are urged to calculate the inverse of the determinant Δ,(40)Δ=xyxyxxΠ2,xΠ2,yΠ21Π4,xΠ4,yΠ41=xyxyxx0y22yyx0x2y22xy2+2x2yyx,where Π2,x,Π2,y,Π4,x, and Π4,y are the components of the symmetries Π2,Π4 shown in the Proposition 4 and Π21,Π41 are its first prolongations. Then, we get Δ=2xy4 which implies that M=1/Δ=x−1y−4/2. Now, from [27], we know that M can also be written as M=Lyxyx which means that Lyxyx=x−1y−4/2; then, integrating twice with respect to yx, we obtain the Lagrangian,(41)Lx,y,yx=yx24xy4+yxf1x,y+f2x,y,where f1,f2 are arbitrary functions. From the preceding expression, we can consider f1=f2=0. It is possible to find more Lagrangians for (3) by considering other vector fields given in the Proposition 4. We then calculate(42)ξx,yLx+ξxx,yL+ηx,yLy+ηxx,yLyx=Dxfx,y,using (41) and (8). Thus, we get(43)ξ−yx2x−2y−44+ξxyx2x−1y−44+η−x−1y−5yx2+ηx+ηy−ξxyx−ξyyx2x−1y−4yx2−fx−yxfy=0.From the preceding expression, rearranging and associating terms with respect to1,yx,yx2 and yx3, we obtain the following determinant equations:(44a)ξy=fx=0,(44b)x−1y−42ηx−fy=0,(44c)−yξ−4xη−2xyξx+2xyηy=0.Solving the preceding system forξ,η and f, we obtain the infinitesimal generators of Noether’s symmetries(45)η=−2a1yx2−a2y2x2+a4y2+a5y,ξ=−4a5x25−8a1x49,fy=a1y−2+a2y−1+a3.with a1,a2,a3,a4, and a5 arbitrary constants. Then, the Noether symmetry group or variational symmetries are(46)V1=−8x49∂∂x−2x2y∂∂y,V2=−x2y2∂∂y,V3=y2∂∂y,V4=−4x25∂∂x+y∂∂y.Remark 1.
Note thatV2=−Π4 and V3=Π2, thus the symmetries of equation (3) have two variational symmetries. According to [28], in order to obtain the conserved quantities or conservation laws, we should solve(47)I=Xyx−YLyx−XL+f,using (41), (45), and (46). Therefore, the conserved quantities are given by(48)I1=−2x3y−4yx29+xy−3yx+a1y−2+a2y−1+a3,I2=xy−2yx2+a1y−2+a2y−1+a3,I3=−x−1y−1yx2+a1y−2+a2y−1+a3,I4=−xy−4yx25−x−1y−3yx2+a1y−2+a2y−1+a3.


## 6. Nonlinear Self-Adjointness

In this section, we present the main definitions in the N. Ibragimov’s approach to nonlinear self-adjointness of differential equations adopted to our specific case. For further details, the interested reader is directed to [29–31].Consider second-order differential equation(49)Fx,y,y1,y2,…,ys=0,with independent variables x and a dependent variable y, where y1,y2,…,ys denote the collection of 1,2,…,s−th order derivatives of y.Definition 1.
LetF be a differential function and ν=νx be the new dependent variable, known as the adjoint variable or nonlocal variable [31]. The formal Lagrangian for F=0 is the differential function defined by(50)L≔νF.Definition 2.
LetF be a differential function and for the differential equation (49), denoted by Fy=0, we define the adjoint differential function to F by(51)F∗≔δLδy,and the adjoint differential equation by(52)F∗y,ν=0,where the Euler operator(53)δδy=∂∂y+∑m=1∞−1mDxi,…,Dxi,m,∂∂yxi1xi2⋯xim,and Dxi is the total derivative operator with respect to xi defined by(54)Dxi=∂xi+yxi∂y+yxixj∂yxj+⋯+yxixi1xi2,…,xin∂yxi1xi2⋯xin…Definition 3.
The differential equation (40) is said to be nonlinearly self-adjoint if there exists a substitution(55)ν=ϕx,y≠0,such that(56)F˜∗|ν=ϕx,y=λF,for some undetermined coefficients λ=λx,y,…. If ν=ϕy in the two previous expressions, equation (49) is called quasi-self-adjoint. If ν=y, we say that equation (49) is strictly self-adjoint.
Now, we shall obtain the adjoint equation to equation (3). For this purpose, we write (3) in the form (49), where(57)F≔yxx−2yx2y−1−x−1yx−x−1y2=0.
Then, the corresponding formal Lagrangian (50) is given by(58)L≔νyxx−2yx2y−1−x−1yx−x−1y2=0,and the Euler operator (53) assumes the following form:(59)δLδy=∂L∂y−Dx∂L∂yx+Dx2∂L∂yxx.
We calculate explicitly the Euler operator previously applied toL determined by (58). In this way, we obtain the adjoint equation (52) to (3):(60)F∗=−ν−2yx2y−2+4yxxy−1−2x−1y−x−2+νx4yxy−1+x−1+νxx=0.
The main result in this section can be stated as follows.Proposition 3.
Equation (3) is nonlinearly self-adjoint, with the substitution given by(61)ϕx,y=y−2k2x1/21+5+k1x1/21−5,where k1,k2 are the arbitrary constants.Proof.
Substituting in (60), and then in (56), ν=ϕx,y and its respective derivatives, and comparing the corresponding coefficients, we get the following five equations:(62a)−ϕy=λ,(62b)4y−1ϕ+2ϕy=0,(62c)4y−1ϕx+2ϕxy=0,(62d)−2y−2ϕ+2y−1ϕy+ϕyy=0,(62e)ϕ−2x−1y−x−2+ϕxx−x−1y2ϕy=0.
We observe that equation (62c) is obtained from equation (62b) by differentiation with respect to x. Therefore, we have to study only equations (62b), (62d), and (62e). Solving for ϕ in (62b), we obtain(63)ϕx,y=c1xy−2,where c1x is the arbitrary function. Using (63) in (62e), we get −c1x/x2+c1,xxx=0, thus solving for c1x, we have c1x=k2x1/21+5+k1x1/21−5; then, substituting in (63), the statement in the theorem is obtained.


## 7. Conservation Laws

In this section, we shall establish some conservation laws for equation (3) using the conservation theorem of N [31]. Since equation (3) is of second order, the formal Lagrangian contains derivatives up to order two. Thus, the general formula in [31] for the component of the conserved vector is reduced to(64)Cx=Wj∂L∂yx−Dx∂L∂yxx+DxWj∂L∂yxx,where(65)Wj=ηj−ξjyx,where j=1,…,5 is the formal Lagrangian (58),(66)L≔νyxx−2yx2y−1−x−1yx−x−1y2,and ηj,ξj are the infinitesimals of a Lie point symmetry admitted by equation (3), given in (4). Using (3), (4), and (3) in (64), we obtain the following conservation vectors for each symmetry stated in (4).(67)C1x=νyx−2+yx2−4xy−1+2y−1−x−1y+y2−νxy+xyx,C2x=νyx−2y−2x−1y2−νxy2,C3x=νyx−2xy+yx2−2xy−1−1+y2−νxxy2+xyx,C4x=νyx−2x2y+xy2−νxx2y2,C5x=νyx−13x2+6x3y+yx24x3y−1−2x3−6xy+3x2y2−νx−2x2y+x3y2−x3yx,where(68)ν=y−2k2x1/21+5+k1x1/21−5,νx=y−2k2x1/21+5−121+5+k1x1/21−5−121−5.


## 8. Classification of Lie Algebra

Generically, a finite-dimensional Lie algebra in a field of characteristic 0 is classified by the Levi’s theorem, which states that any finite-dimensional Lie algebra can be written as a semidirect product of a semisimple Lie algebra and a solvable Lie algebra; the solvable Lie algebra is the radical of that algebra. In other words, there exist two important classes of Lie algebras, the solvable and the semisimple. In each class mentioned above, there are some particular classes that have other classifications, for example, in the solvable one, we have the nilpotent Lie algebra.According the Lie group symmetry of generators given in Table1, we have a five-dimensional Lie algebra. First of all, we remember some basic criteria to classify a Lie algebra. In the case of solvable and semisimple Lie algebra, we will denote K⋅,⋅ as the Cartan-Killing form. The next propositions can be found in [32].Proposition 4.
(Cartan’s theorem). A Lie algebra is semisimple if and only if its Cartan-Killing form is nondegenerate.Proposition 5.
A Lie subalgebrag is solvable if and only if KX,Y=0 for all X∈g,g and Y∈g. Another way to write that is Kg,g,g=0.
We also need the next statements to make the classification.Definition 4.
Letg be a finite-dimensional Lie algebra over an arbitrary field k. Choose a basis ej, 1≤i≤n, in g where n=dimg and set ei,ej=Cijkek. Then, the coefficients Cijk are called structure constants.Proposition 6.
Letg1 and g2 be two Lie algebras of dimension n. Suppose each has a basis with respect to which the structure constants are the same. Then, g1 and g2 are isomorphic.
Letg be the Lie algebra related to the symmetry group of infinitesimal generators of equation (1) as stated by the table of the commutators; it is enough to consider the next relations: Π1,Π2=Π2, Π1,Π4=−Π4, Π1,Π5=−2Π5, Π2,Π5=2Π4, Π3,Π4=−2Π4, and Π3,Π5=−2Π5. Using that, we calculate Cartan-Killing form K as follows:(69)K=6060000000608000000000000,where the determinant vanishes, and thus by Cartan criterion, it is not semisimple (see Proposition 4). Since a nilpotent Lie algebra has a Cartan-Killing form, that is, identically zero, we conclude, using the counter-reciprocal of the last claim, that the Lie algebra g is not nilpotent.
We verify that the Lie algebra is solvable using the Cartan criteria to solvability (Proposition5), and then we have a solvable non-nilpotent Lie algebra. The nilradical of the Lie algebra g, M is generated by Π2,Π4,Π5, and it is isomorphic to h3, the Heisenberg Lie algebra, and so we have a solvable Lie algebra with three-dimensional nilradical. Let m be the dimension of the nilradical M of a solvable Lie algebra. In this case, in fifth-dimensional Lie algebra, we have that 3≤m≤5.
Mubarakzyanov in [33] classified the 5-dimensional solvable non-nilpotent Lie algebras, in particular the solvable non-nilpotent Lie algebra with three-dimensional nilradical. Then, by Proposition 6 and consequently, we establish an isomorphism of Lie algebras with g and the Lie algebra g5,35. In summary, we have the next proposition.Proposition 7.
The 5-dimensional Lie algebrag related to the symmetry group of equation (1) is a solvable non-nilpotent Lie algebra with three-dimensional nilradical; this nilradical is isomorphic to h3, the Heisenberg Lie algebra. Besides that, Lie algebra is isomorphic with g5,35 in the Mubarakzyanov’s classification.


## 9. Conclusion

For a generalized Levinson–Smith equation (3), we obtained the optimal system’s generating operators (see Proposition 2); using those operators, it was possible to characterize all invariant solutions as it is shown in Table 3; these invariant solutions do not appear in the literature known until today.It has been shown in the variational symmetries for (3), as it was shown in (46) with its corresponding conservation laws (48) and all these were using Noether’s theorem, but nontrivial conservation laws were also calculated using the Ibragimov’s method as it is shown in (67) using the nonlinearly self-adjoint of equation (3) as mentioned in Proposition 3.The results obtained in this study are new, and according to the phenomena that govern this equation, which reaches several fields of science, for instance, the nonlinear oscillators, it may be of significant importance for several researchers. Therefore, the goal initially proposed was achieved.The Lie algebra associated to equation (3) is a solvable non-nilpotent Lie algebra with three-dimensional nilradical, and it is isomorphic with g5,35 in the Mubarakzyanov’s classification; therefore, the goal initially proposed was achieved.For future works, equivalence group theory could be also considered to obtain preliminary classifications associated to a complete classification of (3).



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