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

Loaiza, G.; Acevedo, Y.; Duque, O.M.L.; García Hernández, Danilo A.

doi:https://doi.org/10.1155/2021/6628243

International Journal of Differential Equations

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Research Article | Open Access

Volume 2021 | Article ID 6628243 | https://doi.org/10.1155/2021/6628243

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

G. Loaiza,¹Y. Acevedo,¹O.M.L. Duque,¹and Danilo A. García Hernández²

Academic Editor: Peiguang Wang

Received03 Jan 2021

Revised02 Apr 2021

Accepted22 Apr 2021

Published07 May 2021

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.

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:where are the arbitrary functions, for this equation presents the transformation which reduces this equation to a system of two first-order equations. Equation (1) can be written aswhere and . This means that the coefficient of friction, i.e., is a function that depends on , and , and it will almost always be a nonlinear function, and on the other hand, function , 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,is presented; note that equation (3) is a particular case of equation (1), with and ; in [8], Cantwell states that equation (3) has a group of symmetries 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 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:

Proof. A general form of the one-parameter Lie group admitted by (3) is given bywhere is the group parameter. The vector field associated with the group of transformations shown above can be written as where are differentiable functions in . Applying its second prolongation,to equation (3), we must find the infinitesimals , satisfying the symmetry condition,associated with (3). Here, are the coefficients in given byBeing as the total derivative operator, . Replacing (8) into (7) and using (3), we obtainFrom (9), canceling the coefficients of the monomial variables in derivatives , and , we obtain the determining equations for the symmetry group of (3). That is,Solving the system of equations (9a)–(9d) for and , we getThus, the infinitesimal generators of the group of symmetries of (3) are the operators 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 operatorwhere , with , and are the corresponding coefficients of the infinitesimal operators . 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

Making use of this operator, we can construct Table 2, which shows the adjoint representation for each .

Proposition 2. The optimal system associated to equation (3) is given by the vector fields

Proof. To calculate the optimal system, we start with the generators of symmetries (4) and a generic nonzero vector. LetThe objective is to simplify as many coefficients, , as possible, through maps adjoint to , using Table 2.(1)Assuming in (14), we have that . Applying the adjoint operator to and , we do not have any reduction; on the other hand, applying the adjoint operator to , we get (1.1) Case. Using , in (15), is eliminated, therefore , where . Now, applying the adjoint operator to , we get . Case. Using , with , eliminated is , and then . Applying the adjoint operator to , we get Case. Using , with , in (16), is eliminated, therefore . Then, we have the first element of the optimal system. with , , and . This is how the first reduction of the generic element (14) ends. Case. We get . Then, we have the other element of the optimal system. with . This is how the other reduction of the generic element (14) ends. Case. We get . Applying the adjoint operator to , we have Case. Using , with , in (19), is eliminated, therefore . Then, we have the other element of the optimal system. with , , and . This is how the other reduction of the generic element (14) ends. Case. We get . Then, we have the other element of the optimal system. This is how the other reduction of the generic element (14) ends. (1.2) Case. We get , using , then is eliminated, and then . Now, applying the adjoint operator to , we have . It is clear that we do not have any reduction. (1.2.1) Case. Then, using , with , we get . Applying the adjoint operator to , we have Using , with , in (22), is eliminated, therefore . Then, we have the other element of the optimal system. (1.2.2) Case. We get . Applying the adjoint operator to , we have It is clear that we do not have any reduction. Case. Then, using , with , in (24), we get . Then, we have the other element of the optimal system. Case. Then, we get , hence we have the other element of the optimal system.(2)Assuming and in (14), we have that . Applying the adjoint operator to and , we do not have any reduction; on the other hand, applying the adjoint operator to , we get (2.1) Case. Using with , in (27), is eliminated, therefore . Now, applying the adjoint operator to , we get . Case. Using , with , eliminated is , then . Applying the adjoint operator to , we get It is clear that we do not have any reduction. Case. Then, substituting with , we have the other element of the optimal system with , , and . This is how the other reduction of the generic element (14) ends. Case. We get , and then we have the other element of the optimal system, with and , . This is how the other reduction of the generic element (14) ends. Case. We get . Applying the adjoint operator to , we have It is clear that we do not have any reduction; it is also clear that and then ; then, substituting , we have the other element of the optimal system with y. This is how the other reduction of the generic element (14) ends. (2.2) Case. We get . Now, applying the adjoint operator to , we have . Case. Using , with , is eliminated, then . Applying the adjoint operator to , we get It is clear that we do not have any reduction. Case. Then, substituting with , we have other element of the optimal system with . This is how the other reduction of the generic element (14) ends. Case. We get ; we do not have any reduction; then, using , we have the other element of the optimal system This is how the other reduction of the generic element (14) ends. Case. We get . Applying the adjoint operator to , we get Case. Using with , is eliminated, then we have the other element of the optimal system This is how the other reduction of the generic element (14) ends. Case. We get , and then we have the other element of the optimal system 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 for in (14); in (14) and 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 Proposition 2. For this purpose, we use the method of invariant curve condition [23] (presented in Section 4.3), which is given by the following equation:

Using the element from Proposition 2, under the condition (39), we obtain that , which implies ; then, solving this ODE, we have , where 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 being a constant.

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 ,where , and are the components of the symmetries shown in the Proposition 4 and are its first prolongations. Then, we get which implies that . Now, from [27], we know that can also be written as which means that ; then, integrating twice with respect to , we obtain the Lagrangian,where are arbitrary functions. From the preceding expression, we can consider . It is possible to find more Lagrangians for (3) by considering other vector fields given in the Proposition 4. We then calculateusing (41) and (8). Thus, we get

From the preceding expression, rearranging and associating terms with respect to and , we obtain the following determinant equations:

Solving the preceding system for and , we obtain the infinitesimal generators of Noether’s symmetrieswith , and arbitrary constants. Then, the Noether symmetry group or variational symmetries are

Remark 1. Note that and , 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 solveusing (41), (45), and (46). Therefore, the conserved quantities are given by

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 equationwith independent variables and a dependent variable , where denote the collection of th order derivatives of .

Definition 1. Let be a differential function and be the new dependent variable, known as the adjoint variable or nonlocal variable [31]. The formal Lagrangian for is the differential function defined by

Definition 2. Let be a differential function and for the differential equation (49), denoted by , we define the adjoint differential function to byand the adjoint differential equation bywhere the Euler operatorand is the total derivative operator with respect to defined by

Definition 3. The differential equation (40) is said to be nonlinearly self-adjoint if there exists a substitutionsuch thatfor some undetermined coefficients . If in the two previous expressions, equation (49) is called quasi-self-adjoint. If , 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), whereThen, the corresponding formal Lagrangian (50) is given byand the Euler operator (53) assumes the following form:We calculate explicitly the Euler operator previously applied to determined by (58). In this way, we obtain the adjoint equation (52) to (3):The main result in this section can be stated as follows.

Proposition 3. Equation (3) is nonlinearly self-adjoint, with the substitution given bywhere are the arbitrary constants.

Proof. Substituting in (60), and then in (56), and its respective derivatives, and comparing the corresponding coefficients, we get the following five equations:We observe that equation (62c) is obtained from equation (62b) by differentiation with respect to . Therefore, we have to study only equations (62b), (62d), and (62e). Solving for in (62b), we obtainwhere is the arbitrary function. Using (63) in (62e), we get , thus solving for , we have ; 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 towherewhere is the formal Lagrangian (58),and 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).where

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 Table 1, 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 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 subalgebra is solvable if and only if for all and . Another way to write that is .
We also need the next statements to make the classification.

Definition 4. Let be a finite-dimensional Lie algebra over an arbitrary field . Choose a basis , , in where and set . Then, the coefficients are called structure constants.

Proposition 6. Let and be two Lie algebras of dimension . Suppose each has a basis with respect to which the structure constants are the same. Then, and are isomorphic.
Let 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: , , , , , and . Using that, we calculate Cartan-Killing form as follows: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 is not nilpotent.
We verify that the Lie algebra is solvable using the Cartan criteria to solvability (Proposition 5), and then we have a solvable non-nilpotent Lie algebra. The nilradical of the Lie algebra , is generated by , and it is isomorphic to , the Heisenberg Lie algebra, and so we have a solvable Lie algebra with three-dimensional nilradical. Let be the dimension of the nilradical of a solvable Lie algebra. In this case, in fifth-dimensional Lie algebra, we have that .
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 and the Lie algebra . In summary, we have the next proposition.

Proposition 7. The 5-dimensional Lie algebra related to the symmetry group of equation (1) is a solvable non-nilpotent Lie algebra with three-dimensional nilradical; this nilradical is isomorphic to , the Heisenberg Lie algebra. Besides that, Lie algebra is isomorphic with 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 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).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The author Danilo A. García Hernández acknowledges the financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brasil (CAPES), under the finance code 001. In the same way, the EAFIT University is thanked for all the financial supports (Scholarship for Master’s Student). Moreover G. Loaiza and Y. Acevedo also thank the Project of MinScience “Sobre procesos de difusión y simplificación de información” (code 121671250122).

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Copyright © 2021 G. Loaiza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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