Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira. ISSN 0122-1701 y ISSN: 2344-7214
93
Lie algebra classification, conservation laws and
invariant solutions for the a particular case of the
generalized Levinson-Smith equation
Clasificación del álgebra de Lie, leyes de conservación y soluciones invariantes para un
caso particular de la ecuación generalizada e Levinson-Smith
G. I. Loaiza-Ossa ; Y. A. Acevedo-Agudelo ; O. M. Londoño-Duque ; D. A. García-Hernández
DOI: https://doi.org/10.22517/23447214.24960
Scientific and technological research paper
Abstract—In this study, we examine a specific instance of the
generalized Levinson-Smith equation, which is linked to the
Liènard equation and holds significant importance from the
perspectives of physics, mathematics, and engineering. This
underlying equation has practical applications in mechanics and
nonlinear dynamics and has been extensively explored in the
qualitative scheme. Our approach involves applying the Lie group
method to this equation. By doing so, we derive the optimal
generating operators for the system that pertain to the specific
instance of the generalized Levinson-Smith equation. These
operators are then used to define all invariant solutions associated
with the equation. In addition, we demonstrate the variational
symmetries and corresponding conservation laws using Noether's
theorem. Finally, we categorize the Lie algebra related to the given
equation.
Index Terms— Conservation laws; Invariant solutions; Lie
algebra classification; Lie symmetry group; Noether’s theorem;
Optimal system; Variational symmetries.
Resumen—En este estudio, examinamos una instancia
específica de la ecuación generalizada de Levinson-Smith, que está
vinculada con la ecuación de Liènard y tiene una gran importancia
desde las perspectivas de la física, las matemáticas y la ingeniería.
Esta ecuación subyacente tiene aplicaciones prácticas en mecánica
y dinámica no lineal, y ha sido ampliamente explorada en el
esquema cualitativo. Nuestro enfoque implica aplicar el método de
grupos de Lie a esta ecuación. Al hacerlo, obtenemos los
operadores generadores óptimos del sistema que se refieren a la
instancia específica de la ecuación generalizada de Levinson-
Smith. Luego, se utilizan estos operadores para definir todas las
soluciones invariantes asociadas con la ecuación. Además,
demostramos las simetrías variacionales y las leyes de
conservación correspondientes utilizando el teorema de Noether.
Finalmente, categorizamos el álgebra de Lie relacionada con la
ecuación dada.
This manuscript was submitted on October 20, 2021, accepted on March 09,
2023 and published on June 30, 2023.
This work was supported by the EAFIT University, Colombia, with financial
support in the project: Study and applications of diffusion processes of
importance in health and computation, with code 11740052022.
G. I. Loaiza-Ossa. Full-time professor at Universidad EAFIT, Colombia, e-
mail: gloaiza@eafit.edu.co.
Palabras claves— Clasificación del Álgebra de Lie; Grupo de
simetrías de Lie; Leyes de Conservación; Simetrías Variacionales;
Sistema óptimo; Soluciones invariantes; Teorema de Noether.
I.
INTRODUCTION
HE
Lie
group
symmetry
method
is
a
powerful
mathematical tool employed to investigate a wide range of
differential equations, including ODEs, PDEs, FPDEs, and
FODEs. This mathematical theory was first introduced in the
19th century by Sophus Lie, [1], following the principles of
Galois Theory in algebra. The Lie group method has attracted
considerable interest among researchers in various branches of
science, such as mathematics, theoretical and applied physics,
due to the physical interpretations that can be derived from the
underlying equations being studied. As a result, this method
enables the construction of conservation laws, utilizing the
celebrated Noether's theorem [2], and similarity solutions,
which are not achievable using traditional methods, particularly
when utilizing Ibragimov's approach [3].
Furthermore, this method has contributed to establishing
frameworks and the efficacy of certain numerical methods,
leading to the development of numerous software packages
across various computational environments, as exemplified by
[4, 5]. In general, due to the importance of studying equations
such as ODEs, PDEs, and others, the Lie group method is of
interest to a diverse range of scientists. The literature contains
a vast array of references on the Lie group method, including
[6, 7, 8, 9]. Recently, the Lie group method has been employed
to solve and analyze various problems in numerous scientific
fields. For example, in [10], a model with applications in
quantum field and differential geometry theory was studied
using this method. Moreover, [11, 12, 13, 14, 15, 16, 17, 18]
Y. A. Acevedo-Agudelo. Master's Degree in Applied Mathematics,
Universidad EAFIT, Colombia. e-mail: yaceved2@eafit.edu.co.
O. M. Londoño-Duque. PhD of applied mathematics, University of
Campinas, Brazil. e-mail: olondon2@eafit.edu.co.
D. A. García-Hernández. PhD in Mathematics, University of Campinas,
Brazil. e-mail: d190684@dac.unicamp.br
T
94
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira
�
�
�
�
�
�
�
�
�
provide references on the latest advancements in symmetry
analysis.
The equation 𝑦
𝑥𝑥
+ 𝜙
(
𝑥, 𝑦, 𝑦
𝑥
)
𝑦
𝑥
= 𝛾
(
𝑥, 𝑦
)
, which belongs to
the class of generalized Levinson-Smith equations, is closely
connected to the second-order nonlinear differential equation of
the Liènard type [19]. These equations form the foundation for
a wide range of phenomena across various disciplines,
including mechanics, electronics, biology, seismology,
chemistry, and physics. For example, an important model used
in physical is the Van der Pol equation, which describes a non-
conservative oscillator with non-linear damping.
and engineers a better comprehension of the seismic wave
diffusion across these porous media, which is critical for natural
resource exploration and production, geotechnical engineering,
and seismic risk assessment.
II.
CONTINUOUS LIE SYMMETRY GROUPS
In this section, we analyze the Lie symmetry group of (2). The
principal outcome of this section may be formulated as follows:
Proposition 1. The set of vector fields that generate the Lie group
of symmetries for (2) is given by:
𝛱 = −
(
𝑥 + 𝑦
)
−1
𝛛
, 𝛱 = −𝑦
(
𝑥 + 𝑦
)
−1
𝛛
, 𝛱 = 𝑥
(
𝑥 + 2𝑦
)(
𝑥 + 𝑦
)
−1
𝛛
,
1
��
2
��
3
��
(4)
𝛱 = −
𝛛
+
𝛛
, and 𝛱 = 𝑥
𝛛
+ 𝑦
𝛛
.
In
their
work
[20],
Levinson
and
Smith
investigated
a
4
�
�
��
5
�
�
��
generalized equation that describes relaxation oscillations. The
study of such oscillations is fundamental to understanding a
variety of phenomena in fields ranging from electronics and
physics to biology and chemistry. By examining this equation
in depth, Levinson and Smith provided insights into the
dynamics of relaxation oscillations and their role in complex
systems.
Within this paper, we examine the subsequent differential
Proof. The general form for the generator operators of a Lie
group of a parameter admitted by (2) is:
𝑥 → 𝑥 + 𝜖𝜉
(
𝑥, 𝑦
)
+ 𝑂
(
𝜖
2
)
, and 𝑦
→ 𝑦 + 𝜖𝜂
(
𝑥, 𝑦
)
+ 𝑂
(
𝜖
2
)
,
where � is the group parameter. The vector field associated
with this group of transformations is � = �
(
�, �
)
�
+
�
(
�, �
)
�
,
�� ��
equation (1):
�
��
+ �
(
�, �, �
�
)
�
�
= �
(
�, �
)
. (1)
with 𝜉, 𝜂 differentiable functions in ℝ
2
. To find the
infinitesimals 𝜉
(
𝑥, 𝑦
)
and 𝜂
(
𝑥, 𝑦
)
, we applied the second
extension operator:
The differential equation presented in (1), indicates that the
friction coefficient, represented by the function �, is dependent
upon the variables �, �, and �
�
, and is typically a non-linear
function. Additionally, the function �, which is referred to as
�
(2)
= � +
�
[
�
]
+
�
[
��
]
�
�
, (5)
the disturbance function, is also non-linear. It's important to
note that (1) is classified as a generalization of the Levinson-
Smith equation. In, Duarte et al, introduce a particular case of
After applying (5) to (2), we obtain the following symmetry
condition:
𝜉
(
−2𝑦
2
(
𝑥 + 𝑦
)
−2
− 𝑦
(
𝑥 + 𝑦
)
−2
)
+ 𝜂
(
−2𝑦
2
(
𝑥 + 𝑦
)
−2
− 𝑦
(
𝑥 + 𝑦
)
−2
)
(1)
� � � �
𝑦
𝑥𝑥
= −2𝑦
2
(
𝑥 + 𝑦
)
−1
− 𝑦
(
𝑥 + 𝑦
)
−1
, (2)
and its solution (3):
+ 𝜂
[
𝑥
]
(
4𝑦
𝑥
(
𝑥 + 𝑦
)
−1
+
(
𝑥 + 𝑦
)
−1
)
+ 𝜂
[
𝑥𝑥
]
= 0,
(6)
where �
[�
]
and �
[��
]
are the coefficients in �
(2)
given by (see
𝑦
(
𝑥
)
= −
�
2
+ 2�
1
, where � ,
are constants. (3)
[21, 22]):
2� + 2�
2
2 1
𝜂 = 𝐷
[
𝜂
]
−
(
𝐷
[
𝜉
])
𝑦
= + − 𝜉
− 𝜉 𝑦
2
.
[
�
]
� � � � � �
In this work, they present the Lie group of symmetries of (2),
𝜂
[𝑥𝑥
]
= 𝐷
𝑥
[𝜂
[𝑥
]
] −
(
𝐷
𝑥
[
𝜉
])
𝑦
𝑥𝑥
,
using a ODEtools Maple package. thus, the proposal of our
= 𝜂
𝑥𝑥
+ (2𝜂
𝑥𝑦
− 𝜉
𝑥𝑥
)𝑦
𝑥
+ (𝜂
𝑦𝑦
− 2𝜉
𝑥𝑦
)𝑦
2
− 𝜉
�
3
work is: 𝑖) to calculate the 5 −dimensional Lie symmetry group
in all detail, ��) to present the optimal system (optimal
algebra) for (2), ���) making use of all elements of the
optimal algebra,
+(𝜂
𝑦
− 2𝜉
𝑥
)𝑦
𝑥𝑥
− 3𝜉
𝑦
𝑦
𝑥
𝑦
𝑥𝑥
. (7a)
where �
�
is the total derivative operator: �
�
= �
�
+ �
�
�
�
+
to propose invariant solutions for (2), then ��) to construct
the Lagrangian with which we could determine the
variational
�
��
�
�
�
+ ⋯.
symmetries using Noether′s theorem, and thus to present
conservation laws associated, and finally 𝑣) to classify the Lie
algebra that is associated with (2), and corresponds to the Lie
symmetry group.
To conclude this initial section, it is crucial to highlight that the
generalized Levinson-Smith equation is a non-linear partial
differential equation utilized in the elasticity theory to depict
the seismic wave propagation within porous media. This
After applying (7a) in (6) and substitute in the resulting
expression �
��
by (2), is obtained:
(2𝜉
𝑦
(
𝑥 + 𝑦
)
−1
− 𝜉
𝑦𝑦
)𝑦
3
+ (𝜂
𝑦𝑦
− 2𝜉
𝑥𝑦
− 2𝜉
(
𝑥 + 𝑦
)
−2
− 2𝜂
(
𝑥 + 𝑦
)
−2
+2𝜂
𝑦
(
𝑥 + 𝑦
)
−1
+ 3𝜉
𝑥
(
𝑥 + 𝑦
)
−1
+ 2𝜉
𝑦
(
𝑥 + 𝑦
)
−1
)𝑦
2
+(−𝜉
(
𝑥 + 𝑦
)
−2
− 𝜂
(
𝑥 + 𝑦
)
−2
+ 4𝜂
𝑥
(
𝑥 + 𝑦
)
−1
+ 𝜉
𝑥
(
𝑥 + 𝑦
)
−1
+ 2𝜂
𝑥𝑦
− 𝜉
𝑥𝑥
)𝑦
𝑥
+ 𝜂
𝑥
(
𝑥 + 𝑦
)
−1
+ 𝜂
𝑥𝑥
= 0. (7b)
Analyzing the coefficients in (7b) with respect to the
independent variables �
3
, �
2
, � , 1 we get the following system
� � �
equation serves as a model for the propagation of seismic waves
through porous media such as soils, fractured rocks, and
petroleum reservoirs. The equation's solution offers scientists
of determining equations, with
(
𝑥 + 𝑦
)
≠ 0:
�
�
�
�
�
�
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira.
95
13
2𝜉
𝑦
− 𝜉
𝑦𝑦
(
𝑥 + 𝑦
)
=
0, (8a)
(𝜂
𝑦𝑦
− 2𝜉
𝑥𝑦
)
(
𝑥 + 𝑦
)
2
− 2𝜉 − 2𝜂 + (2𝜂
𝑦
+ 3𝜉
𝑥
+ 2𝜉
𝑦
)
(
𝑥 + 𝑦
)
= 0, (8b)
−𝜉 − 𝜂 +
(
4𝜂
𝑥
+ 𝜉
𝑥
)(
𝑥 + 𝑦
)
+ (2𝜂
𝑥𝑦
− 𝜉
𝑥𝑥
)
(
𝑥 + 𝑦
)
2
= 0, (8c)
TABLE II
ADJOINT REPRESENTATION OF THE SYMMETRY GROUP OF (2).
�
�
+
�
��
(
� + �
)
= 0. (8d)
Solving the system in (8a)-(8d), for � and � we get
𝜉 = −𝑘
(
𝑥 + 𝑦
)
−1
− 𝑘 𝑦
(
𝑥 + 𝑦
)
−1
+ 𝑘 𝑥
(
𝑥 + 2𝑦
)(
𝑥 + 𝑦
)
−1
− 𝑘 + 𝑘 𝑥,
1 2 3 4 5
� = �
4
+ �
5
�. (8e)
Note that in (8e), � , � , � , � and
are arbitrary constants.
1 2 3 4 5
Thus, the generators of the group of symmetries of (2) are the
operators �
1
− 𝛱
5
described in the statement of the
Proposition 1; thus achieving the proposed result. ◻
III.
OPTIMAL ALGEBRA
Taking into account [23, 24, 25, 26], we present in this section
Proposition 2. The optimal algebra associated to (2) is given by
the vector fields
the optimal algebra associated to the symmetry group of (2),
� , � , � , �
−
𝑎
1
𝛱
+ 𝛱 , ±
√
−2𝑎
+ � � +
that shows a systematic way to classify the invariant solutions.
1 2 4 2 2
�
2
4 5
2 2 3 3
To obtain the optimal algebra, we should first calculate the
corresponding commutator table, which can be obtained from
the operator
𝛱
4
, −𝑏
12
𝛱
1
+ 𝑎
2
𝛱
2
+ 𝛱
4
, 𝑏
2
𝛱
1
− 2𝑏
13
𝛱
2
+ 𝛱
3
, 𝑎
1
𝛱
1
−
𝑏
11
𝛱
4
+ 𝛱
5
, 𝑏
4
𝛱
1
+ 𝑏
5
𝛱
2
+ 𝑎
3
𝛱
3
+ 𝛱
5
, 𝑏
9
𝛱
1
+ 𝑎
3
𝛱
3
−
�
10
�
4
+ �
5
. (9a)
[� , � ] = � − 𝛱
�
=
∑
(𝛱 (𝜉
𝑖
) − 𝛱
(
𝜉
𝑖
)
)
, (9)
Proof. To calculate the optimal algebra system, we start with
the generators of symmetries (4) and a generic nonzero vector.
� � � �
�=
1
�
�
�
��
�
Let
where 𝑖 = 1,2, with 𝛼, 𝛽 = 1, ⋯ ,5 and 𝜉
𝑖
, 𝜉
𝑖
are the
� = �
1
�
1
+ �
2
�
2
+ �
3
�
3
+ �
4
�
4
+ �
5
�
5
. (10)
�
�
corresponding
coefficients
of
the
infinitesimal
operators
�
�
, �
�
. After applying the operator (9) to the symmetry group
of (2), we obtain the operators that are shown in the table I.
The objective is to simplify as many coefficients �
�
as possible,
through maps adjoint to �, using table II.
TABLE I
CONMMUTATORS TABLE ASSOCIATED TO THE SYMMETRY
GROUP OF (2).
1.
Assuming �
5
= 1 in (10) we have that � = �
1
�
1
+ �
2
�
2
+
𝑎
3
𝛱
3
+ 𝑎
4
𝛱
4
+ 𝛱
5
. Applying the adjoint operator to
(
𝛱
5
, 𝐺
)
,
we don’t have any reduction, on the other hand applying the
adjoint operator to
(
𝛱
1
, 𝐺
)
we get
�
1
= ��(exp
(
�
1
�
1
)
)�
= (𝑎
1
− 𝜆
1
(
1 + 2𝑎
3
)
)𝛱
1
+ 𝑎
2
𝛱
2
+ 𝑎
3
𝛱
3
+ �
4
�
4
+ �
5
. (11)
1.1)
Case 1 + 2𝑎
≠ 0. Using 𝜆
=
𝑎
1
with 1 + 2𝑎
≠ 0, in
3 1
1+2�
3
3
(11), �
is eliminated, therefore �
= � � + � � + � � +
1 1 2 2 3 3 4 4
�
5
. Now, applying the adjoint operator to
(
�
2
, �
1
)
, we get �
2
=
𝐴𝑑(exp
(
𝜆
2
𝛱
2
)
)𝐺
1
= 𝑎
4
𝜆
2
𝛱
1
+ (𝑎
2
− 𝜆
2
(
1 + 2𝑎
3
)
𝛱
2
+
� � + � � + � . As 1 + 2� ≠ 0, we can use
=
�
2
,
3 3 4 4 5 3
2
1+2�
3
then is eliminated �
2
, thus �
2
= �
1
�
1
+ �
3
�
3
+ �
4
�
4
+ �
5
,
To proceed, our next task is to compute the adjoint action
representation of the symmetries for (2). In order to do so, we
with
�
1
=
�
4
�
2
.
Now,
applying
the
adjoint
operator
to
1+2�
3
will refer to table I, and make use of the relevant operator:
��(���
(
��
)
)�
�
�
(
�
3
, �
2
)
, we
get
�
3
=
��(exp
(
�
3
�
3
)
)�
2
= � �
2�
3
�
+ 2�
�
2�
3
�
+ � + � �
�
=
∑
(
)
1 1 4 2 3 3 4 4
�=0
(�� �
)
�!
� for the symmetries � and �. (9.1)
+ �
5
. (12)
It is clear that in (12), we don’t have any reduction, then we get
Making use of the operator (9.1), we can construct the table II,
which shows the adjoint representation for each �
�
.
𝐺
3
= 𝑏
2
𝛱
1
+ 𝑏
3
𝛱
2
+ 𝑎
3
𝛱
3
+ 𝑎
4
𝛱
4
+ 𝛱
5
, with 𝑏
2
= 𝑏
1
𝑒
2𝜆
3
and 𝑏
3
= 2𝑎
4
𝑒
2𝜆
3
. Now applying the adjoint operator to
(
𝛱
4
, 𝐺
3
)
, we get 𝐺
4
= 𝐴𝑑(exp
(
𝜆
4
𝛱
4
)
)𝐺
3
=
(
𝑏
2
− 𝑏
3
𝜆
4
+
� �
2
)
� +
(
� − 2𝑎 𝜆
)
𝛱 + 𝑎 +
(
𝑎
− 𝜆
)
𝛱 + 𝛱 .
∞
[ ; ]
𝜫
𝟏
𝜫
𝟐
𝜫
𝟑
𝜫
𝟒
𝜫
𝟓
𝜫
𝟏
0
0
2𝛱
1
0
𝛱
1
𝜫
𝟐
0
0
2𝛱
2
−𝛱
1
𝛱
2
𝜫
𝟑
−2𝛱
1
−2𝛱
2
0
−2𝛱
2
0
𝜫
𝟒
0
𝛱
1
2𝛱
2
0
𝛱
4
𝜫
𝟓
−𝛱
1
−𝛱
2
0
−𝛱
4
0
Adj[ , ]
𝜫
𝟏
𝜫
𝟐
𝜫
𝟑
𝜫
𝟒
𝜫
𝟓
𝜫
𝟏
𝛱
1
𝛱
2
𝛱
3
− 2𝜆𝛱
1
𝛱
4
𝛱
5
− 𝜆𝛱
1
𝜫
𝟐
𝛱
1
𝛱
2
𝛱
3
− 2𝜆𝛱
2
𝛱
4
+ 𝜆𝛱
1
𝛱
5
− 𝜆𝛱
2
𝜫
𝟑
𝑒
2𝜆
𝛱
1
𝑒
2𝜆
𝛱
2
𝛱
3
𝛱
4
+ 2𝑒
2𝜆
𝛱
2
𝛱
5
𝜫
𝟒
𝛱
1
𝛱
2
− 𝜆𝛱
1
𝛱
3
− 2𝜆𝛱
2
+ 𝜆
2
𝛱
1
𝛱
4
𝛱
5
− 𝜆𝛱
4
𝜫
𝟓
𝑒
𝜆
𝛱
1
𝑒
𝜆
𝛱
2
𝛱
3
𝑒
𝜆
𝛱
4
𝛱
5
96
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira
3 4 1 3 3 4 2 3 3 4 4 4 5
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira.
97
8
Using
�
4
= �
4
,
�
4
is
eliminated,
therefore
�
4
= �
4
�
1
+
1.2. �. 2. � ) Case ≠ 0. Using
=
�
1
, with
≠ 0, is
� + � + � ,
with = − 𝑏
1 2
+ � �
2
and �
=
8
�
2
2
5 2 3 3 5 4 2 3 4 3 4
5
eliminated � , then we have = �
−
𝑎
1
+ � . After,
𝑏
3
− 2𝑎
3
𝜆
4
. Then, we have the first element of the optimal
1
8 2 2
�
2
4 5
algebra
we have other element of the optimal algebra
𝐺
4
= 𝑏
4
𝛱
1
+ 𝑏
5
𝛱
2
+ 𝑎
3
𝛱
3
+ 𝛱
5
, with 𝑎
3
≠ 0. (13)
�
= �
�
1
−
+ � . (16)
This is how the first reduction of the generic element (10) ends.
8 2 2
�
2
4 5
1.2)
Case 1 + 2�
3
= 0. We get, �
1
= �
1
�
1
+ �
2
�
2
+ �
3
�
3
+
𝑎
4
𝛱
4
+ 𝛱
5
. Now, applying the adjoint operator to
(
𝛱
2
, 𝐺
1
)
, we
have 𝐺
5
= 𝐴𝑑(exp
(
𝜆
5
𝛱
2
)
)𝐺
1
=
(
𝑎
1
+ 𝑎
4
𝜆
5
)
𝛱
1
+ (𝑎
2
−
�
(
1 + 2�
3
)
)�
2
+ �
3
�
3
+ �
4
�
4
+ �
5
=
(
�
1
+ �
4
�
5
)
�
1
+
�
2
�
2
+ �
3
�
3
+ �
4
�
4
+ �
5
.
This is how other reduction of the generic element (10) ends.
1.2. 𝐵. 2. 𝐴
2
) Case 𝑎
2
= 0. We have 𝐺
8
= 𝑎
1
𝛱
1
− 𝜆
8
𝛱
4
+ 𝛱
5
.
Then using 𝜆
8
= 𝑏
11
, we have other element of the optimal
algebra
𝐺
8
= 𝑎
1
𝛱
1
− 𝑏
11
𝛱
4
+ 𝛱
5
. (17)
This is how other reduction of the generic element (10) ends.
1.2. 𝐴) Case 𝑎
≠ 0. Using 𝜆
=
−𝑎
1
, then 𝛱
is eliminated,
4 5
�
4
1
2.
Assuming = 0 and = 1 in (10), we have that � = � � +
then we get �
5
= �
2
�
2
+ �
3
�
3
+ �
4
�
4
+ �
5
. Now applying
5 4 1
1
the adjoint operator to
(
� , �
)
, we have �
=
�
2
�
2
+ �
3
�
3
+ �
4
. Applying the adjoint operator to
(
�
5
, �
)
3 5
2�
6
and
(
𝛱
3
, 𝐺
)
we don’t have any reduction, on the other hand
��(exp
(
�
6
�
3
)
)�
5
= �
6
(
�
2
+ 2�
4
)
�
2
+ �
3
�
3
+
�
4
�
4
+
applying the adjoint operator to
(
�
1
, �
)
we get
𝛱
5
. We don’t have any reduction, then we get 𝐺
6
= 𝑏
6
𝛱
2
+
�
3
�
3
+ �
4
�
4
+ �
5
, with �
6
= �
2�
6
(
�
2
+ �
4
)
. Now
applying
�
= ��(exp
(
� �
)
)�
9 9 1
the adjoint operator to
(
�
4
, �
6
)
, we get �
7
=
𝐴𝑑(exp
(
𝜆
7
𝛱
4
)
)𝐺
6
=
(
𝑎 𝜆
2
− 𝑏
6
𝜆
7
)
𝛱
1
+
(
𝑏
6
− 2𝑎
3
𝜆
7
)
𝛱
2
+
=
(
𝑎
1
− 2𝑎
3
𝜆
9
)
𝛱
1
+ 𝑎
2
𝛱
2
+ 𝑎
3
𝛱
3
+ � . (18)
3 7
4
𝑎
3
𝛱
3
+
(
𝑎
4
− 𝜆
7
)
𝛱
4
+ 𝛱
5
.
2.1)
Case 𝑎
≠ 0, in (18). Using 𝜆
=
𝑎
1
with 𝑎
≠ 0, in (18),
3 9
2�
3
3
Using �
7
= �
4
, is elimininated �
4
, then we get �
7
= �
7
�
1
+
𝑏
8
𝛱
2
+ 𝑎
3
𝛱
3
+ 𝛱
5
, with 𝑏
7
= 𝑎 𝜆
2
− 𝑏
6
𝜆
7
and 𝑏
8
= 𝑏
6
−
�
1
is eliminated, therefore �
9
= �
2
�
2
+ �
3
�
3
+ �
4
.
Now, applying the adjoint operator to
(
� , �
)
, we get
� =
3
7
2 9 10
2�
3
�
7
. Thus, we have other element of the optimal
algebra
𝐴𝑑(exp
(
𝜆
10
𝛱
2
)
)𝐺
9
= 𝜆
10
𝛱
1
+
(
𝑎
2
− 2𝑎
3
𝜆
10
)
𝛱
2
+ 𝑎
3
𝛱
3
+
𝛱 . As 𝑎
≠ 0, we can use 𝜆 =
𝑎
2
, then is eliminated 𝛱 ,
�
7
= �
7
�
1
+ �
8
�
2
+ �
3
�
3
+ �
5
. (14)
4 3
10
2�
3
2
thus we have �
=
�
2
+ � � + � . Now, applying the
This is how other reduction of the generic element (10) ends.
10
2�
3
1 3 3 4
adjoint operator to
(
�
4
, �
10
)
, we get �
11
=
��(exp
(
� �
)
)�
= (
�
2
+ � �
2
)
− 2𝑎 � +
1.2. �) Case �
= 0. We get, �
= � � + � � + � � +
� .
11 4 10
2�
3
3 11 1
3 11 2
4 5 1 1 2 2 3 3 5
√
−𝑎
2
Now, applying the adjoint operator to
(
𝛱
3
, 𝐺
5
)
, we don’t have
any reduction, then applying the adjoint operator to
(
� , �
)
we
𝑎
3
𝛱
3
+ 𝛱
4
. Using 𝜆
11
= ±
√
2𝑎
3
, with �
2
< 0, is eliminated �
1
,
4 5
get 𝐺
8
= 𝐴𝑑(exp
(
𝜆
8
𝛱
4
)
)𝐺
5
=
(
𝑎
1
− 𝑎
2
𝜆
8
+ 𝑎
3
𝜆
2
)
𝛱
1
+
(
𝑎
2
− 2𝑎
3
𝜆
8
)
𝛱
2
+ 𝑎
3
𝛱
3
− 𝜆
8
𝛱
4
+ 𝛱
5
.
then we get 𝐺
11
=
√
−2𝑎
2
𝛱
2
+ 𝑎
3
𝛱
3
+ 𝛱
4
, with 𝑎
2
> 0.
After, we have other element of the optimal algebra
𝐺
11
= ±
√
−2𝑎
2
𝛱
2
+ 𝑎
3
𝛱
3
+ 𝛱
4
. (19)
1.2. 𝐵. 1)
Case
𝑎
≠ 0.
Using
𝜆
=
𝑎
2
,
with
𝑎
≠ 0,
is
with �
< 0. This is how other reduction of the generic element
3 8
2�
3
3
�
2
�
2
2
(10) ends.
eliminated 𝛱
2
, we get 𝐺
8
= (𝑎
1
−
2
+
2
) 𝛱
1
+ 𝑎
3
𝛱
3
−
�
2
2�
3
4�
3
�
2
�
2
�
2
𝛱
4
+ 𝛱
5
. Then using 𝑏
9
= 𝑎
1
−
2
+
2
and 𝑏
10
=
, we
2�
3
2�
3
4�
3
2�
3
2.2)
Case �
3
= 0, in (18). We get �
9
= �
1
�
1
+ �
2
�
2
+ �
4
.
have other element of the optimal algebra
𝐺
8
= 𝑏
9
𝛱
1
+ 𝑎
3
𝛱
3
− 𝑏
10
𝛱
4
+ 𝛱
5
. (15)
This is how other reduction of the generic element (10) ends.
Now, applying the adjoint operator to
(
�
2
, �
9
)
, we have �
12
=
��(exp
(
�
12
�
2
)
)�
9
=
(
�
1
+ �
12
)
�
1
+ �
2
�
2
+ �
4
, using
𝜆
12
= −𝑎
1
, is eliminated 𝛱
1
, thus we have 𝐺
12
= 𝑎
2
𝛱
2
+ 𝛱
4
.
Now applying the adjoint operator to
(
𝛱
4
, 𝐺
12
)
, we have 𝐺
13
=
𝐴𝑑(exp
(
𝜆
13
𝛱
4
)
)𝐺
12
= −𝑎
2
𝜆
13
𝛱
1
+ 𝑎
2
𝛱
2
+ 𝛱
4
.
2.2. 𝐴) Case 𝑎
≠ 0. Using 𝜆
=
�
12
, with
≠ 0, we have
1.2. 𝐵. 2) Case 𝑎
3
= 0. We get 𝐺
8
=
(
𝑎
1
− 𝑎
2
𝜆
8
)
𝛱
1
+ 𝑎
2
𝛱
2
−
�
8
�
4
+ �
5
.
2 13
�
2
2
other element of the optimal algebra
𝐺
13
= −𝑏
12
𝛱
1
+ 𝑎
2
𝛱
2
+ 𝛱
4
. (20)
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira.
99
13
2.2. �) Case �
2
= 0. We get �
13
= �
4
. We have other
element of the optimal algebra
�
13
= �
4
. (21)
This is how other reduction of the generic element (10) ends.
IV.
INVARIANT SOLUTIONS THROUGH OPTIMAL ALGEBRA
GENERATORS
In this section, we characterize all invariant solutions taking
into account some operators that generate the optimal algebra
presented in Proposition 2. For this purpose, we use the method
of invariant curve condition [24]
(presented in section 4.3),
Assuming
�
4
=
�
5
= 0 and
�
3
= 1 in (10), we have that � =
which is given by the following equation
𝑎
1
𝛱
1
+ 𝑎
2
𝛱
2
+ 𝛱
3
. Applying the adjoint operator to
(
𝛱
3
, 𝐺
)
and
(
𝛱
5
, 𝐺
)
we don’t have any reduction, on the other hand
applying the adjoint operator to
(
𝛱
1
, 𝐺
)
we get
�
14
= ��(exp
(
�
14
�
1
)
)�
=
(
𝑎
1
− 2𝜆
14
)
𝛱
1
+ 𝑎
2
𝛱
2
+ 𝛱
3
. (22)
𝑄
(
𝑥, 𝑦, 𝑦
𝑥
)
= 𝜂 − 𝑦
𝑥
𝜉 = 0. (27𝑎)
Using the element 𝛱
1
from Proposition 2, under the condition
(27a), we obtain that 𝑄 = 𝜂
1
− 𝑦
𝑥
𝜉
1
= 0, which implies
(
0
)
−
𝑦
𝑥
(
−
(
𝑥 + 𝑦
)
−1
)
= 0, then solving this ODE we have 𝑦
(
𝑥
)
=
�, which is the trivial solution for (2), using an analogous
procedure with all of the elements of the optimal algebra
(Proposition 2), we obtain both implicit and explicit invariant
Using � =
�
1
, in (22), � is eliminated, therefore � =
solutions that are shown in the table III, with � being a constant.
14
2
1 14
�
2
�
2
+ �
3
. Now, applying the adjoint operator to
(
�
2
, �
14
)
,
we
get 𝐺
15
= 𝐴𝑑(exp
(
𝜆
15
𝛱
2
)
)𝐺
14
=
(
𝑎
2
− 2𝜆
15
)
𝛱
2
+ 𝛱
3
, using
TABLE III
� =
�
2
, is eliminated � , therefore
= � . Now, applying
15
2
2 15 3
SOLUTIONS FOR (2) USING INVARIANT CURVE CONDITION.
the adjoint operator to
(
�
4
, �
15
)
, we get �
16
=
��(exp
(
� �
)
)� = �
2
− 2𝜆 𝛱 + 𝛱 . It is clear that
16 4 15
16 1
16 2 3
we don’t have any reduction, then using 𝜆
16
= 𝑏
13
, we have
other element of the optimal algebra
𝐺
16
= 𝑏
2
𝛱
1
− 2𝑏
13
𝛱
2
+ 𝛱
3
. (23)
This is how other reduction of the generic element (10) ends.
Assuming
�
3
=
�
4
=
�
5
= 0 and
�
2
= 1 in (10), we have that
� = � � + � . Applying the adjoint operator to
(
� , �
)
,
1 1 2 1
(
𝛱 , 𝐺
)
,
(
𝛱 , 𝐺
)
and
(
𝛱 , 𝐺
)
we don’t have any reduction, on
2 3 5
the other hand, applying the adjoint operator to
(
�
4
, �
)
we get
𝐺 = 𝐴𝑑(exp
(
𝜆 𝛱
)
)𝐺 =
(
𝑎
− 𝜆
)
𝛱 + 𝛱 . (24)
17 17 4 1 17 1 2
Using �
17
= �
1
, in (24), is eliminated �
1
, therefore �
17
= �
2
.
Then, we have other element of the optimal algebra
�
17
= �
2
. (25)
This is how other reduction of the generic element (10) ends.
Assuming 𝑎
2
= 𝑎
3
= 𝑎
4
= 𝑎
5
= 0 and 𝑎
1
= 1 in (10), we have
that 𝐺 = 𝛱
1
. Applying the adjoint operator to
(
𝛱
1
, 𝐺
)
,
(
𝛱
2
, 𝐺
)
,
(
𝛱
3
, 𝐺
)
,
(
𝛱
4
, 𝐺
)
and
(
𝛱
5
, 𝐺
)
we don’t have any reduction, then
we have other element of the optimal algebra
� = �
1
. (26)
This is how other reduction of the generic element (10) ends.
Taking into account the reductions presented in (13), (14), (15),
V.
EXPLORING CONSERVED QUANTITIES AND VARIATIONAL
SYMMETRIES
In this section, we shall introduce the variational symmetries
associated with (2) and employ them to define conservation
laws based on Noether's theorem [2] (p. 235–267). Our first task
is to identify the Lagrangian by applying the Jacobi Last
Multiplier technique, which was first introduced by Nucci [27].
Consequently, it becomes necessary to evaluate the inverse of
the determinant 𝛥:
(16), (17), (19), (20), (21), (23), (25), and (26), we have all the
∣
𝑥 𝑦
𝑥
𝑦
𝑥𝑥
∣
� � �
elements of the optimal system presented in Proposition 2, (9a).
∣
(
1
)
∣
∣
𝑥 𝑥𝑥
∣
(Note that reductions (13) and (14) are essentially the same). ◻
𝛥 =
∣𝛱
4,𝑥
𝛱
4,𝑦
𝛱
4
∣
=
∣
−1 1 0
∣
, (27b)
∣ ∣
∣ ∣
∣
(
1
)
∣
∣
𝑥 𝑦 0
∣
∣
𝛱
5,𝑥
𝛱
5,𝑦
𝛱
5
∣
where �
4,�
, �
4,�
, �
5,�
, and �
5,�
are the components of the
symmetries � , �
shown in the Proposition 1 and �
(1)
, �
(1)
as
4 5
4 5
Nº
Elements
𝑸(𝒙, 𝒚, 𝒚
𝒙
) = 𝟎
Solutions
Type
Solution
1
𝛱
1
(0) − 𝑦
𝑥
(−(𝑥 + 𝑦)
−1
) = 0
𝑦(𝑥) = 𝑐
Trivial
2
𝛱
2
(0) − 𝑦
𝑥
(−𝑦(𝑥 + 𝑦)
−1
) = 0
𝑦(𝑥) = 0, 𝑦(𝑥) = 𝑐
Trivial
3
𝛱
4
(1) − 𝑦
𝑥
(−1) = 0
𝑦(𝑥) = 𝑐 − 𝑥
Explicit
4
𝛱
2
− 𝛱
4
+ 𝛱
5
(𝑦 − 1)
− 𝑦
𝑥
(𝑥 + 1 − 𝑦(𝑥 + 𝑦)
−1
) = 0
1
𝑦(𝑥) = − 𝑥
𝑥
l
±
1
−
1
−𝑥
2
− 2𝑥
√𝑥
(
𝑥 + 2
)
3/2
√
𝑐 −
2
𝗁
𝑥 + 2
)
− 1
Explicit
5
−√2𝛱
2
+ 𝛱
3
+ 𝛱
4
(1)
− 𝑦
𝑥
(−1
+ (𝑥 + 𝑦)
−1
(√2𝑦 + 𝑥
2
+ 2𝑥𝑦))
= 0
− − −
—
6
−𝛱
1
+ 𝛱
2
+ 𝛱
4
(1)
− 𝑦
𝑥
(−1 + (𝑥 + 𝑦)
−1
(1 − 𝑦))
= 0
√𝑐−𝑥
2
1
(𝑥−1)
2
√𝑐−𝑥
2
1
(𝑥−1)
2
+ +
𝑦(𝑥) =
1−𝑥
−
2
, 𝑦(𝑥) =
2
+
1−𝑥
2 √2 √2 2
Explicit
7
𝛱
1
− 2𝛱
2
+ 𝛱
3
(0)
− 𝑦
𝑥
((𝑥 + 𝑦)
−1
(−1 + 2𝑦 + 𝑥
2
+ 2𝑥𝑦)) = 0
𝑦(𝑥) =
1−𝑥
, 𝑦(𝑥) = 𝑐
2
Explicit
8
𝛱
1
− 𝛱
4
+ 𝛱
5
(𝑦 − 1)
− 𝑦
𝑥
(𝑥 + 1 − (𝑥 + 𝑦)
−1
) = 0
𝑦(𝑥)
1
=
(𝑥 + 1)
l
1
±
𝑥 + 1
𝑥
2
+ 2𝑥
(1 − (𝑥 + 1)
2
)
3/2
√
𝑐 −
2
𝗁
2𝑥
2
+ 4𝑥)
𝑥
2
+ 𝑥 − 1
−
𝑥 + 1
Explicit
9
𝛱
1
+ 𝛱
2
+ 𝛱
3
+ 𝛱
5
(𝑦)
− 𝑦
𝑥
(𝑥
+ (𝑥 + 𝑦)
−1
(−1 − 𝑦 + 𝑥
2
+ 2𝑥𝑦)) = 0
− − −
—
10
𝛱
1
+ 𝛱
3
− 𝛱
4
+ 𝛱
5
(𝑦 − 1)
− 𝑦
𝑥
(𝑥 + 1
+ (𝑥 + 𝑦)
−1
(𝑥
2
+ 2𝑥𝑦 − 1))
= 0
− − −
—
10
0
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira
�
�
�
�
𝑥
−1
�
its first prolongations. Then we get in (27b), 𝛥 = −
(
𝑥 + 𝑦
)
which implies that 𝑀 =
1
= −
(
𝑥 + 𝑦
)
−1
. Now, from [27], we
�
know that � can also be written as � = �
�
�
�
�
which
means
−
that
𝐿
𝑦 𝑦
= −
(
𝑥 + 𝑦
)
, then integrating twice with respect to
�
�
we obtain the Lagrangian
𝑦
2
(
𝑥 + 𝑦
)
−1
VI.
CATEGORIZATION OF LIE ALGEBRA
In the realm of finite dimensional Lie algebras over a field of
characteristic 0, Levi's theorem provides a generic
classification scheme. Specifically, the theorem asserts that
every finite dimensional Lie algebra can be expressed as a
semidirect product of a solvable Lie algebra and a semisimple
Lie algebra, with the solvable Lie algebra serving as the radical
𝐿
(
𝑥, 𝑦, 𝑦
)
= −
𝑥
�
2
+ �
�
�
1
(
�, �
)
+ �
2
(
�, �
)
,
(28)
of the overall algebra. This implies the existence of two major
classes of Lie algebras, namely the solvable and the semisimple.
where 𝑓
1
, 𝑓
2
are arbitrary functions. From the preceding
expression we can consider 𝑓
1
= 𝑓
2
= 0. It’s possible to find
more Lagrangians for (2) by considering other vector fields
given in the Proposition 1. Then, we calculate
�
(
�, �
)
�
�
+ �
�
(
�, �
)
� + �
(
�, �
)
�
�
+ �
[�
]
(
�, �
)
�
�
�
= �
�
[
�
(
�, �
)]
, (28a)
To calculate (28a), we use (28) and (7a). Thus we get
Notably, each of these classes features subclasses that possess
distinct classifications. For instance, within the solvable class,
we encounter the nilpotent Lie algebra.
Based on the Lie group symmetries generated as outlined in
table I, a Lie algebra with five dimensions can be derived. In
order to classify this Lie algebra, we must first recall certain
fundamental criteria. Specifically, in the case of solvable and
semisimple Lie algebras, the Cartan-Killing form K(.,.) serves
as an important tool for classification. Further details and
𝑦
2
(
𝑥 + 𝑦
)
−2
𝜉
( ) + 𝜉 (−
𝑦
2
(
𝑥 + 𝑦
)
−1
𝑦
2
(
𝑥 + 𝑦
)
−2
) + � (
�
)
propositions relating to these criteria can be found in [29].
2
�
2 2
+ (𝜂
𝑥
+ (𝜂
𝑦
− 𝜉
𝑥
)𝑦
𝑥
− 𝜉 𝑦
2
)
(
−𝑦
𝑥
(
𝑥 + 𝑦
)
−1
)
− 𝑓
𝑥
− 𝑦
𝑥
𝑓
𝑦
= 0. (28b)
In (28b), rearranging and associating terms with respect to
1, �
�
, �
2
and �
3
, we obtain the following determinant
equations
Proposition 3. (Cartan’s theorem) If and only if the Killing
form of a Lie algebra is nondegenerate, it is classified as
semisimple.
� �
𝜉
𝑦
= 𝑓
𝑥
= 0, (29a)
(
𝑥 + 𝑦
)
−1
𝜂
𝑥
+ 𝑓
𝑦
= 0, (29b)
Proposition 4. The solvability of a Lie subalgebra 𝔤 can be
determined by evaluating 𝐾
(
𝑋, 𝑌
)
for all 𝑋 ∈
[
𝔤, 𝔤
]
and 𝑌 ∈ 𝔤.
Specifically, if 𝐾
(
𝑋, 𝑌
)
= 0 for all such 𝑋 and 𝑌, then 𝔤 is
𝜉 − 𝜉
𝑥
(
𝑥 + 𝑦
)
+ 𝜂 − 2
(
𝑥 + 𝑦
)
(𝜂
𝑦
− 𝜉
𝑥
) = 0. (29c)
classified as solvable. An alternative representation of this
statement is �
(
�,
[
�, �
])
= 0.
Solving the preceding system (29a, 29b, 29c) for 𝜉, 𝜂, and 𝑓, we
obtain the infinitesimal generators of Noether’s symmetries
The next statements are imperative for the successful
completion of the classification.
�
2
�
2
Definition 1. Let � be a finite-dimensional Lie algebra over an
𝜂 = −
2
𝑎
1
− 𝑥𝑦𝑎
1
+ 𝑦𝑎
2
− 𝑎
3
, 𝜉 = −
2
𝑎
1
+ 𝑎
2
𝑥 + 𝑎
3
,
and �
(
�
)
= ��
1
+ �
4
. (30)
arbitrary field 𝑘. Choose a basis 𝑒
𝑗
, 1 ≤ 𝑖 ≤ 𝑛, in 𝔤 where 𝑛 =
dim � and set [�
�
, �
�
] = �
�
�
�
. The coefficients �
�
are
denoted
with �
1
, �
2
, �
3
and �
4
arbitrary constants. Then, the
Noether
��
as the structure constants.
��
symmetry group or variational symmetries are
Proposition 5. Consider two Lie algebras, �
1
and �
2
, each
with a dimension of �. Assuming that each algebra has a basis
such
�
2
� �
2
�
� � �
𝑉
1
= −
2 𝜕𝑥
−
(
2
+ 𝑥𝑦)
𝜕𝑦
, 𝑉
2
= 𝑥
𝜕𝑦
+ 𝑦
𝜕𝑦
, 𝑉
3
=
𝜕𝑥
−
𝜕𝑦
. (31)
Remarks: Note that in (31), 𝑉
2
= 𝛱
5
and 𝑉
3
= −𝛱
4
thus, the
symmetries in (2) have two variational symmetry. According to
[28], in order to obtain the conserved quantities or conservation
laws, we should solve
𝐼 =
(
𝑋𝑦
𝑥
− 𝑌
)
𝐿
𝑦
𝑥
− 𝑋𝐿 + 𝑓, (32)
so, using (28), (30) and (31). Therefore, the conserved
quantities are given by
that the structure constants are identical, it follows that �
1
and
�
2
are isomorphic.
Consider the Lie algebra �, which corresponds to the
symmetry group of infinitesimal generators of (1) as specified
in the commutator table. In order to proceed, it suffices to
examine the following relations:
[
�
1
, �
3
]
= 2�
1
,
[
�
1
, �
5
]
= �
1
,
[
�
2
, �
3
]
= 2�
2
,
[
�
2
, �
4
]
=
−𝛱
1
,
[
𝛱
2
, 𝛱
5
]
= 𝛱
2
,
[
𝛱
3
, 𝛱
4
]
= −2𝛱
2
,
[
𝛱
4
, 𝛱
5
]
= 𝛱
4
. (34)
Using (34), we calculate Cartan-Killing form � as follows.
𝑦
2
𝑥
2
(
𝑥 + 𝑦
)
−1
𝐼 = − 𝑦
(
𝑥 + 𝑦
)
(
�
2
+ ��) + ��
+ � ,
1
4
�
2
1 4
0
𝑦
2
𝑥
(
𝑥 + 𝑦
)
−1
𝐼 = −
𝑥
+ 𝑦𝑦
(
𝑥 + 𝑦
)
−1
+ 𝑦𝑎 + 𝑎 ,
2
2
� 1 4
𝑦
2
(
𝑥 + 𝑦
)
−1
𝐼
= −
𝑥
− 𝑦
(
𝑥 + 𝑦
)
−1
+ 𝑦𝑎 + 𝑎 . (33)
3
2
� 1 4
In (35) the determinant vanishes, and thus by Proposition 3
(Cartan criterion) it is not semisimple. Given that the Cartan-
Killing form of a nilpotent Lie algebra is uniformly zero, it
0
0
0
0
0
⎡
0
0
0
0
0
⎤
𝐾 =
0
8
0
4
, (35)
⎢0
0
0
0
0⎥
[
0
0
4
0
3
]
Scientia et Technica Año XXVIII, Vol. 28, No. 02, abril-junio de 2023. Universidad Tecnológica de Pereira.
10
1
follows that the Lie algebra � is not nilpotent, by the converse
of the preceding assertion. Next, we confirm the solvability of
the Lie algebra by applying the Cartan criteria for solvability,
as established in Proposition 4. Consequently, we are dealing
with a non-nilpotent, but solvable Lie algebra. The Nilradical
of � is generated by �
1
, �
2
, and �
4
, that is, we have a
Solvable Lie algebra with three dimensional Nilradical.
Let 𝑚 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 [30] classified the 5-
dimensional solvable nonlilpotent Lie algebras, in particular the
solvable nonnilpotent Lie algebra with three dimensional
Nilradical, this Nilradical is isomorphic to 𝔥
3
the Heisenberg
Lie algebra. Then, by the Proposition 5, and consequently we
establish a isomorphism of Lie algebras with 𝔤 and the Lie
algebra 𝔤
5,40
. In summery we have the next proposition.
Let us consider the dimension of the Nilradical, denoted as 𝑀,
of a Solvable Lie algebra, which we denote as 𝑚. In the case of
a fifth-dimensional Lie algebra, we can establish that 3 ≤ 𝑚 ≤
5. In [30], Mubarakzyanov classified the 5-dimensional
solvable non-nilpotent Lie algebras, including the solvable non-
nilpotent Lie algebra with a three-dimensional Nilradical.
Notably, this Nilradical is isomorphic to �
3
, the Heisenberg
Lie algebra. Based on Proposition 5, we can establish an
isomorphism of Lie algebras between � and the Lie
algebra
�
5,40
. In summary, we have the following proposition.
Proposition 6. The 5-dimensional Lie algebra �, which is
associated with the symmetry group of (1), is a solvable non-
nilpotent Lie algebra possessing a three-dimensional Nilradical.
In the classification presented by Mubarakzyanov, it is shown
For future works, equivalence group theory could be also
considered to obtain preliminary classifications associated to a
complete classification of (2).
ACKNOWLEDGMENTS
G. Loaiza, Y. Acevedo and O.M.L. Duque are grateful to
EAFIT University, Colombia, for the financial support in the
project "Study and applications of diffusion processes of
importance in health and computation" with code
11740052022.
DECLARATION INTERESTS
The authors declare that they have no conflict of interest.
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Gabriel Ignacio Loaiza Ossa. PhD D. in
Mathematical Sciences, Universidad
Politécnica de Valencia, Spain. Full-time
professor at Universidad EAFIT, Colombia.
ORCID: https://orcid.org/0000-0003-2413-
1139
Yeisson Alexis Acevedo Agudelo. Master's
Degree in Applied Mathematics, Universidad
EAFIT, Colombia.
ORCID: https://orcid.org/0000-0002-1640-
9084
Oscar Mario Londoño Duque. PhD of
applied mathematics, University of
Campinas, Brazil.
ORCID: https://orcid.org/0000-0002-5666-
8224
Danilo A. García Hernández. PhD in
Mathematics, University of Campinas,
Brazil.
ORCID: https://orcid.org/0000-0002-0807-
2602