1. Introduction
Fractional partial differential equations have proven to be powerful tools for modeling complex nonlinear phenomena across various scientific disciplines, including biology, chemistry, neural networks, viscoelastic mechanics and mathematical physics. Therefore, there exist many effective methods for solving them, such as the shifted Legendre operator matrix method [
1], the Lie group method [
2,
3,
4,
5] and the local discontinuous Galerkin methods [
6], among others. Among the methods mentioned above, the Lie group method has emerged as particularly effective for handling fractional derivative differential equations.
Recently, Rawya Al-Deiakeh, et al. (
J. Geom. Phys. 176(2022)104512) applied the fractional Lie group method to study the (2+1)-dimensional ZK
equation with the following time fractional derivative [
2]:
where
a,
b, and
c are free parameters and
denotes the Riemann–Liouville fractional derivative [
7,
8,
9]. Equation (1) serves as a fundamental model for describing nonlinear ion-acoustic wave evolution in magnetized plasmas with weak dispersion, small wave amplitude and strong magnetic fields. In addition, for this type of model, Valet. F. studied the asymptotic-soliton-like solutions of the Zakharov–Kuznetsov-type equations [
10]. Klein. C. et al. derived the numerical solutions to the Zakharov–Kuznetsov equations in two dimensions [
11]. In [
4], the authors investigated the time fractional generalized (2+1)-dimensional Zakharov–Kuznetsov equation with single power law nonlinearity.
However, we identified several issues in the previous work [
2] regarding the reported symmetries, similarity reductions and conservation laws. This motivated us to revisit the problem and present corrected results. Additionally, we contribute new findings, including the Lie transformation group and the optimal system for Equation (1).
2. Group Analysis for Equation (1)
2.1. Infinitesimal Generators
First of all, this scheme was used to derive the symmetries of the (2+1)-dimensional ZK Equation (1) with time fractional derivative. As a direct conclusion, we found the infinitesimal generators of Equation (1) of the following form.
Theorem 1. The infinitesimal generators of the (2+1)-dimensional ZK Equation (1) with time fractional derivative were given by Proof. Assuming Equation (1) remains invariant under the Lie point transformation group,
where
and
, respectively, are infinitesimals and extended infinitesimals.
and
can be written as follows [
12,
13]:
where
and
are total derivatives.
Thereby, the infinitesimal generator
X is expressed as
Consider the infinitesimal generator
X that needs satisfy the invariance criterion [
14] for Equation (1):
where
is the prolongation and
The
-extended infinitesimal [
14] with the above equation can be shown as
Also, to change Equation (6), the generalized Leibnitz rule [
7] was used as follows:
where
Next, with the aid of the generalized Leibnitz rule (7), one has
It is known that the chain rule for the compound function [
8] can be given by
The combination of the generalized Leibnitz rule (7) and the chain rule for the compound function (9) yields
where
Then, by inserting the
-extended infinitesimal
and the Lie point transformation group (2) into the symmetry determining equation, we collect the coefficients of the function
u and its derivatives, setting each coefficient to zero.
The infinitesimals can be obtained by solving the over-determined system
where
are arbitrary constants. □
At the same time, adding the invariance condition to hold the structure of the fractional derivative of the Riemann–Liouville, we have .
Therefore, the infinitesimal generator (4) becomes
It can be generated from the following three vector fields:
The closed Lie algebra with the Lie brackets [
12,
13]
was seen from the vector fields (13):
The commutation relations of the corresponding Lie algebra are presented in
Table 1.
2.2. Special Cases
We examine several special cases that arise when parameters take specific values.
Case 1. When
, Equation (1) is generated by the following vector fields:
Case 2. When
, Equation (1) is generated by the following vector fields:
Case 3. When
, Equation (1) is generated by the following vector fields:
Case 4. When
, Equation (1) is generated by the following vector fields:
Case 5. When
, Equation (1) is generated by the following vector fields:
These cases demonstrate how the vector fields vary with different parameter values, leading to distinct geometric properties. In particular, when
, it yields the same result as Equation (1).
Remark 1. By comparing the vector fields (13) with the corresponding results in Ref. [2],we find that they are different. According to the Lie symmetry theory, we know that the symmetries, similarity reductions and conservation laws of Equation (1) must consequently differ. This work therefore revises certain conclusions arrived at in the earlier study. 3. Lie Transformation Group for Equation (1)
We construct the Lie transformation group for Equation (1) by solving the following initial value problem:
where
is a small parameter.
This yields the transformation groups
:
As a result, we have
Theorem 2. If is a solution of the (2+1)-dimensional ZK equation of the time fractional derivative, thenare also solutions to Equation (1). 4. Optimal System for Equation (1)
We aim to construct the optimal system for Equation (1) by reducing it as much as possible. The optimal system classifies the orbits in the adjoint representation.
By applying the Lie series [
13], the action of the adjoint operator is given by
where
.
For example,
The infinitesimal generators (13) with the help of the adjoint representation are seen in
Table 2.
As a direct application of the above results, we have
Theorem 3. An optimal system of the Lie subalgebras of infinitesimal generators (13) for Equation (1) is spanned bywhere . The detailed proof follows a process analogous to that applied in Refs. [
13,
15].
In the next section, the optimal system (24) was used to reduce the (2+1)-dimensional ZK Equation (1) of the time fractional derivative.
5. Similarity Reduction for Equation (1)
Here, by considering Theorem 2, the (2+1)-dimensional ZK
Equation (1) was reduced to the (1+1)-dimensional fractional differential equation with the Erdélyi-Kober operators [
16].
Case I:
Applying the symmetry
, Equation (1) can be easily reduced to the following form:
Case II:
Similarly, Equation (1) can be easily reduced to the following form:
If the fractional Lie group method is applied again to Equation (26), then its corresponding vector fields are
Case III:
Utilizing the vector field
, Equation (1) becomes the following form:
Case IV:
For
, it has
Solving Equation (29), we obtain
and
By considering (30) and (31), Equation (1) was reduced into the following (1+1)-dimensional fractional differential equation (see Theorem 4).
Theorem 4. The similarity variables and the group-invariant solution were used to reduce the (2+1)-dimensional ZK equation of the time fractional derivative into the following (1+1)-dimensional fractional-order form:whereare the two parameters of the Erdélyi–Kober fractional differential operator, withandbeing the two parameters of the Erdélyi–Kober fractional integral operator [16]. The detailed proof process of this theorem can be found in [
2,
4,
17].
Remark 2. For the reduced Equation (26), it can be translated into the following result:with the similarity variable and group-invariant solution . 6. Conservation Laws for Equation (1)
The laws of conservation are inherent properties of matter, such as mass and momentum. They play an essential role in the study of integrability and linearization mappings. Because of the importance outlined above, we construct the conservation laws of Equation (1) by using a new conservation theorem [
18,
19,
20]. This implies that the conserved vector
needs to satisfy the following conservation equation:
The formal Lagrangian of Equation (1) was written as follows:
where
is a smooth function.
Further, Equation (36) was expressed by the following integral form:
At the same time, it corresponds to the Euler–Lagrangian operator of the form
where
is the adjoint operator of
.
Equation (1) has the following adjoint equation:
If we are considering the dependent variable
, then we obtain the result
where
is the Euler–Lagrange operator and
I is the identity operator.
The result
and the Lie characteristic equation
W, respectively, are
and
From vector fields (13) of Equation (1), the Lie characteristic equation is expressed by
The fractional generalizations of the Noether operators [
18,
19,
20] about the
t-component of the conserved vector give
with
and where the
x and
y components of the conserved vectors are
where
and
.
By considering Expressions (44) and (45), the
and
components of the conserved vectors of the (2+1)-dimensional ZK
Equation (1) are given by, respectively, the following:
where
and
correspond, respectively, to
(43) as follows:
7. Conclusions and Discussions
In this article, we restudied the (2+1)-dimensional Zakharov0–Kuznetsov equation of time fractional order through the fractional Lie group method. As a direct result, the symmetries, Lie transformation group, optimal system, similarity reduction and conservation laws of this equation were derived. These findings not only revised the mistakes from a previous paper on this topic but also provided additional valuable conclusions. Unfortunately, we were unable to find exact solutions, such as travelling wave solutions and soliton solutions, for the reduced equation. Exploring methods for finding more exact or approximate solutions remains an open problem.
Author Contributions
Methodology, J.-G.L.; Writing—original draft, J.-G.L.; Writing—review & editing, Y.-F.Z. and J.-Q.W.; Funding acquisition, Y.-F.Z. and J.-Q.W. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (No. 12371256), the Natural Science Foundation of Jiangsu Province (No. BK20230708, BK20241071) and the Natural Science Foundation for the Universities in Jiangsu Province (No. 23KJB110003) and the Ministry of Education Key Laboratory of NSLSCS (No. 202412).
Data Availability Statement
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.
Conflicts of Interest
The authors declare no conflicts of interest.
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Table 1.
Commutation table of Lie algebra.
Table 1.
Commutation table of Lie algebra.
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Table 2.
The infinitesimal generators (13) with the help of the adjoint representation.
Table 2.
The infinitesimal generators (13) with the help of the adjoint representation.
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