Revised Lie Group Analysis of the Time Fractional (2+1)-Dimensional Zakharov-Kuznetsov (q, p, r) Equation

Abstract

This article presents a comprehensive study of the (2+1)-dimensional Zakharov–Kuznetsov (ZK) $(q,p,r)$ equation with time fractional derivativeUtilizing the fractional Lie group method, we derive several results, including the symmetries, similarity reductions and conservation laws for this equation. Our findings not only correct previous errors in the literature but also introduce new results, such as the Lie transformation group and optimal system for this model. The study provides a rigorous mathematical framework for analyzing this fundamental model, which describes nonlinear ion-acoustic wave evolution in magnetized plasmas.

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 $(q,p,r)$ equation with the following time fractional derivative [2]:

$$\begin{array}{c}\hfill D_t^{\alpha }u+a{(u^q)}_x+b{(u^p)}_{xxx}+c{(u^r)}_{xyy}=0,0<\alpha \le 1,\end{array}$$

(1)

where a, b, and c are free parameters and

$$\begin{array}{c}\hfill {}^{RL}D_{\tau }^{\alpha }\Theta (\chi ,\tau )=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{{\partial }^n}{\partial {\tau }^n}}\underset{a}{\overset{\tau }{\int }}{\displaystyle \frac{\Theta (\chi ,s)}{{(\tau -s)}^{\alpha +1-n}}}ds,& ifn-1<\alpha <n,n\in N,\\ {\displaystyle \frac{{\partial }^n\Theta (\chi ,\tau )}{\partial {\tau }^n}},& if\alpha =n,n\in N,\end{array}\right.\end{array}$$

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 $(q,p,r)$ 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 $(q,p,r)$ Equation (1) with time fractional derivative were given by

$$\begin{array}{cc}\hfill & X_1=\partial x,X_2=\partial y,\hfill \\ \hfill & X_3={\displaystyle \frac{p-q}{p-3q+2}}x\partial x+{\displaystyle \frac{r-q}{p-3q+2}}y\partial y+{\displaystyle \frac{t}{\alpha }}\partial t+{\displaystyle \frac{2u}{p-3q+2}}\partial u.\hfill \end{array}$$

Proof. 

Assuming Equation (1) remains invariant under the Lie point transformation group,

$$\begin{array}{cc}\hfill & x^{*}\to x+\epsilon {\xi }_x(x,y,t,u)+O({\epsilon }^2),y^{*}\to y+\epsilon {\xi }_y(x,y,t,u)+O({\epsilon }^2),\hfill \\ \hfill & t^{*}\to t+\epsilon {\xi }_t(x,y,t,u)+O({\epsilon }^2),u^{*}\to u+\epsilon {\xi }_u(x,y,t,u)+O({\epsilon }^2),\hfill \\ \hfill & D_t^{\alpha }{u}^{*}\to D_t^{\alpha }u+\epsilon {\xi }_u^{\alpha }(x,y,t,u)+O({\epsilon }^2),{\displaystyle \frac{\partial u^{*}}{\partial x^{*}}}\to {\displaystyle \frac{\partial u}{\partial x}}+\epsilon {\xi }_u^x(x,y,t,u)+O({\epsilon }^2),\hfill \\ \hfill & {\displaystyle \frac{{\partial }^3{u}^{*}}{\partial x^{*3}}}\to {\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\epsilon {\xi }_u^{xxx}(x,y,t,u)+O({\epsilon }^2),\epsilon \ll 1,\hfill \\ \hfill & \dots ,\hfill \end{array}$$

(2)

where ${\xi }_y,{\xi }_u,{\xi }_t,{\xi }_x$ and ${\xi }_u^{\alpha }$, respectively, are infinitesimals and extended infinitesimals.

${\xi }_u^x,{\xi }_u^y,{\xi }_u^t$ and ${\xi }_u^{xxx}$ can be written as follows [12,13]:

$$\begin{array}{cc}\hfill & {\xi }_u^{xxx}=D_{xxx}({\xi }_u-{\xi }_x{u}_x-{\xi }_y{u}_y-{\xi }_t{u}_t)+{\xi }_x{u}_{xxxx}+{\xi }_y{u}_{xxxy}+{\xi }_t{u}_{xxxt},\hfill \\ \hfill & {\xi }_u^x=D_x({\xi }_u-{\xi }_x{u}_x-{\xi }_y{u}_y-{\xi }_t{u}_t)+{\xi }_x{u}_{xx}+{\xi }_y{u}_{xy}+{\xi }_t{u}_{xt},\hfill \\ \hfill & {\xi }_u^t=D_t({\xi }_u-{\xi }_x{u}_x-{\xi }_y{u}_y-{\xi }_t{u}_t)+{\xi }_x{u}_{xt}+{\xi }_y{u}_{xt}+{\xi }_t{u}_{tt},\hfill \\ \hfill & {\xi }_u^y=D_y({\xi }_u-{\xi }_x{u}_x-{\xi }_y{u}_y-{\xi }_t{u}_t)+{\xi }_x{u}_{xy}+{\xi }_y{u}_{yy}+{\xi }_t{u}_{yt},\hfill \\ \hfill & \dots ,\hfill \end{array}$$

(3)

where $D_x,D_t$ and $D_y$ are total derivatives.

Thereby, the infinitesimal generator X is expressed as

$$\begin{array}{c}\hfill X={\xi }_x\partial x+{\xi }_y\partial y+{\xi }_t\partial t+{\xi }_u\partial u.\end{array}$$

(4)

Consider the infinitesimal generator X that needs satisfy the invariance criterion [14] for Equation (1):

$$\begin{array}{c}\hfill P{r}^{(\alpha ,t)}X(\Xi ){|}_{\Xi =0}=0,n=1,2,3,\dots ,\end{array}$$

(5)

where $P{r}^{(\alpha ,t)}X$ is the prolongation and $\Xi =D_t^{\alpha }u+a{(u^q)}_x+b{(u^p)}_{xxx}+c{(u^r)}_{xyy}.$

The $\alpha$-extended infinitesimal [14] with the above equation can be shown as

$$\begin{array}{cc}\hfill & {\xi }_u^{\alpha }=D_t^{\alpha }({\xi }_u)+{\xi }_x{D}_t^{\alpha }(u_x)-D_t^{\alpha }({\xi }_x{u}_x)+{\xi }_y{D}_t^{\alpha }(u_y)-D_t^{\alpha }({\xi }_y{u}_y)+D_t^{\alpha }(D_t({\xi }_t)u)\hfill \\ \hfill & -D_t^{\alpha +1}({\xi }_{t}u)+{\xi }_t{D}_t^{\alpha +1}(u).\hfill \end{array}$$

(6)

Also, to change Equation (6), the generalized Leibnitz rule [7] was used as follows:

$$\begin{array}{c}\hfill D_t^{\alpha }(\Pi (t)\Theta (t))=\sum _{k=0}^{\infty }{D}_t^{\alpha -k}\Pi (t)D_t^k\Theta (t),\alpha >0,\end{array}$$

(7)

where

$$\left(\begin{array}{c}\alpha \\ k\end{array}\right)={\displaystyle \frac{{(-1)}^{k-1}\alpha \Gamma (k-\alpha )}{\Gamma (1-\alpha )\Gamma (k+1)}},\Gamma (t)=\underset{0}{\overset{\infty }{\int }}{e}^{-x}{x}^{t-1}dx,(t>0).$$

Next, with the aid of the generalized Leibnitz rule (7), one has

$$\begin{array}{cc}\hfill & {\xi }_u^{\alpha }=D_t^{\alpha }({\xi }_u)-\alpha D_t^{\alpha }({\xi }_t){\displaystyle \frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}}-\sum _{m=1}^{\infty }\left(\begin{array}{c}\alpha \\ m\end{array}\right)D_t^m({\xi }_x)D_t^{\alpha -m}(u_x)\hfill \\ \hfill & -\sum _{m=1}^{\infty }\left(\begin{array}{c}\alpha \\ m\end{array}\right)D_t^m({\xi }_y)D_t^{\alpha -m}(u_y)-\sum _{m=1}^{\infty }\left(\begin{array}{c}\alpha \\ m+1\end{array}\right)D_t^{m+1}({\xi }_t)D_t^{\alpha -m}(u_t).\hfill \end{array}$$

(8)

It is known that the chain rule for the compound function [8] can be given by

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{m}y(x(t))}{d{t}^m}}=\sum _{k=0}^m\sum _{r=0}^k\left(\begin{array}{c}k\\ r\end{array}\right){\displaystyle \frac{1}{k!}}{(-x(t))}^r{\displaystyle \frac{d^m}{d{t}^m}}(x{(t)}^{k-r}){\displaystyle \frac{d^{k}y(x)}{d{x}^k}}.\end{array}$$

(9)

The combination of the generalized Leibnitz rule (7) and the chain rule for the compound function (9) yields

$$\begin{array}{c}\hfill D_t^{\alpha }({\xi }_u)={\displaystyle \frac{{\partial }^{\alpha }{\xi }_u}{\partial t^{\alpha }}}+{\xi }_u^u{\displaystyle \frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}}-u{\displaystyle \frac{{\partial }^{\alpha }{\xi }_u^u}{\partial t^{\alpha }}}+\sum _{m=1}^{\infty }\left(\begin{array}{c}\alpha \\ m\end{array}\right){\displaystyle \frac{{\partial }^m{\xi }_u^u}{\partial t^m}}{D}_t^{\alpha -m}(u)+\Delta ,\end{array}$$

(10)

where

$$\Delta =\sum _{m=2}^{\infty }\sum _{n=2}^m\sum _{j=2}^n\sum _{r=0}^{j-1}\left(\begin{array}{c}\alpha \\ m\end{array}\right)\left(\begin{array}{c}m\\ n\end{array}\right)\left(\begin{array}{c}j\\ r\end{array}\right){\displaystyle \frac{1}{j!}}{\displaystyle \frac{t^{m-\alpha }}{\Gamma (m+1-\alpha )}}{(-u)}^r{\displaystyle \frac{{\partial }^n}{\partial t^n}}(u^{j-r}){\displaystyle \frac{{\partial }^{m-n+j}{\xi }_u}{\partial t^{m-n}\partial u^j}}.$$

Then, by inserting the $\alpha$-extended infinitesimal

$$\begin{array}{cc}\hfill & {\xi }_u^{\alpha }={\displaystyle \frac{{\partial }^{\alpha }{\xi }_u}{\partial t^{\alpha }}}+\sum _{m=1}^{\infty }[\left(\begin{array}{c}\alpha \\ m\end{array}\right){\displaystyle \frac{{\partial }^{\alpha }{\xi }_u^u}{\partial t^{\alpha }}}-\left(\begin{array}{c}\alpha \\ m+1\end{array}\right)D_t^{m+1}({\xi }_x)]D_t^{\alpha -m}(u)\hfill \\ \hfill & +({\xi }_u^u-\alpha D_t({\xi }_x)){\displaystyle \frac{{\partial }^{\alpha }u}{\partial t^{\alpha }}}-u{\displaystyle \frac{{\partial }^{\alpha }{\xi }_u^u}{\partial t^{\alpha }}}-\sum _{m=1}^{\infty }\left(\begin{array}{c}\alpha \\ m\end{array}\right)D_t^m({\xi }_x)D_t^{\alpha -m}(u_x)\hfill \\ \hfill & -\sum _{m=1}^{\infty }\left(\begin{array}{c}\alpha \\ m\end{array}\right)D_t^m({\xi }_y)D_t^{\alpha -m}(u_y)+\Delta \hfill \end{array}$$

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

$$\begin{array}{cc}\hfill & {\xi }_x=(p-q)c_3\alpha x+c_1,{\xi }_y=(r-q)c_3\alpha y+c_2,\hfill \\ \hfill & {\xi }_t=(p-3q+2)c_3\alpha t+c_4,{\xi }_u=2c_3\alpha u,\hfill \end{array}$$

(11)

where $c_i(i=1,2,3,4)$ are arbitrary constants. □

At the same time, adding the invariance condition ${\xi }_t(x,y,t,u){|}_{t=0}=0$ to hold the structure of the fractional derivative of the Riemann–Liouville, we have $c_4=0$.

Therefore, the infinitesimal generator (4) becomes

$$\begin{array}{c}\hfill X=((p-q)c_3\alpha x+c_1)\partial x+((r-q)c_3\alpha y+c_2)\partial y+(p-3q+2)c_3\alpha t\partial t+2c_3\alpha u\partial u.\end{array}$$

(12)

It can be generated from the following three vector fields:

$$\begin{array}{cc}\hfill & X_1=\partial x,X_2=\partial y,\hfill \\ \hfill & X_3={\displaystyle \frac{p-q}{p-3q+2}}x\partial x+{\displaystyle \frac{r-q}{p-3q+2}}y\partial y+{\displaystyle \frac{t}{\alpha }}\partial t+{\displaystyle \frac{2u}{p-3q+2}}\partial u.\hfill \end{array}$$

(13)

The closed Lie algebra with the Lie brackets [12,13] $[X_i,X_j]=X_i{X}_j-X_j{X}_i(i,j=1,2,3)$ was seen from the vector fields (13):

$$\begin{array}{cc}\hfill & [X_1,X_1]=[X_2,X_2]=[X_3,X_3]=[X_1,X_2]=0,\hfill \\ \hfill & [X_1,X_3]=(p-q)\alpha X_1,[X_2,X_3]=(r-q)\alpha X_2.\hspace{1em}\square \hfill \end{array}$$

(14)

The commutation relations of the corresponding Lie algebra are presented in

Table 1. Commutation table of Lie algebra.

$[{\mathit{X}}_{\mathit{i}},{\mathit{X}}_{\mathit{j}}]$	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$
$X_1$	0	0	$(p-q)\alpha X_1$
$X_2$	0	0	$(r-q)\alpha X_2$
$X_3$	$(q-p)\alpha X_1$	$(q-r)\alpha X_2$	0

.

2.2. Special Cases

We examine several special cases that arise when parameters $(q,p,r)$ take specific values.

Case 1. When $q=p$, Equation (1) is generated by the following vector fields:

$$\begin{array}{c}\hfill X_1=\partial x,X_2=\partial y,X_3=(q-r)y\partial y+{\displaystyle \frac{2t}{\alpha }}(q-1)\partial t-2u\partial u.\end{array}$$

(15)

Case 2. When $q=r$, Equation (1) is generated by the following vector fields:

$$\begin{array}{c}\hfill X_1=\partial x,X_2=\partial y,X_3={\displaystyle \frac{p-q}{p-3q+2}}x\partial x+{\displaystyle \frac{1}{\alpha }}t\partial t+{\displaystyle \frac{2}{p-3q+2}}u\partial u.\end{array}$$

(16)

Case 3. When $p=r$, Equation (1) is generated by the following vector fields:

$$\begin{array}{c}\hfill X_1=\partial x,X_2=\partial y,X_3={\displaystyle \frac{p-q}{p-3q+2}}x\partial x+{\displaystyle \frac{p-q}{p-3q+2}}y\partial y+{\displaystyle \frac{1}{\alpha }}t\partial t+{\displaystyle \frac{2}{p-3q+2}}u\partial u.\end{array}$$

(17)

Case 4. When $q=p=r$, Equation (1) is generated by the following vector fields:

$$\begin{array}{c}\hfill X_1=\partial x,X_2=\partial y,X_3={\displaystyle \frac{t}{\alpha }}(q-1)\partial x-u\partial u.\end{array}$$

(18)

Case 5. When $p-3q+2=0$, Equation (1) is generated by the following vector fields:

$$\begin{array}{c}\hfill X_1=\partial x,X_2=\partial y,X_3=2x(q-1)\partial x+y(r-q)\partial y+2u\partial u.\end{array}$$

(19)

These cases demonstrate how the vector fields vary with different parameter values, leading to distinct geometric properties. In particular, when $p=r$, it yields the same result as Equation (1).

Remark 1. 

By comparing the vector fields (13) with the corresponding results in Ref. [2],

$$\begin{array}{cc}\hfill & V_1={\displaystyle \frac{\partial }{\partial x}},V_2={\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill & V_3=\sigma (q-p)x{\displaystyle \frac{\partial }{\partial x}}+(3q-p-2)t{\displaystyle \frac{\partial }{\partial t}}-2\sigma \vartheta {\displaystyle \frac{\partial }{\partial \vartheta }},\hfill \end{array}$$

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:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{d(x^{*}(\epsilon ),y^{*}(\epsilon ),t^{*}(\epsilon ),u^{*}(\epsilon ))}{d\epsilon }}=\sigma (x^{*}(\epsilon ),y^{*}(\epsilon ),t^{*}(\epsilon ),u^{*}(\epsilon )),\\ (x^{*}(\epsilon ),y^{*}(\epsilon ),t^{*}(\epsilon ),u^{*}(\epsilon )){|}_{\epsilon =0}=(x,y,t,u),\end{array}\right.\end{array}$$

(20)

where $\epsilon$ is a small parameter.

This yields the transformation groups $G_i(\epsilon ),(i=1,2,3)$:

$$\begin{array}{cc}\hfill & G_1(\epsilon ):(x,y,t,u)\to (x+\epsilon ,y,t,u),\hfill \\ \hfill & G_2(\epsilon ):(x,y,t,u)\to (x,y+\epsilon ,t,u),\hfill \\ \hfill & G_3(\epsilon ):(x,y,t,u)\to (e^{(p-q)\epsilon }x,e^{(r-q)\epsilon }y,e^{{\displaystyle \frac{p-3q+2}{\alpha }}\epsilon }t,e^{2\epsilon }u).\hfill \end{array}$$

(21)

As a result, we have

Theorem 2. 

If $u=f(x,y,t)$ is a solution of the (2+1)-dimensional ZK $(q,p,r)$ equation of the time fractional derivative, then

$$\begin{array}{cc}\hfill & u_1=f_1(x-\epsilon ,y,t),\hfill \\ \hfill & u_2=f_2(x,y-\epsilon ,t),\hfill \\ \hfill & u_3=e^{-2\epsilon }{f}_3(e^{(q-p)\epsilon }x,e^{(q-r)\epsilon }y,e^{{\displaystyle \frac{3q-p-2}{\alpha }}\epsilon }t)\hfill \end{array}$$

(22)

are 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

$$A\mathrm{d}(exp(\epsilon X_i))·X_j=X_j-\epsilon [X_i,X_j]+{\displaystyle \frac{{\epsilon }^2}{2}}[X_i,[X_i,X_j]]-\dots ,$$

(23)

where $[X_i,X_j]=X_i{X}_j-X_j{X}_i$.

For example,

$$\begin{array}{cc}\hfill Ad(exp(\epsilon ·X_1))·X_1& =X_1-\epsilon [X_1,X_1]+\dots \hfill \\ \hfill & =X_1,\hfill \end{array}$$

$$\begin{array}{cc}\hfill Ad(exp(\epsilon ·X_1))·X_3& =X_3-\epsilon [X_1,X_3]+{\displaystyle \frac{{\epsilon }^2}{2}}[X_1,[X_1,X_3]]-\dots \hfill \\ \hfill & =X_3-\epsilon (p-q)\alpha X_1,\hfill \end{array}$$

$$\begin{array}{cc}\hfill Ad(exp(\epsilon ·X_3))·X_1& =X_1-\epsilon [X_3,X_1]+{\displaystyle \frac{{\epsilon }^2}{2}}[X_1,[X_3,X_1]]-\dots \hfill \\ \hfill & =e^{\epsilon (p-q)\alpha }{X}_1.\hfill \end{array}$$

The infinitesimal generators (13) with the help of the adjoint representation are seen in

Table 2. The infinitesimal generators (13) with the help of the adjoint representation.

$\mathit{Ad}(exp(\mathit{\epsilon }{\mathit{X}}_{\mathit{i}}))·{\mathit{X}}_{\mathit{j}}$	${\mathit{X}}_1$	${\mathit{X}}_2$	${\mathit{X}}_3$
$X_1$	$X_1$	$X_2$	$X_3-\epsilon (p-q)\alpha X_1$
$X_2$	$X_1$	$X_2$	$X_3-\epsilon (r-q)\alpha X_2$
$X_3$	$e^{\epsilon (p-q)\alpha }{X}_1$	$e^{\epsilon (r-q)\alpha }{X}_2$	$X_3$

.

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 by

$$V_1=X_1,V_2=X_2,V_3=X_3,V_4=k{X}_1\pm X_2,$$

(24)

where $k(\ne 0)\in R$.

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 $(q,p,r)$ Equation (1) of the time fractional derivative.

5. Similarity Reduction for Equation (1)

Here, by considering Theorem 2, the (2+1)-dimensional ZK $(q,p,r)$ Equation (1) was reduced to the (1+1)-dimensional fractional differential equation with the Erdélyi-Kober operators [16].

Case I: $V_1=X_1=\partial x$

Applying the symmetry $\partial x$, Equation (1) can be easily reduced to the following form:

$$\begin{array}{c}\hfill D_t^{\alpha }u(y,t)=0.\end{array}$$

(25)

Case II: $V_2=X_2=\partial y$

Similarly, Equation (1) can be easily reduced to the following form:

$$\begin{array}{c}\hfill D_t^{\alpha }u(x,t)+a{(u{(x,t)}^q)}_x+b{(u{(x,t)}^p)}_{xxx}=0.\end{array}$$

(26)

If the fractional Lie group method is applied again to Equation (26), then its corresponding vector fields are

$$\begin{array}{cc}\hfill & \widetilde{X_1}=\partial x,\hfill \\ \hfill & \widetilde{X_2}={\displaystyle \frac{p-q}{p-3q+2}}x\partial x+{\displaystyle \frac{t}{\alpha }}\partial t+{\displaystyle \frac{2u}{p-3q+2}}\partial u.\hfill \end{array}$$

(27)

Case III: $V_4=k{X}_1\pm X_2$

Utilizing the vector field $V_4$, Equation (1) becomes the following form:

$$\begin{array}{c}\hfill D_t^{\alpha }u(t)=0.\end{array}$$

(28)

Case IV: $V_3=X_3={\displaystyle \frac{p-q}{p-3q+2}}x\partial x+{\displaystyle \frac{r-q}{p-3q+2}}y\partial y+{\displaystyle \frac{t}{\alpha }}\partial t+{\displaystyle \frac{2u}{p-3q+2}}\partial u$

For $X_3$, it has

$$\begin{array}{c}\hfill {\displaystyle \frac{dx}{(p-q)x}}={\displaystyle \frac{dy}{(r-q)y}}={\displaystyle \frac{\alpha dt}{(p-3q+2)t}}={\displaystyle \frac{du}{2u}}.\end{array}$$

(29)

Solving Equation (29), we obtain

$$\begin{array}{c}\hfill {\xi }_1=x{t}^{{\displaystyle \frac{q-p}{p-3q+2}}\alpha },{\xi }_2=y{t}^{{\displaystyle \frac{q-r}{p-3q+2}}\alpha }\end{array}$$

(30)

and

$$\begin{array}{c}\hfill u(x,y,t)=t^{{\displaystyle \frac{2}{p-3q+2}}\alpha }f({\xi }_1,{\xi }_2).\end{array}$$

(31)

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 ${\xi }_1=x{t}^{{\displaystyle \frac{q-p}{p-3q+2}}\alpha },{\xi }_2=y{t}^{{\displaystyle \frac{q-r}{p-3q+2}}\alpha }$ and the group-invariant solution $u(x,y,t)=t^{{\displaystyle \frac{2}{p-3q+2}}\alpha }f({\xi }_1,{\xi }_2)$ were used to reduce the (2+1)-dimensional ZK $(q,p,r)$ equation of the time fractional derivative into the following (1+1)-dimensional fractional-order form:

$$(P_{{\displaystyle \frac{p-3q+2}{\alpha (p-q)}},{\displaystyle \frac{p-3q+2}{\alpha (r-q)}}}^{1-\alpha +{\displaystyle \frac{2\alpha }{p-3q+2}},\alpha }f)({\xi }_1,{\xi }_2)+a{(f^q)}_{{\xi }_1}+b{(f^p)}_{{\xi }_1{\xi }_1{\xi }_1}+c{(f^r)}_{{\xi }_1{\xi }_1{\xi }_2}=0,$$

(32)

where

$$\begin{array}{c}\hfill (P_{{\chi }_1,{\chi }_2}^{\tau ,\alpha }f)({\xi }_1,{\xi }_2)=\prod _{j=0}^{n-1}(\tau +j-{\displaystyle \frac{1}{{\chi }_1}}{\xi }_1{\displaystyle \frac{\partial }{\partial {\xi }_1}}-{\displaystyle \frac{1}{{\chi }_2}}{\xi }_2{\displaystyle \frac{\partial }{\partial {\xi }_2}})(K_{{\chi }_1,{\chi }_2}^{\tau +\alpha ,n-\alpha }f)({\xi }_1,{\xi }_2),\\ \hfill {\chi }_1>0,{\chi }_2>0,\end{array}$$

(33)

are the two parameters of the Erdélyi–Kober fractional differential operator, with

$$\begin{array}{c}\hfill n=\left\{\begin{array}{cc}[\alpha ]+1,& \alpha \ne N,\\ \alpha ,& \alpha \in N,\end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill (K_{{\chi }_1,{\chi }_2}^{\tau ,\alpha }f)({\xi }_1,{\xi }_2)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{1}{\overset{\infty }{\int }}{(\vartheta -1)}^{\alpha -1}{\vartheta }^{-(\tau +\alpha )}f({\xi }_1{\vartheta }^{{\displaystyle \frac{1}{{\chi }_1}}},{\xi }_2{\vartheta }^{{\displaystyle \frac{1}{{\chi }_2}}})d\vartheta ,& \alpha >0;\\ f({\xi }_1,{\xi }_2),& \alpha =0.\end{array}\right.\end{array}$$

(34)

being 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:

$$(P_{{\displaystyle \frac{p-3q+2}{\alpha (p-q)}},{\displaystyle \frac{p-3q+2}{\alpha (r-q)}}}^{1-\alpha +{\displaystyle \frac{2\alpha }{p-3q+2}},\alpha }f)({\xi }_1)+a{(f^q)}_{{\xi }_1}+b{(f^p)}_{{\xi }_1{\xi }_1{\xi }_1}=0,$$

with the similarity variable ${\xi }_1=x{t}^{{\displaystyle \frac{q-p}{p-3q+2}}\alpha }$ and group-invariant solution $u(x,t)=t^{{\displaystyle \frac{2}{p-3q+2}}\alpha }f({\xi }_1)$.

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 $C=(C^t,C^x,C^y)$ needs to satisfy the following conservation equation:

$$\begin{array}{c}\hfill [D_t(C^t)+D_x(C^x)+D_y(C^y)]{|}_{Eq.(1)}=0.\end{array}$$

(35)

The formal Lagrangian of Equation (1) was written as follows:

$$\begin{array}{c}\hfill L=v[D_t^{\alpha }u+a{(u^q)}_x+b{(u^p)}_{xxx}+c{(u^r)}_{xyy}],\end{array}$$

(36)

where $v=v(x,y,t)$ is a smooth function.

Further, Equation (36) was expressed by the following integral form:

$$\begin{array}{c}\hfill {\int }_0^T{\int }_{{\Omega }_x}{\int }_{{\Omega }_y}L(x,y,t,u,v,D_t^{\alpha }u,u_x,u_{xxx},u_{xyy},\dots )dydxdt.\end{array}$$

(37)

At the same time, it corresponds to the Euler–Lagrangian operator of the form

$$\begin{array}{c}\hfill {\displaystyle \frac{\delta }{\delta u}}={\displaystyle \frac{\partial }{\partial u}}+{(D_t^{\alpha })}^{*}{\displaystyle \frac{\partial }{\partial D_t^{\alpha }u}}-D_x{\displaystyle \frac{\partial }{\partial u_x}}-D_x^3{\displaystyle \frac{\partial }{\partial u_{xxx}}}-D_x{D}_y^2{\displaystyle \frac{\partial }{\partial u_{xyy}}},\end{array}$$

(38)

where ${(D_t^{\alpha })}^{*}={}_{t}I_T^{n-\alpha }{D}_t^n$ is the adjoint operator of $D_t^{\alpha }$.

Equation (1) has the following adjoint equation:

$$\begin{array}{c}\hfill {\displaystyle \frac{\delta L}{\delta u}}=0.\end{array}$$

(39)

If we are considering the dependent variable $u=u(x,y,t)$, then we obtain the result

$$\begin{array}{c}\hfill \overline{V}+D_t({\xi }_t)I+D_x({\xi }_x)I+D_y({\xi }_y)I=W{\displaystyle \frac{\delta }{\delta u}}+D_t(C^t)+D_x(C^x)+D_y(C^y),\end{array}$$

(40)

where ${\displaystyle \frac{\delta }{\delta u}}$ is the Euler–Lagrange operator and I is the identity operator.

The result $\overline{V}$ and the Lie characteristic equation W, respectively, are

$$\begin{array}{c}\hfill \overline{V}={\xi }_x\partial x+{\xi }_y\partial y+{\xi }_t\partial t+{\xi }_u\partial u+{\xi }_u^{\alpha }\partial D_t^{\alpha }(u)+{\xi }_u^x\partial u_x+{\xi }_u^{xxx}\partial u_{xxx}+{\xi }_u^{xyy}\partial u_{xyy}\end{array}$$

(41)

and

$$\begin{array}{c}\hfill W={\xi }_u-{\xi }_t{u}_t-{\xi }_x{u}_x-{\xi }_y{u}_y.\end{array}$$

(42)

From vector fields (13) of Equation (1), the Lie characteristic equation is expressed by

$$\begin{array}{cc}\hfill & W_1=-u_x,W_2=-u_y,\hfill \\ \hfill & W_3={\displaystyle \frac{2u}{p-3q+2}}-{\displaystyle \frac{t}{\alpha }}{u}_t-{\displaystyle \frac{p-q}{p-3q+2}}x{u}_x-{\displaystyle \frac{r-q}{p-3q+2}}y{u}_y.\hfill \end{array}$$

(43)

The fractional generalizations of the Noether operators [18,19,20] about the t-component of the conserved vector give

$$\begin{array}{c}\hfill C^t={\xi }_{t}L+\sum _{k=0}^{n-1}{(-1)}^k{}_{0}D_t^{\alpha -1-k}(W_i)D_t^k({\displaystyle \frac{\partial L}{\partial {}_{0}D_t^{\alpha }(u)}})-{(-1)}^{n}J(W_i,D_t^n{\displaystyle \frac{\partial L}{\partial {}_{0}D_t^{\alpha }(u)}}),\end{array}$$

(44)

with

$$J(f,g)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_o^t{\int }_t^T{\displaystyle \frac{f(\tau ,x,y)g(\mu ,x,y)}{{(\mu -\tau )}^{\alpha +1-n}}}d\mu d\tau ,$$

and where the x and y components of the conserved vectors are

$$\begin{array}{cc}\hfill & C^j=X^{j}L+W_i[{\displaystyle \frac{\partial L}{\partial u_J^{\beta }}}-D_l({\displaystyle \frac{\partial L}{\partial u_{jl}^{\beta }}})+D_l{D}_s({\displaystyle \frac{\partial L}{\partial u_{jls}^{\beta }}})-]+D_l(W_i)[{\displaystyle \frac{\partial L}{\partial u_{jl}^{\beta }}}-D_s({\displaystyle \frac{\partial L}{\partial u_{jls}^{\beta }}})+\dots ]\hfill \\ \hfill & +D_l{D}_s(W_i)({\displaystyle \frac{\partial L}{\partial u_{jls}^{\beta }}}-\dots )+\dots ,\hfill \end{array}$$

(45)

where $X^x={\xi }_x,X^y={\xi }_y$ and $\beta =1,2,3$.

By considering Expressions (44) and (45), the $t-,x-$ and $y-$ components of the conserved vectors of the (2+1)-dimensional ZK $(q,p,r)$ Equation (1) are given by, respectively, the following:

$$\begin{array}{cc}\hfill C^t=& T_i·L+\sum _{k=0}^{n-1}{{(-1)}^k}_0{D}_t^{\alpha -1-k}(W_j)D_t^k({\displaystyle \frac{\partial L}{{\partial }_0{D}_t^{\alpha }u}})-{(-1)}^{n}J(W_j,D_t^n({\displaystyle \frac{\partial L}{{\partial }_0{D}_t^{\alpha }u}})),\hfill \\ \hfill C^x=& X_i·L+W_j[{\displaystyle \frac{\partial L}{\partial u_x}}+D_y{D}_y({\displaystyle \frac{\partial L}{\partial u_{xyy}}})+D_x{D}_x({\displaystyle \frac{\partial L}{\partial u_{xxx}}})]+D_x(W_j)[-D_x({\displaystyle \frac{\partial L}{\partial u_{xxx}}})]\hfill \\ \hfill & +D_y(W_j)[-D_y({\displaystyle \frac{\partial L}{\partial u_{xyy}}})]+D_x{D}_x(W_j)({\displaystyle \frac{\partial L}{\partial u_{xxx}}})+D_y{D}_y(W_j)({\displaystyle \frac{\partial L}{\partial u_{xyy}}}),\hfill \\ \hfill C^y=& Y_i·L+W_j[D_x{D}_y({\displaystyle \frac{\partial L}{\partial u_{xyy}}})]-D_x(W_j)[D_y({\displaystyle \frac{\partial L}{\partial u_{xyy}}})]-D_y(W_j)[D_x({\displaystyle \frac{\partial L}{\partial u_{xyy}}})]\hfill \\ \hfill & +D_x{D}_y(W_j)({\displaystyle \frac{\partial L}{\partial u_{xyy}}})\hfill \end{array}$$

(46)

where $T_i,X_i$ and $Y_i(i=1,2,3)$ correspond, respectively, to $W_j(j=1,2,3)$ (43) as follows:

$$\begin{array}{cc}\hfill & T_1=T_2=0,T_3={\displaystyle \frac{t}{\alpha }},\hfill \\ \hfill & X_1=1,X_2=0,X_3={\displaystyle \frac{(p-q)x}{p-3q+2}},\hfill \\ \hfill & Y_1=0,Y_2=1,Y_3={\displaystyle \frac{(r-q)y}{p-3q+2}}.\hfill \end{array}$$

7. Conclusions and Discussions

In this article, we restudied the (2+1)-dimensional Zakharov0–Kuznetsov $(q,p,r)$ 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.