An Analysis of the Lie Symmetry and Conservation Law of a Variable-Coefficient Generalized Calogero–Bogoyavlenskii–Schiff Equation in Nonlinear Optics and Plasma Physics

Abstract

In this paper, the symmetries and conservation laws of a variable-coefficient generalized Calogero–Bogoyavlenskii–Schiff (vcGCBS) equation are investigated by modeling the propagation of long waves in nonlinear optics, fluid dynamics, and plasma physics. A Painlevé analysis is applied using the Kruskal-simplified form of the Weiss–Tabor–Carnevale (WTC) method, which shows that the vcGCBS equation does not possess the Painlevé property. Under the compatibility condition ($a_1\left(t\right)=a_2\left(t\right)$), infinitesimal generators and a symmetry analysis are presented via the symbolic computation program designed. With the Lagrangian, the adjoint equation is analyzed, and the vcGCBS equation is shown to possess nonlinear self-adjointness. Based on its nonlinear self-adjointness, conservation laws for the vcGCBS equation are derived by means of Ibragimov’s conservation theorem for each Lie symmetry.

1. Introduction

Nonlinear science is an emerging interdisciplinary field that studies the common properties of nonlinear phenomena. A significant number of nonlinear phenomena originating from mechanics, physics, or engineering can be abstracted into nonlinear evolution equations. Investigating the symmetries and conservation laws of nonlinear evolution equations is crucial to solving these equations [1,2,3,4].

The generalized Calogero–Bogoyavlenskii–Schiff (GCBS) equation is a (2 + 1)-dimensional nonlinear evolution equation [5]

$$u_{xt}+3u_x{u}_{xy}+3u_y{u}_{xx}+a{u}_{xy}+b{u}_{yy}+u_{xxxy}=0,$$

(1)

where $a,b$ are arbitrary constants; $x,y$ are spatial coordinates; and t is a temporal coordinate, with $(x,y)\in {\mathbb{R}}^2,t\in \mathbb{R}$. The GCBS Equation (1) is a classical two-dimensional nonlinear model involving stronger nonlinear terms and mixed partial derivatives of the fourth order, which can play a key role in the study of two-dimensional nonlinear models with its complex and representative structure. This equation describes the propagation of long waves in nonlinear dispersive media and has many applications in nonlinear optics, fluid dynamics, and plasma physics related to Riemann waves [6,7].

When $a=b=0$, the GCBS equation (1) can be reduced to the CBS equation,

$$u_{xt}+u_x{u}_{xy}+u_y{u}_{xx}+u_{xxxy}=0.$$

(2)

Many novel results in terms of solutions to the GCBS equation have been constructed in recent years. For the GCBS equation and its related forms, explicit solutions such as soliton solutions, localized wave solutions, periodic wave solutions, and lump solutions have been obtained [5,7,8,9]. By employing the Hirota bilinear method, the structure of the solutions can be studied further [6,10,11,12]. Using the Lie symmetry method, the symmetry generators of the GCBS equation can be presented, allowing for the original equation to be reduced into an ordinary differential equation [4]. The Lax pair and multiple conservation laws for a generalized (2 + 1)-dimensional coupled CBS-type equation have been provided in the context of biofluid dynamics [13].

When considering non-uniform media and non-uniform boundaries, variable-coefficie-nt nonlinear evolution equations can capture the complexity of real-world nonlinear phenomena better compared to their constant-coefficient counterparts and have therefore attracted considerable attention [14,15,16]. Based on the GCBS equation (1), in this paper, we will mainly consider the variable-coefficient GCBS equation, a more generalized case of Equation (1),

$$u_{xt}+a_1\left(t\right)u_x{u}_{xy}+a_2\left(t\right)u_y{u}_{xx}+a_3\left(t\right)u_{xy}+a_4\left(t\right)u_{yy}+a_5\left(t\right)u_{xxxy}=0,$$

(3)

where $a_1\left(t\right),a_2\left(t\right),a_3\left(t\right),a_4\left(t\right)$, and $a_5\left(t\right)$ are arbitrary differentiable functions of t.

The variable-coefficient version of the GCBS equation has wider applications to the assortment of models from nonlinear physics with a more complicated structure, and solutions to the vcGCBS equation are used in many branches of science and engineering. Certain properties of the vcGCBS equation, like its conservation laws, have significance for the analytical solutions and reveal traveling wave profiles in different fields of nonlinear science [4]. Thus, we look into an analysis of the Lie symmetry and develop the conservation law for the variable-coefficient GCBS equation further.

The Lie symmetry analysis method is a widely used and important tool in the field of mathematical physics. It provides a systematic and precise approach to obtaining exact or similar solutions to differential equations [17,18,19,20]. This approach not only offers a formal framework for generating solutions through invariance conditions [21] but also enables the derivation of optimal systems and subalgebras using higher-order extensions of the infinitesimal operators [22]. These capabilities make the Lie symmetry analysis method an efficient and structured tool for solving complex differential equations. By applying this method to the variable-coefficient GCBS equation, the symmetry generators can be identified, allowing the original partial differential equation to be reduced into a simpler, ordinary differential equation [4]. Symbolic computation is introduced into the operations of one-parameter Lie groups innovatively, which enhances the computational efficiency and applicability of the method.

The structure of this paper is listed as follows. In Section 2, the variable-coefficient GCBS equation with five variable coefficients will be analyzed using the Weiss–Tabor–Carnevale (WTC) test for Painlevé analysis. In Section 3, the infinitesimal generators and symmetry groups of the constrained variable-coefficient GCBS equation will be presented using the Lie group method. In Section 4, by virtue of symbolic computation, the nonlinear self-adjointness of the variable-coefficient GCBS equation will be presented. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws will be constructed. Conclusions and discussions are given in the final section.

2. Painlevé Integrability

Painlevé integrability plays a crucial role in studying the structure of solutions for nonlinear evolution equations. If a partial differential equation (PDE) cannot be reduced into a Painlevé-type ordinary differential equation (ODE) through similarity reduction, then the original PDE is non-integrable [23].

We analyze the Painlevé integrability of the variable-coefficient GCBS equation (3) using the WTC method. We assume that the solutions to (3) have the form of the following generalized Laurent expansion [23,24,25]:

$$u={\varphi }^p\sum _{j=0}^{\infty }{u}_j{\varphi }^j,$$

where $\varphi (x,y,t)$ is an arbitrary function of $x,y,t$, and $u_j(j=0,1,2,\cdots )$ are analytic functions of $x,y,t$ in the neighborhood of the noncharacteristic movable singularity manifold defined by $\varphi (x,y,t)=0$.

Through leading-order analysis, one can obtain

$$\begin{array}{cc}& p=-1,\hfill \\ & u_0={\displaystyle \frac{12a_5\left(t\right)}{a_1\left(t\right)+a_2\left(t\right)}}.\hfill \end{array}$$

Let $\varphi (x,y,t)=x+\psi (y,t)$, where $\psi (y,t)$ is an auxiliary function. Substituting

$$u={\varphi }^{-1}\sum _{j=0}^{\infty }{u}_j{\varphi }^j,$$

into Equation (3) and organizing the coefficients of $u_j$, we can obtain

$$(j+1)(j-1)(j-4)(j-6)u_j=F_j,$$

where $F_j$ is a composite expression involving $u_i(i<j)$.

It is found that resonance occurs at $-1,1,4,$ and 6. $j=-1$ corresponds to the arbitrariness of the function $\varphi$, and the compatibility condition always holds for $j=1$. For $j=4$, the compatibility condition holds if and only if

$$a_1\left(t\right)=a_2\left(t\right).$$

(4)

Setting $a_1\left(t\right)=a_2\left(t\right)$, the compatibility condition for $j=6$ becomes

$${\psi }_{ty}^2-{\psi }_{tt}{\psi }_{yy}+3{\psi }_{yy}{C}_{1_{yy}}(C_{1_{yt}}-3{\psi }_{ty})=0,$$

where $C_1$ is an arbitrary function of $y,t$. Since this condition does not always hold, the variable-coefficient GCBS equation (3) does not possess the Painlevé property.

3. Lie Symmetry Analysis

3.1. Infinitesimal Generators and Lie Symmetry

Based on the results on the Painlevé integrability, we set $a_1\left(t\right)=a_2\left(t\right)$ and analyse the Lie symmetry of the constrained variable-coefficient GCBS Equation (5)

$$u_{xt}+a_2\left(t\right)u_x{u}_{xy}+a_2\left(t\right)u_y{u}_{xx}+a_3\left(t\right)u_{xy}+a_4\left(t\right)u_{yy}+a_5\left(t\right)u_{xxxy}=0.$$

(5)

We assume that the variable-coefficient GCBS equation (5) is invariant under a one-parameter Lie group of point transformations [18]

$$\left\{\begin{array}{c}{x}^{*}=x+\epsilon {\xi }^x(x,y,t,u)+O\left({\epsilon }^2\right),\hfill \\ y^{*}=y+\epsilon {\xi }^y(x,y,t,u)+O\left({\epsilon }^2\right),\hfill \\ t^{*}=t+\epsilon \tau (x,y,t,u)+O\left({\epsilon }^2\right),\hfill \\ u^{*}=u+\epsilon \eta (x,y,t,u)+O\left({\epsilon }^2\right),\hfill \end{array}\right.$$

(6)

where $\epsilon$ is a small parameter; functions ${\xi }^x,{\xi }^y,\tau$, and $\eta$ are infinitesimals depending on $x,y,t,u$; and $O\left({\epsilon }^2\right)$ denotes the terms of order ${\epsilon }^2$.

The corresponding infinitesimal generator is expressed as

$$V={\xi }^x(x,y,t,u){\displaystyle \frac{\partial }{\partial x}}+{\xi }^y(x,y,t,u){\displaystyle \frac{\partial }{\partial y}}+\tau (x,y,t,u){\displaystyle \frac{\partial }{\partial t}}+\eta (x,y,t,u){\displaystyle \frac{\partial }{\partial u}},$$

(7)

Then, vector field (7) generates the symmetry of the variable-coefficient GCBS equation (5), and V must satisfy the following Lie symmetry condition

$${\mathrm{pr}}^{\left(4\right)}{V\left(F\right)|}_{F=0}=0,$$

(8)

where ${\mathrm{pr}}^{\left(4\right)}V$ is the fourth prolongation of the infinitesimal generator V [19], and F satisfies

$$F=u_{xt}+a_2\left(t\right)u_x{u}_{xy}+a_2\left(t\right)u{u}_{xx}+a_3\left(t\right)u_{xy}+a_4\left(t\right)u_{yy}+a_5\left(t\right)u_{xxxy}.$$

(9)

Expanding Equation (8) and splitting based on the derivatives of u lead to the following system of determining equations via symbolic computation:

$$\left\{\begin{array}{cc}\hfill & {\xi }_y^x={\xi }_{xx}^x=0,\hfill \\ \hfill & {\xi }_x^y=0,\hfill \\ \hfill & {\tau }_x={\tau }_y=0,\hfill \\ \hfill & {\eta }_{xu}={\eta }_{yu}={\eta }_{uu}={\eta }_{ut}=0,\hfill \\ \hfill & 3{\xi }_x^x={\xi }_y^y=-3{\eta }_u,\hfill \\ \hfill & {\tau }_t=2{\xi }_x^x+{\xi }_y^y,\hfill \\ \hfill & a_2\left(t\right){\eta }_{xx}=a_4\left(t\right){\xi }_{yy}^y,\hfill \\ \hfill & a_3\left(t\right){\eta }_{xy}+a_4\left(t\right){\eta }_{yy}+{\eta }_{xt}=0,\hfill \\ \hfill & a_2\left(t\right){\eta }_y={\xi }_t^x,\hfill \\ \hfill & {\xi }_t^y=a_2\left(t\right){\eta }_x+2a_3\left(t\right){\xi }_x^x.\hfill \end{array}\right.$$

(10)

Solving the above equations yields the following infinitesimals,

$$\left\{\begin{array}{cc}\hfill & {\xi }^x=c_{1}x+f\left(t\right),\hfill \\ \hfill & {\xi }^y=3c_{1}y+\int (c_2{a}_2\left(t\right)+2c_1{a}_3\left(t\right))dt+c_3,\hfill \\ \hfill & \tau =5c_{1}t+c_4,\hfill \\ \hfill & \eta =c_{2}x-c_{1}u+{\displaystyle \frac{f^{\prime }\left(t\right)}{a_2\left(t\right)}}y+g\left(t\right),\hfill \end{array}\right.$$

(11)

where $c_i(i=1,2,3,4)$ are arbitrary constants, and $f\left(t\right),g\left(t\right)$ are arbitrary functions of t.

(i)

$f^{\prime }\left(t\right)=0$.

Based on the arbitrary constants $c_i(i=1,2,3,4)$ and $f\left(t\right)$, as well as the arbitrary function $g\left(t\right)$, the Lie algebra of the variable-coefficient GCBS equation is spanned via the following Lie symmetry generators

$$\left\{\begin{array}{cc}\hfill & V_1={\displaystyle \frac{\partial }{\partial x}},\hfill \\ \hfill & V_2={\displaystyle \frac{\partial }{\partial y}},\hfill \\ \hfill & V_3={\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill & V_4=g\left(t\right){\displaystyle \frac{\partial }{\partial u}},\hfill \\ \hfill & V_5=x{\displaystyle \frac{\partial }{\partial u}}+(\int a_2\left(t\right)dt){\displaystyle \frac{\partial }{\partial y}},\hfill \\ \hfill & V_6=x{\displaystyle \frac{\partial }{\partial x}}+(3y+2\int a_3\left(t\right)dt){\displaystyle \frac{\partial }{\partial y}}+5t{\displaystyle \frac{\partial }{\partial t}}-u{\displaystyle \frac{\partial }{\partial u}}.\hfill \end{array}\right.$$

(12)

Hence, the one-parameter Lie symmetry groups $G\left(\epsilon \right)$ generated by $V_i(i=1,2,\cdots ,6)$ are

$$\left\{\begin{array}{cc}{G}_1:(x,y,t,u)\hfill & \to (x+\epsilon ,y,t,u),\hfill \\ G_2:(x,y,t,u)\hfill & \to (x,y+\epsilon ,t,u),\hfill \\ G_3:(x,y,t,u)\hfill & \to (x,y,t+\epsilon ,u),\hfill \\ G_4:(x,y,t,u)\hfill & \to (x,y,t,u+\epsilon g(t\left)\right),\hfill \\ G_5:(x,y,t,u)\hfill & \to (x,y+(\int a_2\left(t\right)dt)\epsilon ,t,u+x\epsilon ),\hfill \\ G_6:(x,y,t,u)\hfill & \to (e^{\epsilon }x,e^{3\epsilon }y+e^{2\epsilon }\int a_3\left(t\right)dt,e^{5\epsilon }t,e^{-\epsilon }u).\hfill \end{array}\right.$$

(13)

(ii)

$f^{\prime }\left(t\right)\ne 0$.

The Lie algebra of the variable-coefficient GCBS equation is spanned via the following Lie symmetry generators:

$$\left\{\begin{array}{cc}\hfill & V_1=f\left(t\right){\displaystyle \frac{\partial }{\partial x}}+{\displaystyle \frac{f^{\prime }\left(t\right)}{a_2\left(t\right)}}y{\displaystyle \frac{\partial }{\partial u}},\hfill \\ \hfill & V_2={\displaystyle \frac{\partial }{\partial y}},\hfill \\ \hfill & V_3={\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill & V_4=g\left(t\right){\displaystyle \frac{\partial }{\partial u}},\hfill \\ \hfill & V_5=x{\displaystyle \frac{\partial }{\partial u}}+(\int a_2\left(t\right)dt){\displaystyle \frac{\partial }{\partial y}},\hfill \\ \hfill & V_6=x{\displaystyle \frac{\partial }{\partial x}}+(3y+2\int a_3\left(t\right)dt){\displaystyle \frac{\partial }{\partial y}}+5t{\displaystyle \frac{\partial }{\partial t}}-u{\displaystyle \frac{\partial }{\partial u}}.\hfill \end{array}\right.$$

(14)

The one-parameter Lie symmetry groups $G\left(\epsilon \right)$ generated by $V_i(i=1,2,\cdots ,6)$ are

$$\left\{\begin{array}{cc}{G}_1:(x,y,t,u)\hfill & \to (x+\epsilon f\left(t\right),y,t,u+{\displaystyle \frac{f^{\prime }\left(t\right)}{a_2\left(t\right)}}\epsilon ),\hfill \\ G_2:(x,y,t,u)\hfill & \to (x,y+\epsilon ,t,u),\hfill \\ G_3:(x,y,t,u)\hfill & \to (x,y,t+\epsilon ,u),\hfill \\ G_4:(x,y,t,u)\hfill & \to (x,y,t,u+\epsilon g(t\left)\right),\hfill \\ G_5:(x,y,t,u)\hfill & \to (x,y+(\int a_2\left(t\right)dt)\epsilon ,t,u+x\epsilon ),\hfill \\ G_6:(x,y,t,u)\hfill & \to (e^{\epsilon }x,e^{3\epsilon }y+e^{2\epsilon }\int a_3\left(t\right)dt,e^{5\epsilon }t,e^{-\epsilon }u).\hfill \end{array}\right.$$

(15)

3.2. The Lie Algebra Commutation Table

When $f^{\prime }\left(t\right)=0$, the Lie bracket $[V_i,V_j]=V_i{V}_j-V_j{V}_i$ [18] can be used to derive the Lie algebra commutation table.

If $g^{\prime }\left(t\right)=0$, the Lie algebra commutator table is as shown in

Table 1. Commutator table when $g^{\prime }\left(t\right)=0$.

$[{\mathit{V}}_{\mathit{i}},{\mathit{V}}_{\mathit{j}}]$	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$	${\mathit{V}}_5$	${\mathit{V}}_6$
$V_1$	0	0	0	0	${\displaystyle \frac{1}{g\left(t\right)}}{V}_4$	$V_1$
$V_2$	0	0	0	0	0	$3V_2$
$V_3$	0	0	0	0	$a_2\left(t\right)V_2$	$2a_3\left(t\right)V_2+5V_3$
$V_4$	0	0	0	0	0	$-V_4$
$V_5$	$-{\displaystyle \frac{1}{g\left(t\right)}}{V}_4$	0	$-a_2\left(t\right)V_2$	0	0	$-{\displaystyle \frac{5x}{g\left(t\right)}}{V}_4+3V_5-5a_2\left(t\right)t{V}_2$
$V_6$	$-V_1$	$-3V_2$	$-2a_3\left(t\right)V_2-5V_3$	$V_4$	${\displaystyle \frac{5x}{g\left(t\right)}}{V}_4-3V_5+5a_2\left(t\right)t{V}_2$	0

.

If $g^{\prime }\left(t\right)\ne 0$, the Lie algebra commutator table is as shown in

Table 2. Commutator table when $g^{\prime }\left(t\right)\ne 0$.

$[{\mathit{V}}_{\mathit{i}},{\mathit{V}}_{\mathit{j}}]$	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$	${\mathit{V}}_5$	${\mathit{V}}_6$
$V_1$	0	0	0	0	${\displaystyle \frac{1}{g\left(t\right)}}{V}_4$	$V_1$
$V_2$	0	0	0	0	0	$3V_2$
$V_3$	0	0	0	${\displaystyle \frac{g^{\prime }\left(t\right)}{g\left(t\right)}}{V}_4$	$a_2\left(t\right)V_2$	$-2a_3\left(t\right)V_2-5V_3$
$V_4$	0	0	$-{\displaystyle \frac{g^{\prime }\left(t\right)}{g\left(t\right)}}{V}_4$	0	0	$-{\displaystyle \frac{5t{g}^{\prime }\left(t\right)+g\left(t\right)}{g\left(t\right)}}{V}_4$
$V_5$	$-{\displaystyle \frac{1}{g\left(t\right)}}{V}_4$	0	$-a_2\left(t\right)V_2$	0	0	$-{\displaystyle \frac{5x}{g\left(t\right)}}{V}_4+3V_5-5a_2\left(t\right)t{V}_2$
$V_6$	$-V_1$	$-3V_2$	$-2a_3\left(t\right)V_2-5V_3$	${\displaystyle \frac{5t{g}^{\prime }\left(t\right)+g\left(t\right)}{g\left(t\right)}}{V}_4$	${\displaystyle \frac{5x}{g\left(t\right)}}{V}_4-3V_5+5a_2\left(t\right)t{V}_2$	0

.

4. Nonlinear Self-Adjointness and Conservation Laws

Ibragimov proposed the concept of adjoint equations based on variational derivatives, forming a new system by combining the adjoint equation with the original equation to study conservation laws [26,27]. Using the new Lagrangian operator, we can derive the conservation laws. We will construct the conservation vectors for the variable-coefficient GCBS equation (5) and verify its conservation laws.

4.1. Nonlinear Self-Adjointness

To investigate the self-adjointness of Equation (5), we first introduce the following formal Lagrangian [26]:

$$\mathcal{L}=v\left(a_2\left(t\right)u_x{u}_{xy}+a_2\left(t\right)u_y{u}_{xx}+a_5\left(t\right)u_{xxxy}+u_{xt}+a_3\left(t\right)u_{xy}+a_4\left(t\right)u_{yy}\right),$$

(16)

where v is a new independent variable. The adjoint equation is defined as

$$F^{*}={\displaystyle \frac{\delta \mathcal{L}}{\delta u}}=0,$$

(17)

where

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta }{\delta u}}=& {\displaystyle \frac{\partial }{\partial u}}-D_x{\displaystyle \frac{\partial }{\partial u_x}}-D_y{\displaystyle \frac{\partial }{\partial u_y}}+D_x{D}_x{\displaystyle \frac{\partial }{\partial u_{xx}}}+D_x{D}_t{\displaystyle \frac{\partial }{\partial u_{xt}}}+D_x{D}_y{\displaystyle \frac{\partial }{\partial u_{xy}}}+D_y{D}_y{\displaystyle \frac{\partial }{\partial u_{yy}}}\hfill \\ \hfill & +D_x{D}_x{D}_x{D}_y{\displaystyle \frac{\partial }{\partial u_{xxxy}}},\hfill \end{array}$$

(18)

is the Euler–Lagrange operator. $D_t,D_x$ and $D_y$ are the total derivatives with respect to $t,x,$ and y, respectively [27].

Substituting (5) and (17) into (18), one can obtain the adjoint equation

$$F^{*}=2a_2\left(t\right)v_x{u}_{xy}+a_2\left(t\right)v_{xx}{u}_y+a_2\left(t\right)v_{xy}{u}_x+v_{xt}+a_3\left(t\right)v_{xy}+a_4\left(t\right)v_{yy}+a_5\left(t\right)v_{xxxy}=0.$$

(19)

To find the relationship between v and $u,x,y,t$, restricting u to a solution to Equation (5), let

$$F^{*}{|}_{v=\varphi (x,y,t,u)}=\lambda F,$$

where $\varphi (x,y,t,u)\ne 0,\lambda$ is an undetermined coefficient. Substituting the expressions of F and $F^{*}$, one can obtain

$$\left\{\begin{array}{cc}\hfill & {\varphi }_u=\lambda =0,\hfill \\ \hfill & {\varphi }_x=0,\hfill \\ \hfill & b{\varphi }_{yy}=0.\hfill \end{array}\right.$$

(20)

Thus, we can derive

$$\varphi (x,y,t,u)=c_1\left(t\right)y+c_2\left(t\right).$$

(21)

Since $\varphi (x,y,t,u)\ne 0$, the equation is nonlinearly self-adjoint.

4.2. Conservation Laws

The conservation law for Equation (5) is defined by a vector field $(C^t,C^x,C^y)$, where $(C^t=C^t(x,y,t,u,v,\cdots ),$$C^x=C^x(x,y,t,u,v,\cdots ),C^y=C^y(x,y,t,u,v,\cdots ))$ is called a conserved vector for Equation (5) if it satisfies the conservation equation [26,27,28]

$$D_t\left(C^t\right)+D_x\left(C^x\right)+D_y\left(C^y\right)=0$$

(22)

upon all solutions of Equation (5).

According to the technique in Ref. [28], Equation (5) has a second-order Lagrangian function of the following form:

$$\mathcal{L}=v\left(a_2\left(t\right)u_x{u}_{xy}+a_2\left(t\right)u_y{u}_{xx}+u_{xt}+a_3\left(t\right)u_{xy}+a_4\left(t\right)u_{yy}\right)+a_5\left(t\right)u_{xx}{v}_{xy}.$$

(23)

Based on the second-order Lagrangian (23), the conserved vector $C^i$ can be constructed as

$$C^i={\xi }^i\mathcal{L}+w_1\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_i}}-D_k\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{ik}}}\right)\right]+D_k\left(w_1\right){\displaystyle \frac{\partial \mathcal{L}}{\partial u_{ik}}}+w_2\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial v_i}}-D_k\left({\displaystyle \frac{\partial \mathcal{L}}{\partial v_{ik}}}\right)\right]+D_k\left(w_2\right){\displaystyle \frac{\partial \mathcal{L}}{\partial v_{ik}}},$$

(24)

where $w_1=\eta -{\xi }^j{u}_j,w_2={\eta }^{*}-{\xi }^j{v}_j,{\eta }^{*}=-[\lambda +D_i\left({\xi }^i\right)]v$, and $\lambda$ is determined by $X\left(F\right)=\lambda F$, with X being the extended generator of the infinitesimal generator $V_i$ of Equation (5).

For Equation (5), the conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& \tau \mathcal{L}-w_1{D}_x\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{tx}}}\right)+D_x\left(w_1\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_{tx}}}\right],\hfill \\ \hfill C^x=& {\xi }^x\mathcal{L}+w_1\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_x}}-D_x\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{xx}}}\right)-D_t\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{xt}}}\right)-D_y\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{xy}}}\right)\right]+D_x\left(w_1\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_{xx}}}\right]\hfill \\ \hfill & +D_t\left(w_1\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_{xt}}}\right]+D_y\left(w_1\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_{xy}}}\right]+w_2\left[-D_y\left({\displaystyle \frac{\partial \mathcal{L}}{\partial v_{xy}}}\right)\right]+D_y\left(w_2\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial v_{xy}}}\right],\hfill \\ \hfill C^y=& {\xi }^y\mathcal{L}+w_1\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_y}}-D_x\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{yx}}}\right)-D_y\left({\displaystyle \frac{\partial \mathcal{L}}{\partial u_{yy}}}\right)\right]+D_x\left(w_1\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_{yx}}}\right]+D_y\left(w_1\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial u_{yy}}}\right]\hfill \\ \hfill & +w_2\left[-D_x\left({\displaystyle \frac{\partial \mathcal{L}}{\partial v_{yx}}}\right)\right]+D_x\left(w_2\right)\left[{\displaystyle \frac{\partial \mathcal{L}}{\partial v_{yx}}}\right],\hfill \end{array}$$

(25)

where

$$w_1=\eta -\tau u_t-{\xi }^x{u}_x-{\xi }^y{u}_y,w_2={\eta }^{*}-\tau v_t-{\xi }^x{v}_x-{\xi }^y{v}_y.$$

Calculating the derivatives of each term containing u in $\mathcal{L}$ and substituting them into the expression for the conserved vector (25), we have

$$\begin{array}{cc}\hfill C^t& =\tau \mathcal{L}-{\displaystyle \frac{1}{2}}{w}_1{v}_x+{\displaystyle \frac{1}{2}}{D}_x\left(w_1\right)v,\hfill \\ \hfill C^x& ={\xi }^x\mathcal{L}-w_1\left(a_2\left(t\right)v_x{u}_y+a_5\left(t\right)v_{xxy}+{\displaystyle \frac{1}{2}}{v}_t+{\displaystyle \frac{1}{2}}{a}_2\left(t\right)v_y{u}_x+{\displaystyle \frac{1}{2}}{a}_2\left(t\right)v{u}_{xy}+{\displaystyle \frac{1}{2}}{a}_3\left(t\right)v_y\right)\hfill \\ & +D_x\left(w_1\right)\left(a_2\left(t\right)v{u}_y+a_5\left(t\right)v_{xy}\right)+{\displaystyle \frac{1}{2}}{D}_y\left(w_1\right)\left(a_2\left(t\right)v{u}_x+a_3\left(t\right)v\right)+{\displaystyle \frac{1}{2}}{D}_t\left(w_1\right)v\hfill \\ & +{\displaystyle \frac{1}{2}}\left(D_y\left(w_2\right)a_5\left(t\right)u_{xx}-w_2{a}_5\left(t\right)u_{xxy}\right),\hfill \\ \hfill C^y& ={\xi }^y\mathcal{L}+{\displaystyle \frac{1}{2}}{w}_1\left(a_2\left(t\right)v{u}_{xx}-a_2\left(t\right)v_x{u}_x-a_3\left(t\right)v_x-2a_4\left(t\right)v_y\right)\hfill \\ & +{\displaystyle \frac{1}{2}}{D}_x\left(w_1\right)\left(a_2\left(t\right)v{u}_x+a_3\left(t\right)v\right)+D_y\left(w_1\right)a_4\left(t\right)v+{\displaystyle \frac{1}{2}}\left(D_x\left(w_2\right)a_5\left(t\right)u_{xx}-w_2{a}_5\left(t\right)u_{xxx}\right).\hfill \end{array}$$

where $f^{\prime }\left(t\right)=0$ and $g^{\prime }\left(t\right)\ne 0$, we solve for the conservation laws corresponding to each symmetry.

Case 1 

$V_1={\displaystyle \frac{\partial }{\partial x}}$

The corresponding infinitesimal generators and Lie characteristic functions are

$$w_1=-u_x,w_2=-v_x.$$

The corresponding conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& {\displaystyle \frac{1}{2}}{u}_x{v}_x-{\displaystyle \frac{1}{2}}{u}_{xx}v,\hfill \\ \hfill C^x=& {\displaystyle \frac{1}{2}}[a_2\left(t\right)u_x^2{v}_y+a_3\left(t\right)u_x{v}_y+2a_2\left(t\right)u_y{u}_x{v}_x+2a_5\left(t\right)u_x{v}_{xxy}-a_5\left(t\right)u_{xx}{v}_{xy}+a_5\left(t\right)u_{xxy}{v}_x\hfill \\ & +v\left(2a_4\left(t\right)u_{yy}+u_{xy}(2a_2\left(t\right)u_x+a_3\left(t\right))+u_{xt}\right)+u_x{v}_t],\hfill \\ \hfill C^y=& {\displaystyle \frac{1}{2}}[a_2\left(t\right)u_x^2{v}_x+2a_4\left(t\right)(u_x{v}_y-u_{xy}v)+a_3\left(t\right)(u_x{v}_x-u_{xx}v)-a_5\left(t\right)u_{xx}{v}_{xx}\hfill \\ & +a_5\left(t\right)u_{xxx}{v}_x-2a_2\left(t\right)u_{xx}{u}_{x}v]\hfill \end{array}$$

Case 2 

$V_2={\displaystyle \frac{\partial }{\partial y}}$

The corresponding infinitesimal generators and Lie characteristic functions are

$$w_1=-u_y,w_2=-v_y.$$

The corresponding conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& {\displaystyle \frac{1}{2}}{u}_y{v}_x-{\displaystyle \frac{1}{2}}{u}_{xy}v,\hfill \\ \hfill C^x=& {\displaystyle \frac{1}{2}}[-2u_{xy}\left(a_2\left(t\right)u_{y}v+a_5\left(t\right)v_{xy}\right)-a_5\left(t\right)u_{xx}{v}_{yy}+a_5\left(t\right)u_{xxy}{v}_y\hfill \\ & +u_y(a_2\left(t\right)u_x{v}_y+2a_2\left(t\right)u_y{v}_x+a_2\left(t\right)u_{xy}v\hfill \\ & +a_3\left(t\right)v_y+2a_5\left(t\right)v_{xxy}+v_t)-u_{yy}v(a_2\left(t\right)u_x+a_3\left(t\right))-u_{yt}v],\hfill \\ \hfill C^y=& {\displaystyle \frac{1}{2}}[2a_4\left(t\right)u_y{v}_y+a_5\left(t\right)u_{xxx}{v}_y+a_2\left(t\right)u_y{u}_x{v}_x+a_3\left(t\right)(u_y{v}_x+u_{xy}v)\hfill \\ & +a_5\left(t\right)u_{xx}{v}_{xy}+a_2\left(t\right)u_x{u}_{xy}v+a_2\left(t\right)u_y{u}_{xx}v+2u_{xt}v].\hfill \end{array}$$

Case 3 

$V_3={\displaystyle \frac{\partial }{\partial t}}$

The corresponding infinitesimal generators and Lie characteristic functions are

$$w_1=-u_t,w_2=-v_t.$$

The corresponding conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& \mathcal{L}+{\displaystyle \frac{1}{2}}{u}_t{v}_x-{\displaystyle \frac{1}{2}}{u}_{xt}v,\hfill \\ \hfill C^x=& u_t(a_2\left(t\right)v_x{u}_y+a_5\left(t\right)v_{xxy}+{\displaystyle \frac{1}{2}}{v}_t+{\displaystyle \frac{1}{2}}{a}_2\left(t\right)v_y{u}_x+{\displaystyle \frac{1}{2}}{a}_2\left(t\right)v{u}_{xy}\hfill \\ & +{\displaystyle \frac{1}{2}}{a}_3\left(t\right)v_y)-u_{xt}\left(a_2\left(t\right)v{u}_y+a_5\left(t\right)v_{xy}\right)-{\displaystyle \frac{1}{2}}{u}_{yt}\left(a_2\left(t\right)v{u}_x+a_3\left(t\right)v\right)\hfill \\ & -{\displaystyle \frac{1}{2}}{u}_{tt}v+{\displaystyle \frac{1}{2}}\left(-v_{yt}{a}_5\left(t\right)u_{xx}+v_t{a}_5\left(t\right)u_{xxy}\right),\hfill \\ \hfill C^y=& -{\displaystyle \frac{1}{2}}{u}_t\left(a_2\left(t\right)v{u}_{xx}-a_2\left(t\right)v_x{u}_x-a_3\left(t\right)v_x-2a_4\left(t\right)v_y\right)\hfill \\ & -{\displaystyle \frac{1}{2}}{u}_{xt}\left(a_2\left(t\right)v{u}_x+a_3\left(t\right)v\right)-u_{yt}{a}_4\left(t\right)v\hfill \\ & +{\displaystyle \frac{1}{2}}\left(-v_{xt}{a}_5\left(t\right)u_{xx}+a_5\left(t\right)v_t{a}_{xxx}\right).\hfill \end{array}$$

Case 4 

$V_4=g\left(t\right){\displaystyle \frac{\partial }{\partial u}}$

The corresponding infinitesimal generators and Lie characteristic functions are

$$w_1=g\left(t\right),w_2=0.$$

The corresponding conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& -{\displaystyle \frac{1}{2}}g\left(t\right)v_x,\hfill \\ \hfill C^x=& g\left(t\right)(a_2\left(t\right)v_x{u}_y+a_5\left(t\right)v_{xxy}+{\displaystyle \frac{1}{2}}{v}_t+{\displaystyle \frac{1}{2}}{a}_2\left(t\right)v_y{u}_x\hfill \\ & +{\displaystyle \frac{1}{2}}{a}_2\left(t\right)v{u}_{xy}+{\displaystyle \frac{1}{2}}{a}_3\left(t\right)v_y)+{\displaystyle \frac{1}{2}}{g}^{\prime }\left(t\right)v-{\displaystyle \frac{1}{2}}g\left(t\right)a_5\left(t\right)u_{xxy},\hfill \\ \hfill C^y=& {\displaystyle \frac{1}{2}}g\left(t\right)\left(a_2\left(t\right)v{u}_{xx}-a_2\left(t\right)v_x{u}_x-a_3\left(t\right)v_x-2a_4\left(t\right)v_y\right)-{\displaystyle \frac{1}{2}}{a}_5\left(t\right)g\left(t\right)u_{xxx}.\hfill \end{array}$$

Case 5 

$V_5=x{\displaystyle \frac{\partial }{\partial u}}+(\int a_2\left(t\right)dt){\displaystyle \frac{\partial }{\partial y}}$

The corresponding infinitesimal generators and Lie characteristic functions are

$$w_1=x-(\int a_2\left(t\right)dt)u_y,w_2=(-\int a_2\left(t\right)dt)v_y.$$

The corresponding conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& -{\displaystyle \frac{1}{2}}\left[\left(-x+(\int a_2\left(t\right)dt)u_y\right)v_x+v\left(1-(\int a_2\left(t\right)dt)u_{xy}\right)\right],\hfill \\ \hfill C^x=& {\displaystyle \frac{1}{2}}[-v\left(a_2\left(t\right)u_y+(\int a_2\left(t\right)dt)u_{yt}\right)\hfill \\ & -(\int a_2\left(t\right)dt)v{u}_{yy}\left(a_3\left(t\right)+a_2\left(t\right)u_x\right)-2(-1+(\int a_2\left(t\right)dt)u_{xy})],\hfill \\ \hfill C^y=& {\displaystyle \frac{1}{2}}[a_3\left(t\right)\left(-x+(\int a_2\left(t\right)dt)u_y\right)v_x-x{a}_2\left(t\right)u_x{v}_x+(\int a_2\left(t\right)dt)a_2\left(t\right)u_y{u}_x{v}_x\hfill \\ & +(\int a_2\left(t\right)dt)a_5\left(t\right)v_{xy}{u}_{xx}+v(a_3\left(t\right)\left(1+(\int a_2\left(t\right)dt)u_{xy}\right)+a_2\left(t\right)(u_x+x{u}_{xx})\hfill \\ & +(\int a_2\left(t\right)dt)\left(2a_4\left(t\right)u_{yy}+2u_{xt}+a_2\left(t\right)(u_x{u}_{xy}+u_y{u}_{xx})\right))\hfill \\ & +(\int a_2\left(t\right)dt)a_5\left(t\right)v_y{u}_{xxx}].\hfill \end{array}$$

Case 6 

$V_6=x{\displaystyle \frac{\partial }{\partial x}}+(3y+2\int a_3\left(t\right)dt){\displaystyle \frac{\partial }{\partial y}}+5t{\displaystyle \frac{\partial }{\partial t}}-u{\displaystyle \frac{\partial }{\partial u}}$

The corresponding infinitesimal generators and Lie characteristic functions are

$$\begin{array}{cc}& w_1=-u-5t{u}_t-\left(3y+2\int a_3\left(t\right)dt\right)u_y-x{u}_x,\hfill \\ & w_2=-2v-5t{v}_t-\left(3y+2\int a_3\left(t\right)dt\right)v_y-x{v}_x.\hfill \end{array}$$

The corresponding conserved vector can be expressed as

$$\begin{array}{cc}\hfill C^t=& {\displaystyle \frac{1}{2}}[\left(u+5t{u}_t+3y{u}_y+2(\int a_3\left(t\right)dt)u_y+x{u}_x\right)v_x\hfill \\ & +v\left(-2u_x-5t{u}_{xt}-(3y+2\int a_3\left(t\right)dt)u_{xy}-x{u}_{xx}\right)\hfill \\ & +10t\left(a_5\left(t\right)v_{xy}{u}_{xx}+v\left(a_4\left(t\right)u_{yy}+u_{xt}+a_3\left(t\right)u_{xy}+a_2\left(t\right)u_x{u}_{xy}+a_2\left(t\right)u_y{u}_{xx}\right)\right)],\hfill \\ \hfill C^x=& {\displaystyle \frac{1}{2}}[v\left(-6u_t-5t{u}_{tt}-2a_3\left(t\right)u_y-(3y+2\int a_3\left(t\right)dt)u_{yt}-x{u}_{xt}\right)\hfill \\ & +v(a_3\left(t\right)+a_2\left(t\right)u_x)\left(-4u_y-5t{u}_{yt}-(3y+2\int a_3\left(t\right)dt)u_{yy}-x{u}_{xy}\right)\hfill \\ & +a_5\left(t\right)\left(-5v_y-5t{v}_{yt}-(3y+2\int a_3\left(t\right)dt)v_{yy}-x{v}_{xy}\right)u_{xx}\hfill \\ & +2(v{a}_2\left(t\right)u_y+a_5\left(t\right)v_{xy})\left(-2u_x-5t{u}_{xt}-(3y+2\int a_3\left(t\right)dt)u_{xy}-x{u}_{xx}\right)\hfill \\ & +2x(a_5\left(t\right)v_{xy}{u}_{xx}+v(a_4\left(t\right)u_{yy}+u_{xt}+a_3\left(t\right)u_{xy}+a_2\left(t\right)u_x{u}_{xy}\hfill \\ & +a_2\left(t\right)u_y{u}_{xx}\left)\right)+a_5\left(t\right)\left(2v+5t{v}_t+3y{v}_y+2(\int a_3\left(t\right)dt)v_y+x{v}_x\right)u_{xxy}\hfill \\ & +\left(u+5t{u}_t+3y{u}_y+2(\int a_3\left(t\right)dt)u_y+x{u}_x\right)(v_t+a_3\left(t\right)v_y+a_2\left(t\right)v_y{u}_x\hfill \\ & +2a_2\left(t\right)u_y{v}_x+v{a}_2\left(t\right)u_{xy}+2a_5\left(t\right)v_{xxy}\left)\right],\hfill \\ \hfill C^y=& {\displaystyle \frac{1}{2}}\left[v(a_3\left(t\right)+a_2\left(t\right)u_x)\left(-2u_x-5t{u}_{xt}-(3y+2\int a_3\left(t\right)dt)u_{xy}-x{u}_{xx}\right)\right.\hfill \\ & +\left(u+5t{u}_t+3y{u}_y+2(\int a_3\left(t\right)dt)u_y+x{u}_x\right)\left(a_3\left(t\right)v_x+a_2\left(t\right)(u_x{v}_x-v{u}_{xx})\right)\hfill \\ & +2(3y+2\int a_3\left(t\right)dt)(a_5\left(t\right)v_{xy}{u}_{xx}+v(a_4\left(t\right)u_{yy}+u_{xt}+a_3\left(t\right)u_{xy}+a_2\left(t\right)u_x{u}_{xy}\hfill \\ & +a_2\left(t\right)u_y{u}_{xx}\left)\right)+a_5\left(t\right)u_{xx}\left(-3v_x-5t{v}_{xt}-(3y+2\int a_3\left(t\right)dt)v_{xy}-x{v}_{xx}\right)\hfill \\ & \left.+a_5\left(t\right)\left(2v+5t{v}_t+3y{v}_y+2(\int a_3\left(t\right)dt)v_y+x{v}_x\right)u_{xxx}\right].\hfill \end{array}$$

Finally, all the conservation laws obtained in this section are verified to be validated by the symbolic computation.

5. Conclusions

The variable-coefficient GCBS equation is a classical two-dimensional nonlinear model which describes the propagation of long waves in nonlinear optics, fluid dynamics, and plasma physics in non-uniform media and with non-uniform boundaries. Through a Painlevé analysis, it was shown that the variable-coefficient GCBS equation (3) does not possess the Painlevé property. By means of an analysis of the Lie symmetry, one-parameter Lie group transformation was applied to the variable-coefficient GCBS equation. Under invariance conditions, infinitesimal generators and one-parameter Lie transformation groups were obtained through symbolic computation for two distinct cases, and the specific transformations between the solutions to the equation were also provided. Additionally, a Lie algebra commutation table was derived.

Subsequently, based on Ibragimov’s theorem, an adjoint equation corresponding to the GCBS equation was presented, and its nonlinear self-adjointness was proven. Then, conservation laws corresponding to each Lie group symmetry were also constructed.

The Painlevé property and Lie symmetry are two important properties for nonlinear evolution equations, while the results obtained above show the weak relationship between these two properties, which is worthy of further investigation.

It is expected that future research will extend the GCBS equation to higher-dimensional systems, study the structure of the solutions to the (3 + 1)-dimensional variable-coefficient GCBS equation, and explore its physical significance. Furthermore, the variable-coefficient GCBS equation could be applied to specific physical contexts, thereby expanding its practical applications in engineering and physics.