Lie Group Classification for a Reduced Burgers System

Abstract

We consider a variable coefficient Burgers system in two spatial variables. A similarity mapping is used to reduce the number of independent variables. Lie symmetry analysis is applied to the reduced system. We present the equivalence groups for this system and the complete Lie group classification. Lie symmetries are employed to reduce the system into ordinary differential equations. In certain cases, we derive exact solutions. A special case of the system admits a hidden symmetry. Furthermore, a differential substitution is applied to this special system that enables us to construct additional exact solutions.

1. Introduction

Since the appearance of the well known Burgers equation, several variations (equations or systems) of it have been used to describe problems in nonlinear acoustics, hydrodynamics, and oceanography. One such variation is the system [1,2,3,4,5,6]

$${\tilde{u}}_{\tilde{t}}=A\tilde{u}{\tilde{u}}_{\tilde{x}}+B{\tilde{v}}_{\tilde{x}\tilde{x}},{\tilde{v}}_{\tilde{t}\tilde{y}}=A\tilde{u}{\tilde{v}}_{\tilde{x}\tilde{y}}+A{\tilde{u}}_{\tilde{x}}{\tilde{v}}_{\tilde{y}}+C{\tilde{u}}_{\tilde{x}\tilde{x}\tilde{y}},$$

(1)

where $A,B$, and C are nonzero constants. In most cases, these constants are fixed to specific values. In fact, (1) can be mapped into a system with fixed constants. For example, if we assume that the constants are positive, the point transformation

$$\tilde{t}={\displaystyle \frac{\sqrt{BC}t}{\sqrt{\mu }{A}^2}},\tilde{x}={\displaystyle \frac{\sqrt{BC}x}{\sqrt{\mu }A}},\tilde{y}=y,\tilde{u}=u,\tilde{v}={\displaystyle \frac{\sqrt{C}v}{\sqrt{\mu B}}}$$

connects system (1) with

$$u_t=u{u}_x+v_{xx},v_{ty}=u{v}_{xy}+u_x{v}_y+\mu u_{xxy},$$

(2)

where $\mu$ is a positive constant that, without loss of generality, can be taken equal to one. We point out that appropriate mapping and connecting systems can be derived if some or all constants are negative.

The transformation

$$t={\displaystyle \frac{\tau }{\sqrt{\mu }}},u=2\sqrt{\mu }\varphi ,v_y=2\mu (2{\psi }_x-{\varphi }_y)$$

maps (2) into the known Boiti–Leon–Pempinelli system

$${\varphi }_{\tau y}={\left({\varphi }^2-{\varphi }_x\right)}_{xy}+2{\psi }_{xxx},{\psi }_{\tau }={\psi }_{xx}+2\varphi {\psi }_x.$$

Considering that physical phenomena can be described more accurate by differential equations with non-constant coefficients, it is obvious that coefficients can be functions of time or of a spatial variable or of both variables. For this reason, we have the appearance of the variable coefficient Burgers system [7,8]

$$u_t=a\left(t\right)u{u}_x+b\left(t\right)v_{xx},v_{ty}=c\left(t\right)u{v}_{xy}+d\left(t\right)u_x{v}_y+f\left(t\right)u_{xxy}.$$

(3)

Lie symmetry methods have been extensively used for past decades in the study of nonlinear partial differential equations. In the absence of general theory for solving such equations, these methods have been very powerful. Today, there are many powerful algebraic computation software programs, like Maple and Reduce, that can perform many calculations automatically. For this reason, one might think that Lie symmetry methods are easy to use because they are algorithmic. However, the problem is not so simple. When partial differential equations have arbitrary functions, such as variable coefficients, the determining system that we need to solve to find the symmetries becomes nonlinear. This means that using Lie symmetry methods still requires a lot of theoretical and analytical work, especially for equations with variable coefficients. The problem of deriving the Lie symmetries of a system with variable coefficients (unknown functions) is called Lie group classification. The problem consists, first, in deriving the Lie symmetries of the system when the coefficient functions are arbitrary, and second, deriving the additional (if any) symmetries when the functions have specific forms.

Group classifications for various forms of variable coefficient Burgers equations/systems can be found in the literature; see, for example, references [7,9,10,11]. The difference and the difficulty for each classification depends on the number of coefficient functions that appear in the equation/system. The Lie group classification for the system (3) was considered in [7]; however, the results are incomplete, since a number of cases that lead to additional Lie symmetries are missing. Here we use the Lie symmetries of the system (3) for arbitrary coefficient functions to reduce it into a system with two independent variables. We perform the Lie group classification for this reduced system. Prior to the classification, we present the equivalence group of the reduced system. In the study of differential equations, equivalence transformations have an important role. These transformations change an equation into another one of the same type, but perhaps with different coefficient functions. They are useful for the Lie group classification because it reduces the number of cases that need to be analyzed separately. Lie symmetries are used to derive a number of exact solutions.

The structure and the contribution of the present work is as follows. In Section 2, we perform a symmetry reduction of system (3), reducing it from a (2+1)-dimensional system to a (1+1)-dimensional one, which we denote as system (7). We also determine the equivalence transformations for this reduced system. In Section 3, we carry out the Lie group classification of system (7). In Section 4, we use Lie symmetries to reduce system (7) into ordinary differential equations in specific cases. From these reductions, we derive several exact solutions. In Section 5, we focus on a special case, system (9), which corresponds to the constant-coefficient version of system (7). Finally, Section 6 contains the conclusions. For readability, we include Appendix A where we list the longer cases omitted in the main text.

2. Reduction of the System (3)

In this section, we begin by applying a change of variables to the (2+1)-dimensional Burgers system (3) with variable coefficients. This change reduces the number of variable coefficients in the system. After that, we use a symmetry transformation to reduce the number of independent variables. In this way, we obtain a new system that has only two independent variables. We study this reduced system in the next sections. It is simpler than the original one, but it still keeps the main features.

The change of the variable $\tilde{t}=\int b\left(t\right)\mathrm{d}t$ maps (3) into

$$u_{\tilde{t}}={\displaystyle \frac{a\left(\tilde{t}\right)}{b\left(\tilde{t}\right)}}u{u}_x+v_{xx},v_{\tilde{t}y}={\displaystyle \frac{c\left(\tilde{t}\right)}{b\left(\tilde{t}\right)}}u{v}_{xy}+{\displaystyle \frac{d\left(\tilde{t}\right)}{b\left(\tilde{t}\right)}}{u}_x{v}_y+{\displaystyle \frac{f\left(\tilde{t}\right)}{b\left(\tilde{t}\right)}}{u}_{xxy}.$$

Hence, in the subsequent analysis, we can equivalently study the class (3) with $b\left(t\right)=1$,

$$u_t=a\left(t\right)u{u}_x+v_{xx},v_{ty}=c\left(t\right)u{v}_{xy}+d\left(t\right)u_x{v}_y+f\left(t\right)u_{xxy}.$$

(4)

The detailed procedure for finding Lie symmetries is well known and will be recalled in Section 3 in the context of the Lie group classification of a reduced system. Hence, without presenting any detailed analysis, we state that system (4) with $a\left(t\right),c\left(t\right),d\left(t\right),$ and $f\left(t\right)$ being arbitrary functions admits an infinite-dimensional Lie algebra that is spanned by the operators

$$X_1={\partial }_x,X\left(\psi \right)=\psi \left(y\right){\partial }_y,X\left({\varphi }_1\right)={\varphi }_1\left(t\right)x{\partial }_v,X\left({\varphi }_2\right)={\varphi }_2\left(t\right){\partial }_v,$$

(5)

where $\psi \left(y\right),{\varphi }_1\left(t\right)$, and ${\varphi }_2\left(t\right)$ are arbitrary functions in their arguments. The general symmetry

$${\partial }_x+\psi \left(y\right){\partial }_y+[{\varphi }_1\left(t\right)x+{\varphi }_2\left(t\right)]{\partial }_v$$

produces the reduction

$$u=\Phi (t,x-h\left(y\right)),v=\frac{1}{2}{\varphi }_1\left(t\right)x^2+{\varphi }_2\left(t\right)x+\Psi (t,x-h\left(y\right)).$$

The mapping

$$\tilde{t}=t,\tilde{x}=x,\tilde{y}=S\left(y\right),\tilde{u}=u,\tilde{v}=v+{\varphi }_1\left(t\right)x+{\varphi }_2\left(t\right)$$

(6)

leaves the system (4) invariant. Therefore, we can take $h\left(y\right)=y,{\varphi }_1\left(t\right)={\varphi }_2\left(t\right)=0$, without loss of generality. Hence, we have the reduction mapping

$$u=\Phi (t,\xi ),v=\Psi (t,\xi ),\xi =x-y$$

that reduces (4) into the system with two independent variables

$${\Phi }_t-a\left(t\right)\Phi {\Phi }_{\xi }-{\Psi }_{\xi \xi }=0,{\Psi }_{t\xi }-c\left(t\right)\Phi {\Psi }_{\xi \xi }-d\left(t\right){\Phi }_{\xi }{\Psi }_{\xi }-f\left(t\right){\Phi }_{\xi \xi \xi }=0.$$

(7)

The usual equivalence group for (7) consists of the point transformations

$$\tilde{t}={\alpha }_{1}t+{\alpha }_2,\tilde{\xi }={\alpha }_3\xi +{\alpha }_4,\tilde{\Phi }={\alpha }_1{\alpha }_5\Phi ,\tilde{\Psi }={\alpha }_3^2{\alpha }_5\Psi +{\varphi }_1\left(t\right),$$

$$\tilde{a}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_{3}a\left(t\right)}{{\alpha }_1^2{\alpha }_5}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_{3}c\left(t\right)}{{\alpha }_1^2{\alpha }_5}},\tilde{d}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_{3}d\left(t\right)}{{\alpha }_1^2{\alpha }_5}},\tilde{f}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_3^{4}f\left(t\right)}{{\alpha }_1^2}},$$

where ${\alpha }_1,\dots ,{\alpha }_5$ are arbitrary constants, ${\varphi }_1\left(t\right)$ is an arbitrary function, and ${\alpha }_1{\alpha }_3{\alpha }_5\ne 0$.

Additionally, if $c\left(t\right)=a\left(t\right)$ ($\tilde{c}\left(\tilde{t}\right)=\tilde{a}\left(\tilde{t}\right)$), we have the generalized equivalence group

$$\begin{array}{ccc}& & \tilde{t}=Q\left(t\right),\tilde{\xi }={\displaystyle \frac{{\alpha }_3\xi }{{\varphi }_2\left(t\right)Q^{\prime }\left(t\right)}}+{\varphi }_3\left(t\right),\hfill \\ & & \tilde{\Phi }={\varphi }_2\left(t\right)Q^{\prime }\left(t\right)\Phi -{\displaystyle \frac{Q^{″}\left(t\right){\varphi }_2\left(t\right)\xi }{d\left(t\right)}}-{\displaystyle \frac{{\varphi }_2^2\left(t\right){Q^{\prime }}^2\left(t\right){\varphi }_3^{\prime }\left(t\right)}{{\alpha }_{3}a\left(t\right)}},\tilde{\Psi }={\displaystyle \frac{{\alpha }_3^2\Psi }{{\varphi }_2\left(t\right){Q^{\prime }}^2\left(t\right)}}+{\varphi }_1\left(t\right),\hfill \\ & & \tilde{a}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_{3}a\left(t\right)}{{\varphi }_2^2\left(t\right){Q^{\prime }}^3\left(t\right)}},\tilde{d}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_{3}d\left(t\right)}{{\varphi }_2^2\left(t\right){Q^{\prime }}^3\left(t\right)}},\tilde{f}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_3^{4}f\left(t\right)}{{\varphi }_2^4\left(t\right){Q^{\prime }}^6\left(t\right)}},\hfill \end{array}$$

where ${\alpha }_3$ is an arbitrary constant, ${\varphi }_1\left(t\right)$ is an arbitrary function,

$${\varphi }_2\left(t\right)=exp\left[-\int {\displaystyle \frac{a\left(t\right)+d\left(t\right)}{d\left(t\right)}}{\displaystyle \frac{Q^{″}\left(t\right)}{Q^{\prime }\left(t\right)}}dt\right],{\varphi }_3\left(t\right)={\alpha }_1\int \left[a\left(t\right)exp\left(2\int {\displaystyle \frac{a\left(t\right)}{d\left(t\right)}}{\displaystyle \frac{Q^{″}\left(t\right)}{Q^{\prime }\left(t\right)}}dt\right)\right]dt,$$

with ${\alpha }_1$ being an arbitrary constant, and $Q\left(t\right)$ satisfies the differential equation

$$d\left(t\right)Q^{\prime }\left(t\right)Q^{‴}\left(t\right)-d^{\prime }\left(t\right)Q^{\prime }\left(t\right)Q^{″}\left(t\right)-[a\left(t\right)+d\left(t\right)]{Q^{″}}^2\left(t\right)=0.$$

We require ${\alpha }_3{\varphi }_2\left(t\right)Q^{\prime }\left(t\right)\ne 0$ for non-degenerate point transformations.

We note from the usual equivalence group, that if $(\Phi (t,\xi ),\Psi (t,\xi \left)\right)$ is a solution of (7), then $(\Phi (t,\xi ),\Psi (t,\xi )+\varphi (t\left)\right)$ is also a solution.

3. Lie Symmetry Classification of the System (7)

The Lie algorithm for finding symmetries is well known, and it is presented in many textbooks (see, for example, in [12,13,14,15]). Here, we perform the Lie group classification for the reduced system (7). Based on the results in reference [16], we deduce that the Lie operator has the restricted form

$$\Gamma =\tau \left(t\right){\partial }_t+X(t,\xi ){\partial }_{\xi }+U(t,\xi ,\Phi ){\partial }_{\Phi }+V(t,\xi ,\Psi ){\partial }_{\Psi }.$$

(8)

We apply the appropriate extension of the Lie operator $\Gamma$ to system (7) to obtain a multivariable polynomial in the derivatives of $\Phi$ and $\Psi$. Coefficients of these derivatives lead to the determining system.

Further calculations that have been manipulated independently by the algebraic packages MAPLE and REDUCE showed that the infinitesimal components X, U, V of the Lie operator $\Gamma$ take the specific simplified forms:

$$\begin{array}{ccc}& & X=\left[{\varphi }_1\left(t\right)+{\tau }^{\prime }\left(t\right)+{\beta }_1\right]\xi +{\varphi }_2\left(t\right),\hfill \\ & & U=-\left[{\varphi }_1\left(t\right)+{\tau }^{\prime }\left(t\right)+2{\beta }_1\right]\Phi -{\displaystyle \frac{1}{a\left(t\right)}}\left[{\tau }^{″}\left(t\right)+{\varphi }_1^{\prime }\left(t\right)\right]\xi -{\displaystyle \frac{{\varphi }_2^{\prime }\left(t\right)}{a\left(t\right)}},\hfill \\ & & V={\varphi }_1\left(t\right)\Psi +\varphi \left(t\right),\hfill \end{array}$$

where ${\beta }_1$ is a constant and $\tau \left(t\right),{\varphi }_1\left(t\right),{\varphi }_2\left(t\right),$ and $\varphi \left(t\right)$ are functions to be determined by solving the remaining equations of the determining system:

$$\begin{array}{ccc}& & a\left(t\right){\tau }^{\prime }\left(t\right)-\tau \left(t\right)a^{\prime }\left(t\right)+\left[2{\varphi }_1\left(t\right)+3{\beta }_1\right]a\left(t\right)=0,\hfill \\ & & a\left(t\right)\left[{\tau }^{‴}\left(t\right)+{\varphi }_1^{″}\left(t\right)\right]-a^{\prime }\left(t\right)\left[{\tau }^{″}\left(t\right)+{\varphi }_1^{\prime }\left(t\right)\right]=0,\hfill \\ & & a\left(t\right){\varphi }_2^{″}\left(t\right)-a^{\prime }\left(t\right){\varphi }_2^{\prime }\left(t\right)=0,\hfill \\ & & 2f\left(t\right){\tau }^{\prime }\left(t\right)-f^{\prime }\left(t\right)\tau \left(t\right)+4[{\varphi }_1\left(t\right)+{\beta }_1]f\left(t\right)=0,\hfill \\ & & c\left(t\right)f\left(t\right){\tau }^{\prime }\left(t\right)+[f\left(t\right)c^{\prime }\left(t\right)-c\left(t\right)f^{\prime }\left(t\right)]\tau \left(t\right)+[2{\varphi }_1\left(t\right)+{\beta }_1]c\left(t\right)f\left(t\right)=0,\hfill \\ & & d\left(t\right)f\left(t\right){\tau }^{\prime }\left(t\right)+[f\left(t\right)d^{\prime }\left(t\right)-d\left(t\right)f^{\prime }\left(t\right)]\tau \left(t\right)+[2{\varphi }_1\left(t\right)+{\beta }_1]d\left(t\right)f\left(t\right)=0,\hfill \\ & & [d\left(t\right)-a\left(t\right)]{\tau }^{″}\left(t\right)+{\varphi }_1^{\prime }\left(t\right)d\left(t\right)=0,\hfill \\ & & [a\left(t\right)-c\left(t\right)]\left[{\tau }^{″}\left(t\right)+{\varphi }_1^{\prime }\left(t\right)\right]=0,\hfill \\ & & [a\left(t\right)-c\left(t\right)]{\varphi }_2^{\prime }\left(t\right)=0.\hfill \end{array}$$

The solution of this system provides us the forms of $\tau \left(t\right),\varphi \left(t\right),{\varphi }_1\left(t\right),{\varphi }_2\left(t\right),a\left(t\right),c\left(t\right),$ $d\left(t\right)$, and $f\left(t\right)$. Consequently, the desired Lie symmetries are derived and the complete Lie group classification is presented. In order to solve the system, we are required to split the analysis into five exclusive cases:

• $c\left(t\right)\ne a\left(t\right)$;
• $c\left(t\right)=a\left(t\right)$;
• $c\left(t\right)=a\left(t\right),d\left(t\right)={\mu }_{2}a\left(t\right),f\left(t\right)\ne {\mu }_3{a}^2\left(t\right)$;
• $c\left(t\right)=a\left(t\right),d\left(t\right)={\mu }_{2}a\left(t\right),f\left(t\right)={\mu }_3{a}^2\left(t\right)$;
• $c\left(t\right)=a\left(t\right),d\left(t\right)=2a\left(t\right),f\left(t\right)={\mu }_3{a}^2\left(t\right)$.

Case 1: We have four subcases.

(i)

For arbitrary functions $a\left(t\right),c\left(t\right),d\left(t\right)$, and $f\left(t\right)$, Equation (7) admits an infinite dimensional Lie algebra that is spanned by the operators

$$Y_1={\partial }_{\xi },Y\left(\varphi \right)=\varphi \left(t\right){\partial }_{\Psi }.$$

Additionally, we have the following forms for functions $a\left(t\right),c\left(t\right),d\left(t\right)$, and $f\left(t\right)$ of (7) that admit additional Lie symmetries:

(ii)

$a\left(t\right)=t^n,c\left(t\right)={\mu }_1{t}^n,d\left(t\right)={\mu }_2{t}^n,f\left(t\right)={\mu }_3{t}^m$,

$$Y_1,Y\left(\varphi \right),Y_3=4t{\partial }_t+(m+2)\xi {\partial }_{\xi }+(m-4n-2)\Phi {\partial }_{\Phi }+(3m-4n-2)\Psi {\partial }_{\Psi }.$$

(iii)

$a\left(t\right)=e^{nt},c\left(t\right)={\mu }_1{e}^{nt},d\left(t\right)={\mu }_2{e}^{nt},f\left(t\right)={\mu }_3{e}^{mt}$,

$$Y_1,Y\left(\varphi \right),Y_4=4{\partial }_t+m\xi {\partial }_{\xi }+(m-4n)\Phi {\partial }_{\Phi }+(3m-4n)\Psi {\partial }_{\Psi }.$$

(iv)

$a\left(t\right)=1,c\left(t\right)={\mu }_1,d\left(t\right)={\mu }_2,f\left(t\right)={\mu }_3$ (system (7) has constant coefficients),

$$Y_1,Y\left(\varphi \right),Y_3(m=n=0)=2t{\partial }_t+\xi {\partial }_{\xi }-\Phi {\partial }_{\Phi }-\Psi {\partial }_{\Psi },Y_4(m=n=0)={\partial }_t.$$

Case 2: Here we have $c\left(t\right)=a\left(t\right)$, for arbitrary functions $a\left(t\right)$, $d\left(t\right)$, and $f\left(t\right)$. Equation (7) admits an infinite dimensional Lie algebra that is spanned by the operators

$$Y_1,Y\left(\varphi \right),Y_2=\left(\int a\left(t\right)dt\right){\partial }_{\xi }-{\partial }_{\Phi }.$$

Case 3: We have the conditions $c\left(t\right)=a\left(t\right),d\left(t\right)={\mu }_{2}a\left(t\right),f\left(t\right)\ne {\mu }_3{a}^2\left(t\right)$.

(i)

For arbitrary functions $a\left(t\right)$ and $f\left(t\right)$, Equation (7) admits an infinite dimensional Lie algebra that is spanned by the operators of Case 2. An additional Lie symmetry is obtained if the functions $a\left(t\right)$ and $f\left(t\right)$ satisfy a system of two ordinary differential equations. This system is tabulated in Appendix A. Here, we present two special solutions of the system. Namely, we have the following two subcases:

(ii)

$a\left(t\right)=c\left(t\right)=t^n,d\left(t\right)={\mu }_2{t}^n,f\left(t\right)={\mu }_3{t}^m$,

$$Y_1,Y\left(\varphi \right),Y_2(a=t^n),Y_3.$$

(iii)

$a\left(t\right)=c\left(t\right)=e^{nt},d\left(t\right)={\mu }_2{e}^{nt},f\left(t\right)={\mu }_3{e}^{mt}$,

$$Y_1,Y\left(\varphi \right),Y_2(a=e^{nt}),Y_4.$$

Case 4: Here, we have $c\left(t\right)=a\left(t\right),d\left(t\right)={\mu }_{2}a\left(t\right),f\left(t\right)={\mu }_3{a}^2\left(t\right)$.

(i)

For arbitrary function $a\left(t\right)$, system (7) admits an infinite dimensional Lie algebra that is spanned by the same operators as in Case 2.

(ii)

An additional Lie symmetry is admitted when $a\left(t\right)$ satisfies a specific ordinary differential equation. This subcase is tabulated in Appendix A.

(iii)

In the subcase $a\left(t\right)=c\left(t\right)=t^n,d\left(t\right)={\mu }_2{t}^n,f\left(t\right)={\mu }_3{t}^{2n},{\mu }_2\ne 1,n={\displaystyle \frac{2-{\mu }_2}{{\mu }_2-1}}$, we have two additional Lie symmetries:

$$\begin{array}{ccc}& & Y_1,Y\left(\varphi \right),Y_2(a=t^n),Y_3,\hfill \\ & & Y_5=({\mu }_2-1)t^{{\displaystyle \frac{{\mu }_2}{{\mu }_2-1}}}{\partial }_t+\xi t^{{\displaystyle \frac{1}{{\mu }_2-1}}}{\partial }_{\xi }-\left({\displaystyle \frac{\xi }{{\mu }_2-1}}+\Phi t^{{\displaystyle \frac{1}{{\mu }_2-1}}}\right){\partial }_{\Phi }-({\mu }_2-1)\Psi t^{{\displaystyle \frac{1}{{\mu }_2-1}}}{\partial }_{\Psi }.\hfill \end{array}$$

(iv)

Also, we obtain two additional Lie symmetries if $a\left(t\right)=c\left(t\right)=d\left(t\right)=e^{nt},f\left(t\right)={\mu }_3{e}^{2nt}$:

$$Y_1,Y\left(\varphi \right),Y_2(a=e^{nt}),Y_4,Y_6=e^{nt}{\partial }_t+n\xi e^{nt}{\partial }_{\xi }-n\left(n\xi +\Phi e^{nt}\right){\partial }_{\Phi }.$$

(v)

The constant subcase $a\left(t\right)=c\left(t\right)=1,d\left(t\right)={\mu }_2,f\left(t\right)={\mu }_3$ also admits two additional Lie symmetries:

$$Y_1,Y\left(\varphi \right),Y_2(a=1),Y_3(m=n=0),Y_4(m=n=0).$$

Case 5: Here, we have $c\left(t\right)=a\left(t\right),d\left(t\right)=2a\left(t\right),f\left(t\right)={\mu }_3{a}^2\left(t\right)$, and, therefore, only the function $a\left(t\right)$ appears in Equation (7). For arbitrary $a\left(t\right)$, it admits an infinite dimensional Lie algebra that is spanned by the same operators as in Case 2.

(i)

If $a\left(t\right)$ satisfies a specific ordinary differential equation, then system (7) admits an additional Lie symmetry. This subcase is also presented in Appendix A.

(ii)

Additional Lie symmetries exist in the subcase where $a\left(t\right)=1$. Specifically, the constant coefficient system

$${\Phi }_t-\Phi {\Phi }_{\xi }-{\Psi }_{\xi \xi }=0,{\Psi }_{t\xi }-\Phi {\Psi }_{\xi \xi }-2{\Phi }_{\xi }{\Psi }_{\xi }-{\mu }_3{\Phi }_{\xi \xi \xi }=0$$

(9)

admits the infinite dimensional Lie algebra, which is spanned by the operators

$$\begin{array}{ccc}& & Y_1,Y\left(\varphi \right),Y_2(a=1),Y_3(m=n=0),Y_4(m=n=0),\hfill \\ & & Y_7=t^2{\partial }_t+t\xi {\partial }_{\xi }-(\xi +t\Phi ){\partial }_{\Phi }-t\Psi {\partial }_{\Psi }.\hfill \end{array}$$

We use the obtained Lie symmetries to derive similarity mappings. These mappings reduce system (7) into a system with two ordinary differential equations. In some cases, the reduced systems are solved to provide exact solutions for the original system.

4. Lie Reductions and Exact Solutions for Certain Cases of System (7)

An important application of Lie symmetries is the construction reduction (similarity) mappings, which have the property to reduce the number of independent variables by one in a system of partial differential equations. We use certain of the derived Lie symmetries admitted by system (7) to construct reduction mappings. In some cases, the reduced systems of ordinary differential equations are solved, and, consequently, we have exact solutions for the original systems. The desired similarity mappings are obtained by solving the characteristic system

$${\displaystyle \frac{\mathrm{d}t}{\tau }}={\displaystyle \frac{\mathrm{d}\xi }{X}}={\displaystyle \frac{\mathrm{d}\Phi }{U}}={\displaystyle \frac{\mathrm{d}\Psi }{V}},$$

which corresponds to the Lie operator (8).

Initially, we consider the form of (7) that has arbitrary coefficient functions, then with coefficient functions with specific forms, and, finally, we examine (7) with constant coefficients. As a first example, we consider Case 2 where the coefficient functions of system (7) are arbitrary. By using the general symmetry

$$Y=Y\left(\varphi \right)+Y_2=\left(\int a\left(t\right)dt\right){\partial }_{\xi }-{\partial }_{\Phi }+\varphi \left(t\right){\partial }_{\Psi }$$

we find the trivial solution of (7)

$$\Phi (t,\xi )={\displaystyle \frac{g_0-\xi }{\left(\int a\left(t\right)dt\right)}},\Psi (t,\xi )=h\left(t\right)+k_0{e}^{-\int {\displaystyle \frac{d\left(t\right)}{\left(\int a\left(t\right)dt\right)}}dt}\xi .$$

In Case 1 (ii), we use the Lie symmetry $Y_3$ that produces the similarity mapping

$$\Phi =t^{{\displaystyle \frac{m-4n-2}{4}}}g\left(\sigma \right),\Psi =t^{{\displaystyle \frac{3m-4n-2}{4}}}h\left(\sigma \right),\sigma =\xi t^{-{\displaystyle \frac{m+2}{4}}}$$

for the Burgers system

$${\Phi }_t-t^n\Phi {\Phi }_{\xi }-{\Psi }_{\xi \xi }=0,{\Psi }_{t\xi }-{\mu }_1{t}^n\Phi {\Psi }_{\xi \xi }-{\mu }_2{t}^n{\Phi }_{\xi }{\Psi }_{\xi }-{\mu }_3{t}^m{\Phi }_{\xi \xi \xi }=0$$

and the corresponding reduced system takes the form

$$\begin{array}{ccc}& & h^{″}\left(\sigma \right)+g{g}^{\prime }+{\displaystyle \frac{1}{4}}(m+2)\sigma g^{\prime }+{\displaystyle \frac{1}{4}}(4n-m+2)g=0,\hfill \\ & & {\mu }_3{g}^{‴}\left(\sigma \right)+{\mu }_{1}g{h}^{″}+{\displaystyle \frac{1}{4}}(m+2)\sigma h^{″}+{\mu }_2{g}^{\prime }{h}^{\prime }+{\displaystyle \frac{1}{2}}(2n-m+2)h^{\prime }=0.\hfill \end{array}$$

We present some exact solutions of the above system of ordinary differential equations for specific values or relations of the parameters.

(i)

If ${\mu }_2=-2{\mu }_1,n=0,m=-6$, we have the exact solution

$$g\left(\sigma \right)=c_1{\sigma }^2,h\left(\sigma \right)=-{\displaystyle \frac{1}{10}}{c}_1^2{\sigma }^5.$$

(ii)

If ${\mu }_1={\displaystyle \frac{3}{2}}(n+2),{\mu }_2=-3(n+2),m=-2(2n+5)$, then

$$g\left(\sigma \right)=c_1{\sigma }^2,h\left(\sigma \right)=-{\displaystyle \frac{1}{10}}{c}_1^2{\sigma }^5+{\displaystyle \frac{1}{12}}{c}_1{\sigma }^4.$$

(iii)

If $m=-2,n=-1,{\mu }_3={\displaystyle \frac{1}{8}}{c}_1(2c_1{\mu }_1-1),{\mu }_2={\displaystyle \frac{1}{2c_1}}(2c_1{\mu }_1-3)$, the similarity mapping simplifies to $\Phi =g\left(\xi \right),\Psi ={\displaystyle \frac{h\left(\xi \right)}{t}}$, and we find the exact solution

$$g\left(\xi \right)=c_{1}tan\left(\sigma \right),h\left(\xi \right)=-{\displaystyle \frac{1}{2}}{c}_1^{2}tan\left(\sigma \right).$$

(iv)

If $m=-2,n=-1,{\mu }_3={\displaystyle \frac{1}{8}}{c}_1(2c_1{\mu }_1+1),{\mu }_2={\displaystyle \frac{1}{2c_1}}(2c_1{\mu }_1+3)$, we have

$$g\left(\xi \right)=c_{1}cot\left(\sigma \right),h\left(\xi \right)={\displaystyle \frac{1}{2}}{c}_1^{2}cot\left(\sigma \right).$$

(v)

If $m=2n,{\mu }_2={\displaystyle \frac{1}{n+1}},{\mu }_3={\displaystyle \frac{c_1^2}{12}}(2{\mu }_1+{\displaystyle \frac{1}{n+1}})$, we have the solution

$$g\left(\sigma \right)=-(n+1)\sigma +{\displaystyle \frac{c_1}{\sigma }},h\left(\sigma \right)={\displaystyle \frac{c_1^2}{2\sigma }}+{\displaystyle \frac{(1-{\mu }_1){(n+1)}^2{c}_1}{2}}\sigma .$$

(vi)

If $m=2n,{\mu }_2=2{\mu }_1-{\displaystyle \frac{n}{n+1}},{\mu }_3={\displaystyle \frac{c_1^2}{12}}(4{\mu }_1-{\displaystyle \frac{n}{n+1}})$, we find the exact solution

$$g\left(\sigma \right)=-(n+1)\sigma +{\displaystyle \frac{c_1}{\sigma }},h\left(\sigma \right)={\displaystyle \frac{c_1^2}{2\sigma }}.$$

We consider Case 1 (iii), which corresponds to the system

$${\Phi }_t-e^{nt}\Phi {\Phi }_{\xi }-{\Psi }_{\xi \xi }=0,{\Psi }_{t\xi }-{\mu }_1{e}^{nt}\Phi {\Psi }_{\xi \xi }-{\mu }_2{e}^{nt}{\Phi }_{\xi }{\Psi }_{\xi }-{\mu }_3{e}^{mt}{\Phi }_{\xi \xi \xi }=0.$$

(10)

We use the Lie symmetry

$$Y=Y\left(\varphi \right)+Y_4=4{\partial }_t+m\xi {\partial }_{\xi }+(m-4n)\Phi {\partial }_{\Phi }+\left((3m-4n)\Psi +\varphi \left(t\right)\right){\partial }_{\Psi }$$

to obtain the similarity reduction

$$\Phi (t,\xi )=e^{{\displaystyle \frac{m-4n}{4}}t}g\left(\sigma \right),\Psi (t,\xi )=e^{{\displaystyle \frac{3m-4n}{4}}t}h\left(\sigma \right)+{\displaystyle \frac{\int e^{-\frac{3m-4n}{4}t}\varphi \left(t\right)dt}{4e^{\frac{-3m+4n}{4}t}}},\sigma =\xi e^{-{\displaystyle \frac{m}{4}}t},$$

which reduces system (10) into the system of ordinary differential equations

$$(4g-m\sigma )g^{\prime }-(m-4n)g+4h^{″}=0,(m\sigma +4{\mu }_{1}g)h^{″}+2(2{\mu }_2{g}^{\prime }-m+2n)h^{\prime }+4{\mu }_3{g}^{‴}=0.$$

We deduce two special solutions of the above system.

(i)

If ${\mu }_2=-1,{\mu }_1={\displaystyle \frac{1}{2}},m=0,n={\lambda }^2\sqrt{{\mu }_3}$, then

$$g\left(\sigma \right)=e^{\lambda \sigma },h\left(\sigma \right)=-\sqrt{{\mu }_3}{e}^{\lambda \sigma }-{\displaystyle \frac{e^{2\lambda \sigma }}{2\lambda \sigma }}.$$

(ii)

If ${\mu }_2=-1,{\mu }_1={\displaystyle \frac{1}{2}},m=0,n=-{\lambda }^2\sqrt{{\mu }_3}$, then

$$g\left(\sigma \right)=e^{\lambda \sigma },h\left(\sigma \right)=\sqrt{{\mu }_3}{e}^{\lambda \sigma }-{\displaystyle \frac{e^{2\lambda \sigma }}{2\lambda \sigma }}.$$

We use the corresponding similarity transformation to obtain the desired exact solutions for system (10).

Now, we consider Case 3 (iii), where system (7) takes the form

$${\Phi }_t-t^{{\displaystyle \frac{2-{\mu }_2}{{\mu }_2-1}}}\Phi {\Phi }_{\xi }-{\Psi }_{\xi \xi }=0,{\Psi }_{t\xi }-t^{{\displaystyle \frac{2-{\mu }_2}{{\mu }_2-1}}}\Phi {\Psi }_{\xi \xi }-{\mu }_2{t}^{{\displaystyle \frac{2-{\mu }_2}{{\mu }_2-1}}}{\Phi }_{\xi }{\Psi }_{\xi }-{\mu }_3{t}^{2{\displaystyle \frac{2-{\mu }_2}{{\mu }_2-1}}}{\Phi }_{\xi \xi \xi }=0.$$

(11)

We use the Lie symmetry

$$Y_5=({\mu }_2-1)t^{{\displaystyle \frac{{\mu }_2}{{\mu }_2-1}}}{\partial }_t+\xi t^{{\displaystyle \frac{1}{{\mu }_2-1}}}{\partial }_{\xi }-\left({\displaystyle \frac{\xi }{{\mu }_2-1}}+\Phi t^{{\displaystyle \frac{1}{{\mu }_2-1}}}\right){\partial }_{\Phi }-({\mu }_2-1)\Psi t^{{\displaystyle \frac{1}{{\mu }_2-1}}}{\partial }_{\Psi }$$

to derive the similarity mapping

$$\Phi (t,\xi )=t^{{\displaystyle \frac{1}{1-{\mu }_2}}}\left({\displaystyle \frac{\xi }{1-{\mu }_2}}+g\left(\sigma \right)\right),\Psi (t,\xi )={\displaystyle \frac{h\left(\sigma \right)}{t}},\sigma =\xi t^{{\displaystyle \frac{1}{1-{\mu }_2}}},$$

which reduces system (11) into

$$g{g}^{\prime }+h^{″}=0,h^{″}g+{\mu }_2{h}^{\prime }{g}^{\prime }+{\mu }_3{g}^{‴}=0.$$

This system has the general solution

$$\int {\left[{\displaystyle \frac{1}{12{\mu }_3}}({\mu }_2+2)g^4-{\displaystyle \frac{{\lambda }_1{\mu }_2}{{\mu }_3}}{g}^2+{\lambda }_{2}g+{\lambda }_3\right]}^{-{\displaystyle \frac{1}{2}}}\mathrm{d}g=\sigma +{\lambda }_4,h\left(\sigma \right)=\int \left(-{\displaystyle \frac{1}{2}}{g}^2+{\lambda }_1\right)\mathrm{d}\sigma .$$

Special solutions are obtained using the results of the next case. In particular, we set $\gamma =0$ and $\mu =1$. We point out that the form of the similarity variable $\sigma$ in this case is different from the form in the next case.

We consider Case 1 (iv), in which the system (7) has constant coefficients,

$${\Phi }_t-\Phi {\Phi }_{\xi }-{\Psi }_{\xi \xi }=0,{\Psi }_{t\xi }-{\mu }_1\Phi {\Psi }_{\xi \xi }-{\mu }_2{\Phi }_{\xi }{\Psi }_{\xi }-{\mu }_3{\Phi }_{\xi \xi \xi }=0$$

(12)

and the Lie symmetry $Y={\partial }_t+\gamma {\partial }_{\xi }$. We derive the similarity mapping

$$\Phi =g\left(\sigma \right),\Psi =h\left(\sigma \right),\sigma =\xi -\gamma t,$$

which reduces (12) to

$$\gamma g^{\prime }+g{g}^{\prime }+h^{″}=0,{\mu }_3{g}^{‴}+{\mu }_2{g}^{\prime }{h}^{\prime }+\gamma h^{″}+{\mu }_{1}g{h}^{″}=0.$$

(13)

The general solution of this system can be written in implicit form:

$$\int {\left[{\displaystyle \frac{1}{12{\mu }_3}}({\mu }_2+2{\mu }_1)g^4+{\displaystyle \frac{1}{3{\mu }_3}}\gamma ({\mu }_1+{\mu }_2+1)g^3+{\displaystyle \frac{1}{{\mu }_3}}({\gamma }^2-{\lambda }_1{\mu }_2)g^2+{\lambda }_{2}g+{\lambda }_3\right]}^{-{\displaystyle \frac{1}{2}}}\mathrm{d}g=\sigma +{\lambda }_4,$$

$$h\left(\sigma \right)=\int \left(-{\displaystyle \frac{1}{2}}{g}^2-\gamma g+{\lambda }_1\right)\mathrm{d}\sigma .$$

We list some special solutions giving only the form of $g\left(\sigma \right)$, where $h\left(\sigma \right)$ can be found from the above integral. If ${\mu }_1=1,{\mu }_2=-2$, we have the following exact solutions:

(i)

$$g\left(\sigma \right)=c_2{\mathrm{e}}^{k\sigma }+c_3{\mathrm{e}}^{k\sigma }+c_4,k=\sqrt{{\displaystyle \frac{{\gamma }^2+2{\lambda }_1}{{\mu }_3}}},$$

(ii)

$$g\left(\sigma \right)=c_{2}cos\left(k\sigma \right)+c_{3}sin\left(k\sigma \right)+c_4,k=\sqrt{-{\displaystyle \frac{{\gamma }^2+2{\lambda }_1}{{\mu }_3}}}.$$

For arbitrary constants ${\mu }_1,{\mu }_2$, and ${\mu }_3$, we obtain the following special solutions:

(iii)

$$g\left(\sigma \right)=c_{1}cot\left(k\sigma \right)-{\displaystyle \frac{\gamma ({\mu }_1+{\mu }_2+1)}{2{\mu }_1+{\mu }_2}}+ϵc_{1}tan\left(k\sigma \right),$$

where

$$k^2={\displaystyle \frac{c_1^2(2{\mu }_1+{\mu }_2)}{12{\mu }_3}},{\lambda }_1={\displaystyle \frac{(\frac{3}{2}ϵ-\frac{1}{2})(2{\mu }_1+{\mu }_2)c_1^2}{3{\mu }_2}}-{\displaystyle \frac{[{({\mu }_1+{\mu }_2)}^2+1-2{\mu }_1]{\gamma }^2}{2{\mu }_2(2{\mu }_1+{\mu }_2)}},ϵ=\pm 1.$$

(iv)

$$g\left(\sigma \right)=c_{1}coth\left(k\sigma \right)-{\displaystyle \frac{\gamma ({\mu }_1+{\mu }_2+1)}{2{\mu }_1+{\mu }_2}}+ϵc_{1}tanh\left(k\sigma \right),$$

where

$${\lambda }_1={\displaystyle \frac{(\frac{3}{2}ϵ+\frac{1}{2})(2{\mu }_1+{\mu }_2)c_1^2}{3{\mu }_2}}-{\displaystyle \frac{[{({\mu }_1+{\mu }_2)}^2+1-2{\mu }_1]{\gamma }^2}{2{\mu }_2(2{\mu }_1+{\mu }_2)}}$$

and $k,ϵ$ as in (iii).

(v)

$$g\left(\sigma \right)=c_{1}tan\left(k\sigma \right)-{\displaystyle \frac{\gamma ({\mu }_1+{\mu }_2+1)}{2{\mu }_1+{\mu }_2}},$$

where

$$k^2={\displaystyle \frac{c_1^2(2{\mu }_1+{\mu }_2)}{12{\mu }_3}},{\lambda }_1=-{\displaystyle \frac{(2{\mu }_1+{\mu }_2)c_1^2}{6{\mu }_2}}-{\displaystyle \frac{[{({\mu }_1+{\mu }_2)}^2+1-2{\mu }_1]{\gamma }^2}{2{\mu }_2(2{\mu }_1+{\mu }_2)}}.$$

(vi)

$$g\left(\sigma \right)=c_{1}cot\left(k\sigma \right)-{\displaystyle \frac{\gamma ({\mu }_1+{\mu }_2+1)}{2{\mu }_1+{\mu }_2}},$$

where k and ${\lambda }_1$ as in Case (v).

(vii)

$$g\left(\sigma \right)=c_{1}tanh\left(k\sigma \right)-{\displaystyle \frac{\gamma ({\mu }_1+{\mu }_2+1)}{2{\mu }_1+{\mu }_2}},$$

where

$$k^2={\displaystyle \frac{c_1^2(2{\mu }_1+{\mu }_2)}{12{\mu }_3}},{\lambda }_1={\displaystyle \frac{(2{\mu }_1+{\mu }_2)c_1^2}{6{\mu }_2}}-{\displaystyle \frac{[{({\mu }_1+{\mu }_2)}^2+1-2{\mu }_1]{\gamma }^2}{2{\mu }_2(2{\mu }_1+{\mu }_2)}}.$$

(viii)

$$g\left(\sigma \right)=c_{1}coth\left(k\sigma \right)-{\displaystyle \frac{\gamma ({\mu }_1+{\mu }_2+1)}{2{\mu }_1+{\mu }_2}},$$

where k and ${\lambda }_1$ as in Case (vii).

Using the above solutions and the similarity reductions, we can derive the corresponding solutions for system (12).

In this section, we have derived a number of exact solutions for the Burgers system (7) using Lie similarity reductions. In most cases, the reduced systems cannot be solved analytically in general, and numerical or approximation methods must be used. Numerical similarity solutions can be found, for example, in ref. [17]. Also in the present work, we have not considered boundary/initial problems. In certain cases, such problems can be solved using Lie symmetries; see, for example, [18].

5. The Special System (9)

We have listed the Lie symmetries for system (4) in the case where the coefficient functions are arbitrary. If the functions $a\left(t\right),c\left(t\right),d\left(t\right)$, and $f\left(t\right)$ are constants, then the Burgers system (4), in addition to the symmetries given by Equation (5), it admits the additional Lie symmetries:

(i)

$$X_2={\partial }_t,X_3=2t{\partial }_t+x{\partial }_x-u{\partial }_u-v{\partial }_v,a,c,d,f\mathrm{arbitrary},$$

(ii)

$$X_2,X_3,X\left({\varphi }_3\right)={\varphi }_3\left(t\right){\partial }_x-{\displaystyle \frac{1}{a}}{\varphi }_3^{\prime }\left(t\right){\partial }_u-{\displaystyle \frac{1}{2a}}{\varphi }_3^{″}\left(t\right)x^2{\partial }_v,c=a$$

(iii)

$$\begin{array}{ccc}& & X\left({\varphi }_3\right),X\left(\tau \right)=\tau \left(t\right){\partial }_t+{\displaystyle \frac{1}{2}}{\tau }^{\prime }\left(t\right)x{\partial }_x-{\displaystyle \frac{1}{2a}}\left[a{\tau }^{\prime }\left(t\right)u+{\tau }^{″}\left(t\right)x\right]{\partial }_u\hfill \\ & & -{\displaystyle \frac{1}{12a}}\left[6a{\tau }^{\prime }\left(t\right)v+{\tau }^{‴}\left(t\right)x^3\right]{\partial }_v,d=c=a.\hfill \end{array}$$

We point out that the constant coefficient Burgers system (2) falls into Case (iii). However, system (9) falls into Case (ii), and we deduce the admitted Lie symmetry

$$Y_7=t^2{\partial }_t+t\xi {\partial }_{\xi }-(\xi +t\Phi ){\partial }_{\Phi }-t\Psi {\partial }_{\Psi }$$

is a hidden symmetry. It is a Lie symmetry that would be lost if system (9) reduces directly into a system of ordinary differential equations. This hidden symmetry, which is new, provides the reduction

$$\Phi (t,\xi )={\displaystyle \frac{1}{t}}g\left(\sigma \right)-{\displaystyle \frac{\xi }{t}},\Psi (t,\xi )={\displaystyle \frac{h\left(\sigma \right)}{t}},\sigma ={\displaystyle \frac{\xi }{t}}$$

that maps (9) into

$$g{g}^{\prime }+h^{″}=0,2g^{\prime }{h}^{\prime }+g{h}^{″}+{\mu }_3{g}^{‴}=0,$$

which is a special case of system (13) with $\gamma =0,{\mu }_1=1,{\mu }_2=2$. Therefore, using the solutions of (13) with $\gamma =0,{\mu }_1=1,{\mu }_2=2$ and the above similarity reduction, we derive the corresponding solutions of (9).

Now, we apply the differential substitution

$$\Phi =w_{\xi },{\Psi }_{\xi }=z_{\xi }-{\displaystyle \frac{1}{2}}{w}_{\xi }^2$$

(14)

to system (9). The first equation becomes $w_{t\xi }=z_{\xi \xi }$. Ignoring the function of integration, we have

$$z_{\xi }=w_t$$

(15)

and the second equation takes the form

$$w_{tt}+{\left({\displaystyle \frac{2}{3}}{w}_{\xi }^3-2w_{\xi }{w}_t-{\mu }_3{w}_{\xi \xi \xi }\right)}_{\xi }=0.$$

(16)

Solutions of Equations (15) and (16) provide solutions for system (9) using mapping (14). We observe that, in Equation (16), only the dependent variable w appears. We apply the Lie symmetry analysis for Equation (16). It turns out that Equation (16) admits a six-dimensional Lie algebras, which is spanned by the operators

$$W_1={\partial }_t,W_2={\partial }_{\xi },W_3={\partial }_w,W_4=2t{\partial }_t+\xi {\partial }_{\xi },W_5=t{\partial }_{\xi }-\xi {\partial }_w,W_6=2t^2{\partial }_t+2t\xi {\partial }_{\xi }-{\xi }^2{\partial }_w.$$

We use certain Lie symmetries that lead to exact solutions. Lie symmetry $W_4+k{W}_5$ produces the mapping

$$w(t,\xi )={\displaystyle \frac{k^2}{2}}t-k\xi +g\left(\sigma \right),\sigma ={\displaystyle \frac{\xi -kt}{\sqrt{t}}}$$

that reduces (16) into the ordinary differential equation

$$4{\mu }_3{g}^{\left(iv\right)}=(8g^{\prime 2}+8\sigma g^{\prime }+{\sigma }^2)g^{″}+(4g^{\prime }+3\sigma )g^{\prime }.$$

A special solution of the above equation is $g\left(\sigma \right)=-{\displaystyle \frac{{\sigma }^2}{4}}+\sqrt{3{\mu }_3}ln\sigma +g_0$. Using that $z_{\xi }=w_t$ and mapping (14), we obtain the exact solution of system (7):

$$\Phi (t,\xi )={\displaystyle \frac{2\sqrt{3{\mu }_3}t-{\xi }^2+k^{2}t}{2t(\xi -kt)}},\Psi (t,\xi )={\displaystyle \frac{{\xi }^3}{24t^2}}+{\displaystyle \frac{3{\mu }_3}{2(\xi -kt)}}-{\displaystyle \frac{k\xi }{8t}}(\xi -kt).$$

Lie symmetry $W_6$ leads to the reduction mapping

$$w(t,\xi )=-{\displaystyle \frac{{\xi }^2}{2t}}+g\left(\sigma \right),\sigma ={\displaystyle \frac{\xi }{t}}$$

and the reduced equation has the form

$${\mu }_3{g}^{\left(iv\right)}-2{g^{\prime }}^2{g}^{″}=0$$

with a general solution in the implicit form

$$\int {\left({\displaystyle \frac{1}{3{\mu }_3}}{h}^4+c_{1}h+c_2\right)}^{-{\displaystyle \frac{1}{2}}}\mathrm{d}h=\sigma +c_3,g^{\prime }\left(\sigma \right)=h\left(\sigma \right).$$

The special solution $g\left(\sigma \right)=ϵ\sqrt{3{\mu }_3}ln\sigma$ produces the solution

$$\Phi (t,\xi )={\displaystyle \frac{ϵ\sqrt{3{\mu }_3}}{\xi }}-{\displaystyle \frac{\xi }{t}},\Psi (t,\xi )={\displaystyle \frac{3{\mu }_3}{2\xi }},ϵ=\pm 1.$$

Finally, we use the Lie symmetry $W_4+k_1{W}_5+k_2{W}_6$, which produces the mapping

$$w(t,\xi )={\displaystyle \frac{k_1^2}{2}}t-k_1\xi +{\displaystyle \frac{{(\xi -k_{1}t)}^2}{2(k_{2}t+1)}}{k}_2+g\left(\sigma \right),\sigma ={\displaystyle \frac{\xi -k_{1}t}{\sqrt{t}\sqrt{k_{2}t+1}}}$$

that reduces (16) into the ordinary differential equation

$$4{\mu }_3{g}^{\left(iv\right)}=(8g^{\prime 2}+8\sigma g^{\prime }+{\sigma }^2)g^{″}+(4g^{\prime }+3\sigma )g^{\prime }.$$

A special solution of the above equation is $g\left(\sigma \right)=-{\displaystyle \frac{{\sigma }^2}{4}}+\sqrt{3{\mu }_3}ln\sigma +g_0$. Using that $z_{\xi }=w_t$ and mapping (14), we obtain the exact solution of system (7):

$$\begin{array}{ccc}& & \Phi (t,\xi )={\displaystyle \frac{2\sqrt{3{\mu }_3}t(k_{2}t+1)+k_1{t}^2(2k_2\xi +k_1)-{\xi }^2(2k_{2}t+1)}{2t(\xi -kt)(k_{2}t+1)}},\hfill \\ & & \Psi (t,\xi )={\displaystyle \frac{36{\mu }_3{t}^2{(k_{2}t+1)}^2-3{(\xi -k_{1}t)}^2{k}_{1}t\xi +(\xi -k_{1}t){\xi }^3}{24t^2{(k_{2}t+1)}^2(\xi -k_{1}t)}}.\hfill \end{array}$$

6. Conclusions

When conducting research in the area of nonlinear partial differential equations, one often needs to derive exact solutions, a task that can be quite challenging. The complexity of such equations increases when the coefficients are functions of time and/or spatial variables. Over the past decades, Lie methods have proven to be very effective in tackling such problems.

In the present work, we have performed the Lie group classification for the variable coefficient Burgers system (7). Based on the classification results, we have derived a list of exact solutions. It is important to note that system (7) results from a similarity reduction of the (1+2)-dimensional Burgers system (3). Consequently, the solutions obtained for system (7) can be projected onto solutions of system (3), thereby extending their applicability. Furthermore, the analysis presented here has served as a partial contribution toward the Lie group classification of system (3). In other words, the comprehensive Lie group classification of the variable coefficient Burgers system (3) remains an open problem, which we intend to address in the near future.

The challenging problem of Lie group classification can, in many cases, be simplified by using equivalence transformations. In the present work, we fixed one coefficient function equal to one with the aid of a transformation that is a member of the equivalence group admitted by system (3). In particular, we transformed system (3) to the simplified system (4). Hence, it is convenient and simpler to study the equivalent system (4).

For partial differential equations involving two independent variables, we can use one-dimensional subalgebras of the obtained Lie symmetries to construct similarity mappings that reduce the system to ordinary differential equations, effectively lowering the number of independent variables by one. This has been precisely the approach taken in the present work, where we have used one-dimensional subalgebras to reduce the Burgers system (7) to a system of ordinary differential equations. For partial differential equations in three independent variables, one-dimensional subalgebras similarly allow reductions to systems with two independent variables. Lie symmetry analysis is then applied to these reduced systems, with the ultimate goal of obtaining further reductions to ordinary differential equations. An alternative approach involves the use of two-dimensional subalgebras, which yeld similarity transformations that reduce partial differential equations in three independent variables directly to ordinary differential equations. The first approach has the additional advantage of potentially uncovering hidden symmetries that cannot be identified using the second method. Once the Lie group classification for the (1+2)-dimensional Burgers system (3) is obtained, both of these reduction techniques can be systematically applied.