Lie Group Classification of a Class of Variable Coefficient Boiti–Leon–Manna–Pempinelli Equations

Abstract

The Boiti–Leon–Manna–Pempinelli (BLMP) equation with coefficients being functions of time is considered. Since the coefficient functions are arbitrary, we have a class of BLMP equations. Symmetry analysis is carried out for this class. We derive the equivalence group admitted by the class and we present the enhanced Lie group classification. Lie symmetries are used to construct similarity reductions. Reduction operators that are not equivalent to Lie ones are also constructed.

1. Introduction

The system

$$w_t-3{\left(wv\right)}_x+w_{xxx}=0,w_x=v_y$$

(1)

has appeared in the literature as a generalization of a two-dimensional KdV equation [1]. If we introduce the transformation $w=u_y,v=u_x$ then Equation (1) can be written as a single non-linear partial differential equation:

$$u_{ty}-3u_x{u}_{xy}-3u_y{u}_{xx}+u_{xxxy}=0$$

(2)

which is known as the Boiti–Leon–Manna–Pempinelli equation and has different applications in mathematical physics. For example, it can be used to describe the two-dimensional interaction of the Riemann wave propagated along the y-axis with a long wave propagated along the x-axis. Since the appearance of Equation (2), it has been studied by several authors from different points of view. In particular, there is a continued interest in finding exact solutions [2,3,4,5,6,7,8,9]. For example, in the literature one can find soliton solutions [2,4,5], similarity solutions [5,6,8], periodic solutions [3] and solutions that are obtained using Bäcklund transformations [7,9].

Considering that physical phenomena can be described more accurately 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 Boiti–Leon–Manna–Pempinelli equation in the literature [10,11,12,13]:

$$u_{ty}+c\left(t\right)u_x{u}_{xy}+b\left(t\right)u_y{u}_{xx}+a\left(t\right)u_{xxxy}=0,$$

(3)

where $a\left(t\right),b\left(t\right)$ and $c\left(t\right)$ are non-zero smooth functions of time. Throughout the analysis, these functions are considered to be non-zero. The corresponding system has the form

$$w_t+c\left(t\right)v{w}_x+b\left(t\right)w{v}_x+a\left(t\right)w_{xxx}=0,w_x=v_y.$$

(4)

In the study of non-linear partial differential equations, Lie symmetry methods have been extensively used for the last 50 years. In this work, we consider the Lie group classification for class (3). This problem was also considered in [13], but the analysis was incomplete. The Lie group classification of (4) with $b\left(t\right)=c\left(t\right)$, but in a potential form, was presented in reference [10]. The appearance of the variable coefficient makes the analysis difficult. However, using the equivalence group admitted by class (3), we can fix one function equal to a constant. This shows that prior to the Lie symmetry analysis it is very useful to derive the equivalence group of transformations. In particular, using a special case of the equivalence group, we can fix $a\left(t\right)=1$, and class (3) becomes

$$u_{ty}+c\left(t\right)u_x{u}_{xy}+b\left(t\right)u_y{u}_{xx}+u_{xxxy}=0$$

(5)

and the corresponding system has the form

$$w_t+c\left(t\right)v{w}_x+b\left(t\right)w{v}_x+w_{xxx}=0,w_x=v_y.$$

(6)

In the spirit of the work [14], where a different class of variable coefficient Boiti–Leon–Manna–Pempinelli equations was considered, we derive the equivalence groups of transformations and the Lie group classification for class (3). In Section 2, we present the equivalence groups admitted by (3) and we show that this class is equivalent to class (5). The enhanced Lie group classification for the simplified class (5) is presented in Section 3 and the corresponding results for the general class (3) are tabulated in Appendix A. Equivalence groups and the Lie group classification for system (4) are presented in Appendix B. Lie symmetries are used to construct reduction mappings in Section 4. Reduced equations are solved, to provide exact solutions for the original equations. Finally, we derive a number of non-Lie reduction operators that also lead to some exact solutions.

2. Equivalence Transformations and Their Applications

Equivalence transformations are non-degenerate point transformations that preserve the differential structure of the equation and form a group. An equivalence transformation might change only the form of the coefficient functions (arbitrary elements). The notion of these transformations was introduced by Ovsiannikov [15], by defining the usual equivalence group. In this simple group, the transformations of independent and dependent variables do not depend on arbitrary elements. If these transformations depend on arbitrary elements then it is called a generalized equivalence group [16]. In a case where we have dependence on nonlocalities with respect to arbitrary elements then it is called an extended equivalence group [17]. If the properties of both generalized and extended equivalence groups are satisfied then it is called a generalized extended equivalence group.

Class (3) admits usual, extended and generalized extended equivalence groups. The results are summarized in the following theorem. The point transformations that are presented connect class (3) and a similar class with tilded variables.

Theorem 1. 

The usual equivalence group of class (3) consists of the point transformations

$$\tilde{t}=Q\left(t\right),\tilde{x}={\alpha }_{1}x+{\beta }_1,\tilde{y}=\psi \left(y\right),\tilde{u}={\alpha }_{2}u+{\varphi }_1\left(t\right),$$

$$\tilde{a}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_1^{3}a\left(t\right)}{Q^{\prime }\left(t\right)}},\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_1^{2}b\left(t\right)}{{\alpha }_2{Q}^{\prime }\left(t\right)}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_1^{2}c\left(t\right)}{{\alpha }_2{Q}^{\prime }\left(t\right)}},$$

where $Q\left(t\right),{\varphi }_1\left(t\right),\psi \left(y\right)$ are arbitrary functions, ${\alpha }_1,{\alpha }_2,{\beta }_1$ are arbitrary constants and ${\alpha }_1{\alpha }_2{Q}^{\prime }\left(t\right){\psi }^{\prime }\left(y\right)\ne 0$ for non-degenerate transformations.

Class (3) admits the extended equivalence group

$$\tilde{t}=Q\left(t\right),\tilde{x}={\alpha }_{1}x+{\varphi }_2\left(t\right),\tilde{y}=\psi \left(y\right),\tilde{u}={\alpha }_{2}u+{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}{\displaystyle \frac{{\varphi }_2^{\prime }\left(t\right)}{c\left(t\right)}}x+{\varphi }_1\left(t\right),$$

$$\tilde{a}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_1^{3}a\left(t\right)}{Q^{\prime }\left(t\right)}},\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_1^{2}b\left(t\right)}{{\alpha }_2{Q}^{\prime }\left(t\right)}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{{\alpha }_1^{2}c\left(t\right)}{{\alpha }_2{Q}^{\prime }\left(t\right)}},$$

where $Q\left(t\right),{\varphi }_1\left(t\right),{\varphi }_2\left(t\right),\psi \left(y\right)$ are arbitrary functions, ${\alpha }_1,{\alpha }_2$ are arbitrary constants and ${\alpha }_1{\alpha }_2{Q}^{\prime }\left(t\right){\psi }^{\prime }\left(y\right)\ne 0$.

Finally, class (3) admits the generalized extended equivalence group, which consists of the point transformations

$$\tilde{t}=Q\left(t\right),\tilde{x}={\varphi }_3\left(t\right)x+{\varphi }_2\left(t\right),\tilde{y}=\psi \left(y\right),$$

$$\tilde{u}={\displaystyle \frac{{\varphi }_4\left(t\right)}{c\left(t\right){\varphi }_3\left(t\right)}}\left[c\left(t\right){\varphi }_3\left(t\right)u+{\displaystyle \frac{1}{2}}{\varphi }_3^{\prime }\left(t\right)x^2+{\varphi }_2^{\prime }\left(t\right)x\right]+{\varphi }_1\left(t\right),$$

$$\tilde{a}\left(\tilde{t}\right)={\displaystyle \frac{{\varphi }_3^{3}a\left(t\right)}{Q^{\prime }\left(t\right)}},\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{{\varphi }_3^{2}b\left(t\right)}{{\varphi }_4{Q}^{\prime }\left(t\right)}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{{\varphi }_3^{2}c\left(t\right)}{{\varphi }_4{Q}^{\prime }\left(t\right)}},$$

where $b\left(t\right){\varphi }_3^{\prime }\left(t\right){\varphi }_4\left(t\right)+c\left(t\right){\varphi }_4^{\prime }\left(t\right){\varphi }_3\left(t\right)=0$ and for non-degenerate transformations the condition $Q^{\prime }\left(t\right){\psi }^{\prime }\left(y\right){\varphi }_3\left(t\right){\varphi }_4\left(t\right)\ne 0$ must hold.

In the problem of Lie group classification, one of the main applications of equivalence groups is the simplification of the class of equations. In particular, variable coefficients of the equation can be fixed into constants. In the present work, the member of the usual equivalence group

$$\tilde{t}=\int a\left(t\right)dt,\tilde{x}=x,\tilde{y}=y,\tilde{u}=u,\tilde{a}=1,\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{b\left(t\right)}{a\left(t\right)}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{c\left(t\right)}{a\left(t\right)}}$$

(7)

maps Equation (3) onto (5) with the variables being tilded. In other words, we can fix $a\left(t\right)=1$. Therefore, for the symmetry analysis, we can use the equivalent Equation (5). The corresponding results for the original Equation (3) can be recalled using the inverse transformation of (7).

From transformation (7), we deduce that class (3) can be mapped onto the corresponding constant coefficient equation if $b\left(t\right)={\mu }_{1}a\left(t\right)$ and $c\left(t\right)={\mu }_{2}a\left(t\right)$, where ${\mu }_1$ and ${\mu }_2$ are constants. Additionally, if $c\left(t\right)=\mu b\left(t\right),\mu \ne 1$ then, by using the generalized extended equivalence group, class (3) can be transformed into a constant coefficient equation. Specifically, the member of the generalized extended equivalence group

$$\begin{array}{ccc}\hfill & & \tilde{t}=\int {\left[a\left(t\right)\right]}^{{\displaystyle \frac{2\mu +1}{1-\mu }}}{\left[b\left(t\right)\right]}^{{\displaystyle \frac{3\mu }{\mu -1}}}dt,\tilde{x}={\left[{\displaystyle \frac{b\left(t\right)}{a\left(t\right)}}\right]}^{{\displaystyle \frac{\mu }{\mu -1}}}x,\tilde{y}=y,\hfill \\ & & \tilde{u}={\left[{\displaystyle \frac{b\left(t\right)}{a\left(t\right)}}\right]}^{{\displaystyle \frac{1}{1-\mu }}}\left[u+{\displaystyle \frac{a\left(t\right)}{2(\mu -1){\left[b\left(t\right)\right]}^2}}{\left[{\displaystyle \frac{b\left(t\right)}{a\left(t\right)}}\right]}^{\prime }{x}^2\right]\hfill \end{array}$$

(8)

connects class (3) with $c\left(t\right)=\mu b\left(t\right),\mu \ne 1$ and the constant coefficient equation

$${\tilde{u}}_{\tilde{t}\tilde{y}}+\mu {\tilde{u}}_{\tilde{x}}{\tilde{u}}_{\tilde{x}\tilde{y}}+{\tilde{u}}_{\tilde{y}}{\tilde{u}}_{\tilde{x}\tilde{x}}+{\tilde{u}}_{\tilde{x}\tilde{x}\tilde{x}\tilde{y}}=0.$$

(9)

We set $\tilde{a}\left(\tilde{t}\right)=a\left(t\right)=1$ in Theorem 1, to obtain the corresponding equivalence groups for class (5). The usual equivalence group consists of the transformations

$$\tilde{t}={\alpha }_1^{3}t+{\beta }_2,\tilde{x}={\alpha }_{1}x+{\beta }_1,\tilde{y}=\psi \left(y\right),\tilde{u}={\alpha }_{2}u+{\varphi }_1\left(t\right),$$

$$\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{b\left(t\right)}{{\alpha }_1{\alpha }_2}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{c\left(t\right)}{{\alpha }_1{\alpha }_2}},$$

where ${\alpha }_1{\alpha }_2{\psi }_y\ne 0$. The extended equivalence group admitted by class (5) is defined by the point transformations

$$\tilde{t}={\alpha }_1^{3}t+{\beta }_2,\tilde{x}={\alpha }_{1}x+{\varphi }_2\left(t\right),\tilde{y}=\psi \left(y\right),\tilde{u}={\alpha }_{2}u+{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}{\displaystyle \frac{{\varphi }_2^{\prime }\left(t\right)}{c\left(t\right)}}x+{\varphi }_1\left(t\right),$$

$$\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{b\left(t\right)}{{\alpha }_1{\alpha }_2}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{c\left(t\right)}{{\alpha }_1{\alpha }_2}},$$

where ${\alpha }_1{\alpha }_2{\psi }_y\ne 0$. The generalized extended equivalence group consists of the transformations

$$\tilde{t}=\int {\left[{\varphi }_3\left(t\right)\right]}^{3}dt,\tilde{x}={\varphi }_3\left(t\right)x+{\varphi }_2\left(t\right),\tilde{y}=\psi \left(y\right),$$

$$\tilde{u}={\displaystyle \frac{{\varphi }_4\left(t\right)}{c\left(t\right){\varphi }_3\left(t\right)}}\left[c\left(t\right){\varphi }_3\left(t\right)u+{\displaystyle \frac{1}{2}}{\varphi }_3^{\prime }\left(t\right)x^2+{\varphi }_2^{\prime }\left(t\right)x\right]+{\varphi }_1\left(t\right),$$

$$\tilde{b}\left(\tilde{t}\right)={\displaystyle \frac{b\left(t\right)}{{\varphi }_3\left(t\right){\varphi }_4\left(t\right)}},\tilde{c}\left(\tilde{t}\right)={\displaystyle \frac{c\left(t\right)}{{\varphi }_3\left(t\right){\varphi }_4\left(t\right)}},$$

where ${\varphi }_4\left(t\right)=\nu exp\left(-\int {\displaystyle \frac{b\left(t\right)}{c\left(t\right)}}{\displaystyle \frac{{\varphi }_3^{\prime }\left(t\right)}{{\varphi }_3\left(t\right)}}dt\right)$ and ${\psi }^{\prime }\left(y\right){\varphi }_3\left(t\right)\ne 0$. Finally, we state that class (5) with $c\left(t\right)=\mu b\left(t\right),\mu \ne 1$ is connected with the constant coefficient Equation (9) under the mapping (8) with $a\left(t\right)=1$.

3. Lie Group Classification

We perform the group classification for Equation (5). The Lie algorithm for finding symmetries is well known, and it is described in the relevant textbooks (see, for example, [18,19,20,21,22]). We omit the detailed analysis, and we give, briefly, the basic steps of the method. We derive Lie operators of the form

$$\mathsf{\Gamma }=\tau (t,x,y,u){\mathsf{\partial }}_t+\xi (t,x,y,u){\mathsf{\partial }}_x+\zeta (t,x,y,u){\mathsf{\partial }}_y+\eta (t,x,y,u){\mathsf{\partial }}_u$$

that generate one-parameter Lie groups of transformations

$$\tilde{t}=t+ϵ\tau ,\tilde{x}=x+ϵ\xi ,\tilde{y}=y+ϵ\zeta ,\tilde{u}=u+ϵ\eta$$

to the first order of $ϵ$. These transformations leave Equation (5) invariant. We apply the fourth extension ${\mathsf{\Gamma }}^{\left(4\right)}$ of the operator $\mathsf{\Gamma }$, since Equation (5) is of the fourth order. We require ${\mathsf{\Gamma }}^{\left(4\right)}\left(E\right)=0$ modulus Equation (5), where E is the left-hand side of (5). This leads to a multi-variable polynomial in the derivatives u. We set the coefficients of these derivatives equal to zero, to obtain an over-determined system of linear partial differential equations with unknowns the functions $\tau ,\xi ,\zeta ,\eta$ of the operator $\mathsf{\Gamma }$ and the coefficient function $b\left(t\right),c\left(t\right)$ of Equation (5). The solution of the over-determined system leads to the desired Lie group classification.

The calculations involved in the analysis have been greatly facilitated by the computer algebraic manipulation package REDUCE. Straightforward calculations show that the Lie operator has the restricted form

$$\mathsf{\Gamma }=\tau \left(t\right){\mathsf{\partial }}_t+\left[\frac{1}{3}{\tau }^{\prime }\left(t\right)x+{\varphi }_1\left(t\right)\right]{\mathsf{\partial }}_x+\psi \left(y\right){\mathsf{\partial }}_y+\left[\frac{1}{3}{\tau }^{\prime }\left(t\right)u+{\varphi }_2\left(t\right)u+F(t,x)\right]{\mathsf{\partial }}_u$$

and the over-determined system reduces to four equations:

$$\begin{array}{ccc}& & 3b^{\prime }\left(t\right)\tau \left(t\right)+2b\left(t\right){\tau }^{\prime }\left(t\right)+3b\left(t\right){\varphi }_2\left(t\right)=0,\hfill \\ & & 3c^{\prime }\left(t\right)\tau \left(t\right)+2c\left(t\right){\tau }^{\prime }\left(t\right)+3c\left(t\right){\varphi }_2\left(t\right)=0,\hfill \\ & & 3c\left(t\right)F_x-{\tau }^{″}\left(t\right)x-3{\varphi }_1\left(t\right)=0,\hfill \\ & & 3b\left(t\right)F_{xx}+{\tau }^{″}\left(t\right)+3{\varphi }_2\left(t\right)=0.\hfill \end{array}$$

The solution of this system leads to four exclusive cases, which are listed below:

1. In the first case, $b\left(t\right),c\left(t\right)$ are arbitrary functions. Equation (5) admits an infinite dimensional Lie algebra, which is spanned by the Lie operators

$$X_1={\varphi }_1\left(t\right){\mathsf{\partial }}_x+{\displaystyle \frac{1}{c\left(t\right)}}{\varphi }_1^{\prime }\left(t\right)x{\mathsf{\partial }}_u,X_2=\psi \left(y\right){\mathsf{\partial }}_y,X_3={\varphi }_2\left(t\right){\mathsf{\partial }}_u,$$

where ${\varphi }_1\left(t\right),{\varphi }_2\left(t\right)$ and $\psi \left(y\right)$ are arbitrary functions. Additional Lie symmetries exist in the following three cases:

2. In the second case, $c\left(t\right)=\mu b\left(t\right)$, $\mu \ne 1$, where $b\left(t\right)$ is an arbitrary function. Equation (5) admits two additional Lie point symmetries, which are spanned by the operators

$$\begin{array}{ccc}& & X_4=6\mu b^n{\mathsf{\partial }}_t+2\mu x{\left(b^n\right)}^{\prime }{\mathsf{\partial }}_x+\left[b^{-1}{\left(b^n\right)}^{″}{x}^2-2\mu {\displaystyle \frac{n+3}{n}}{\left(b^n\right)}^{\prime }u\right]{\mathsf{\partial }}_u,\hfill \\ & & X_5=6\mu b^n\left(\int b^{-n}dt\right){\mathsf{\partial }}_t+2\mu x\left[1+{\left(b^n\right)}^{\prime }\left(\int b^{-n}dt\right)\right]{\mathsf{\partial }}_x\hfill \\ & & +\left\{-2\mu \left[1+(n+3)b^{n-1}{b}^{\prime }\left(\int b^{-n}dt\right)\right]u\right.\hfill \\ & & \left.+n\left[b^{-2}{b}^{\prime }+b^{n-2}{b}^{″}\left(\int b^{-n}dt\right)+(n-1)b^{n-3}{b^{\prime }}^2\left(\int b^{-n}dt\right)\right]x^2\right\}{\mathsf{\partial }}_u,n={\displaystyle \frac{3\mu }{1-\mu }}.\hfill \end{array}$$

3. In the third case, $c\left(t\right)=b\left(t\right)\ne$ constant. Equation (5) admits the additional Lie point symmetry

$$Y_4=6{\displaystyle \frac{b}{b^{\prime }}}{\mathsf{\partial }}_t+2x{\left({\displaystyle \frac{b}{b^{\prime }}}\right)}^{\prime }{\mathsf{\partial }}_x+\left\{-2\left[{\left({\displaystyle \frac{b}{b^{\prime }}}\right)}^{\prime }+3\right]u+{\displaystyle \frac{1}{b}}{\left({\displaystyle \frac{b}{b^{\prime }}}\right)}^{″}{x}^2\right\}{\mathsf{\partial }}_u.$$

4. In the fourth case, $c\left(t\right)=b\left(t\right)=\lambda$, where $\lambda$ is an arbitrary constant. Equation (5) admits an infinite dimensional Lie algebra, which is spanned by the Lie operators $X_1,X_2,X_3$ and

$$Z_4=6\lambda \tau \left(t\right){\mathsf{\partial }}_t+2\lambda x{\tau }^{\prime }\left(t\right){\mathsf{\partial }}_x+\left[-2\lambda {\tau }^{\prime }\left(t\right)u+{\tau }^{″}\left(t\right)x^2\right]{\mathsf{\partial }}_u.$$

In the previous section, we have shown that in the case $c\left(t\right)=\mu b\left(t\right),\mu \ne 1$ Equation (5) can be mapped onto the counterpart with $b\left(t\right)=1,c\left(t\right)=\mu$ using the transformation (8) with $a\left(t\right)=1$. Hence, case 2 is equivalent to $b\left(t\right)=1$ and $c\left(t\right)=\mu$ and the corresponding additional Lie symmetries are

$$X_4={\mathsf{\partial }}_t,X_5=3t{\mathsf{\partial }}_t+x{\mathsf{\partial }}_x-u{\mathsf{\partial }}_u.$$

The above cases consist of the enhanced group classification of class (5). The present analysis completes the existing results in the literature [13]. The corresponding group classification for the general class (3) can be derived using the above results and transformation (7). We present this group classification in Appendix A. The Lie group classification for the corresponding system (6) is given in Appendix B. Furthermore, in Appendix B we present the generalized extended equivalence group for class (4).

4. Examples of Similarity Reductions

The main application of Lie symmetries is the construction of reduction mappings. These have the property of reducing the number of independent variables in the equation under study. These mappings are obtained by solving the characteristic system

$${\displaystyle \frac{dt}{\tau }}={\displaystyle \frac{dx}{\xi }}={\displaystyle \frac{dy}{\zeta }}={\displaystyle \frac{du}{\eta }}.$$

We present some examples. Initially, we take the general symmetry (the linear combination of $X_1,X_2,X_3$), which corresponds to Equation (5) with arbitrary coefficient functions:

$$\mathsf{\Gamma }={\varphi }_1\left(t\right){\mathsf{\partial }}_x+\psi \left(y\right){\mathsf{\partial }}_y+\left({\displaystyle \frac{1}{c\left(t\right)}}{\varphi }_1^{\prime }\left(t\right)x+{\varphi }_2\left(t\right)\right){\mathsf{\partial }}_u.$$

This symmetry leads to the similarity reduction

$$u={\displaystyle \frac{{\varphi }_1^{\prime }\left(t\right)}{2c\left(t\right){\varphi }_1\left(t\right)}}{x}^2+{\varphi }_2\left(t\right)x+F(t,\xi ),\xi ={\displaystyle \frac{x}{{\varphi }_1\left(t\right)}}+\mathsf{\Psi }\left(y\right),$$

where the $\mathsf{\Psi }\left(y\right)$ arbitrary function relates to $\psi \left(y\right)$, which reduces (5) to the partial differential equations in two independent variables:

$$c\left(t\right){\varphi }_1^3\left(t\right)F_{t\xi }+c\left(t\right)F_{\xi \xi \xi \xi }+{\varphi }_1\left(t\right)(b\left(t\right)+c\left(t\right))c\left(t\right)F_{\xi }{F}_{\xi \xi }+c^2\left(t\right){\varphi }_1^2\left(t\right){\varphi }_2\left(t\right)F_{\xi \xi }$$

$$+b\left(t\right){\varphi }_1^2\left(t\right){\varphi }_1^{\prime }\left(t\right)F_{\xi }=0.$$

We note that this equation is a special case of the class

$$u_{tx}+{\beta }_1\left(t\right)u_{xxxx}+{\beta }_2\left(t\right)u_x{u}_{xx}+{\beta }_3\left(t\right)u_{xx}+{\beta }_4\left(t\right)u_x=0$$

which is connected with

$${\tilde{u}}_{\tilde{t}\tilde{x}}+{\tilde{\beta }}_1\left(\tilde{t}\right){\tilde{u}}_{\tilde{x}\tilde{x}\tilde{x}\tilde{x}}+{\tilde{\beta }}_2\left(\tilde{t}\right){\tilde{u}}_{\tilde{x}}{\tilde{u}}_{\tilde{x}\tilde{x}}=0$$

under the mapping

$$\tilde{t}=t,\tilde{x}=x-\int {\beta }_3\left(t\right)dt,\tilde{u}={\mathrm{e}}^{\int {\beta }_4\left(t\right)dt}u.$$

Hence, we can take ${\varphi }_1\left(t\right)=1$ and ${\varphi }_2\left(t\right)=0$, and the reduced equation becomes

$$F_{t\xi }+F_{\xi \xi \xi \xi }+\left[b\left(t\right)+c\left(t\right)\right]F_{\xi }{F}_{\xi \xi }=0.$$

Furthermore, we can take $\psi \left(y\right)=y$. We point out that we do not lose the generality for restricting these three functions, since any results for arbitrary ${\varphi }_1\left(t\right),{\varphi }_2\left(t\right)$ and $\psi \left(y\right)$ can be recalled using the equivalence transformations.

We integrate, once, the above reduced equation with respect to $\xi$, to obtain the potential form of the variable coefficient KdV equation

$$F_t+F_{\xi \xi \xi }+{\displaystyle \frac{1}{2}}d\left(t\right)F_{\xi }^2=0,$$

(10)

where $d\left(t\right)=b\left(t\right)+c\left(t\right)$. The arbitrary function of integration can be taken as equal to zero, since the mapping $F\mapsto F+\int k\left(t\right)dt$ transforms the nonhomogeneous equation into the homogeneous, where $k\left(t\right)$ is the integrating function. Equation (10) and the variable coefficient KdV equation

$$G_t+G_{\xi \xi \xi }+d\left(t\right)G{G}_{\xi }=0$$

(11)

are connected by the Bäcklund transformations

$$F_{\xi }=G,F_t=-G_{\xi \xi }-{\displaystyle \frac{1}{2}}d\left(t\right)G^2.$$

We note that in the case $c\left(t\right)=-b\left(t\right)$, Equation (5) can be reduced to the linear equation $F_t+F_{\xi \xi \xi }=0$. The Lie group classification for an equivalent equation to (10) can be found in [23], and for (11) it is presented in [24].

As a second example, we consider case 2, where $c\left(t\right)=\mu b\left(t\right)$, $\mu \ne 1$. We have shown, using equivalence transformations, that this case is equivalent to $b\left(t\right)=1,c\left(t\right)=\mu \ne 1$. We use the Lie symmetries

$${\mathsf{\Gamma }}_1=\lambda X_1+\lambda X_2+X_4=\lambda {\mathsf{\partial }}_x+\lambda y{\mathsf{\partial }}_y+{\mathsf{\partial }}_t,{\mathsf{\Gamma }}_2=X_5=3t{\mathsf{\partial }}_t+x{\mathsf{\partial }}_x-u{\mathsf{\partial }}_u,$$

where we have taken ${\varphi }_1\left(t\right)=\psi \left(y\right)=1$. The symmetry ${\mathsf{\Gamma }}_1$ produces the reduction

$$u=F(\xi ,\eta ),\xi =x-\lambda t,\eta =y-\lambda t$$

that reduces (5) to

$$F_{\eta \xi \xi \xi }+\mu F_{\xi }{F}_{\eta \xi }-\lambda (F_{\eta \eta }+F_{\eta \xi })+F_{\eta }{F}_{\xi \xi }=0.$$

The above equation admits the Lie symmetries ${\mathsf{\partial }}_{\eta },{\mathsf{\partial }}_{\xi },{\mathsf{\partial }}_F,3\mu \eta {\mathsf{\partial }}_{\eta }+\mu \xi {\mathsf{\partial }}_{\xi }+(2\lambda \xi -\mu F){\mathsf{\partial }}_F$. Combining the first two Lie symmetries and the above mapping, we derive the double reduction

$$u=F\left(w\right),w=x+y-\lambda t$$

that reduces (5) with $b\left(t\right)=1,c\left(t\right)=\mu \ne 1$ to the ordinary differential equation

$$F^{\left(iv\right)}+(\mu +1)F^{\prime }{F}^{″}-\lambda F^{″}=0.$$

We note again that if $\mu =-1$ ($c\left(t\right)=-b\left(t\right)$) then the above equation can be solved in terms of elementary functions. For $\mu \ne -1$, we have the solution

$$\int {\displaystyle \frac{dG}{\sqrt{\lambda G^2-\frac{1}{3}(\mu +1)G^3+c_{1}w+c_2}}}=w+c_3,G=F^{\prime }.$$

The symmetry ${\mathsf{\Gamma }}_2$ produces the similarity mapping

$$u=F(\xi ,\eta ),\xi ={\displaystyle \frac{x}{t^{\frac{1}{3}}}},\eta =y$$

that reduces (5) with $b\left(t\right)=1,c\left(t\right)=\mu \ne 1$ to

$$F_{\eta \xi \xi \xi }+\mu F_{\xi }{F}_{\eta \xi }-\frac{1}{3}{\left[\xi F_{\eta }\right]}_{\xi }+F_{\eta }{F}_{\xi \xi }=0.$$

This equation admits an infinite dimensional Lie algebra, which is spanned by the Lie operators $\psi \left(\eta \right){\mathsf{\partial }}_{\eta },3\mu {\mathsf{\partial }}_{\xi }+\xi {\mathsf{\partial }}_F,{\mathsf{\partial }}_F$. Motivated by these Lie symmetries and the above reduction mapping, we obtain the double reduction

$$u={\displaystyle \frac{1}{6\mu }}{\displaystyle \frac{x^2}{t}}+{\displaystyle \frac{1}{t^{\frac{1}{3}}}}F\left(w\right),w={\displaystyle \frac{x}{t^{\frac{1}{3}}}}-3\mu y$$

that reduces (5) with $b\left(t\right)=1,c\left(t\right)=\mu \ne 1$ to the ordinary differential equation

$$F^{\left(iv\right)}\left(w\right)+(\mu +1)F^{\prime }\left(w\right)F^{″}\left(w\right)+{\displaystyle \frac{1}{3\mu }}(1-\mu )F^{\prime }\left(w\right)=0.$$

As before, we note that if $\mu =-1$ ($c\left(t\right)=-b\left(t\right)$) then the above equation can be solved in terms of elementary functions. Also, for $\mu =1$ we obtain the solution

$$\int {\displaystyle \frac{dG}{\sqrt{-\frac{2}{3}{G}^3+c_{1}w+c_2}}}=w+c_3,G=F^{\prime }.$$

Similarity solutions for the case $c\left(t\right)=b\left(t\right)=\lambda$, which is the BLMP equation, can be found in [5,6,8]. If we set $b\left(t\right)=c\left(t\right)=\lambda$ in the first example of the section, the results correspond to the BLMP equation. In other words, BLMP can be reduced to KdV using similarity reductions. Now, we use the additional symmetry that is admitted by (5) in the case where $c\left(t\right)=b\left(t\right)=\lambda$:

$$\mathsf{\Gamma }=6\lambda \tau \left(t\right){\mathsf{\partial }}_t+2\lambda x{\tau }^{\prime }\left(t\right){\mathsf{\partial }}_x+\psi \left(y\right){\mathsf{\partial }}_y+\left(-2\lambda {\tau }^{\prime }\left(t\right)u+{\tau }^{″}\left(t\right)x^2\right){\mathsf{\partial }}_u.$$

We derive the similarity mapping

$$u={\displaystyle \frac{x^2}{6\lambda }}{\displaystyle \frac{{\tau }^{\prime }\left(t\right)}{\tau \left(t\right)}}+{\tau }^{-{\displaystyle \frac{1}{3}}}\left(t\right)F(\eta ,\xi ),\eta =\int {\displaystyle \frac{dt}{\tau \left(t\right)}}+A\left(y\right),\xi ={\tau }^{-{\displaystyle \frac{1}{3}}}\left(t\right)x$$

that reduces (5) with $c\left(t\right)=b\left(t\right)=\lambda$ to

$$F_{\eta \eta }+\lambda {\left(F_{\eta }{F}_{\xi }\right)}_{\xi }+F_{\eta \xi \xi \xi }=0.$$

(12)

The reduced equation (12) admits four Lie symmetries:

$$X_1={\mathsf{\partial }}_{\eta },X_2={\mathsf{\partial }}_{\xi },X_3={\mathsf{\partial }}_F,X_4=3\eta {\mathsf{\partial }}_{\eta }+\xi {\mathsf{\partial }}_{\xi }-F{\mathsf{\partial }}_F.$$

Using these Lie symmetries and the above similarity mapping, we present two double reductions. The first is the mapping

$$u={\displaystyle \frac{x^2}{6\lambda }}{\displaystyle \frac{{\tau }^{\prime }\left(t\right)}{\tau \left(t\right)}}+{\tau }^{-{\displaystyle \frac{1}{3}}}\left(t\right)F\left(w\right),w=\int {\displaystyle \frac{dt}{\tau \left(t\right)}}+A\left(y\right)+{\tau }^{-{\displaystyle \frac{1}{3}}}\left(t\right)x$$

that reduces (5) with $c\left(t\right)=b\left(t\right)=\lambda$ to the ordinary differential equation

$$F^{\left(iv\right)}\left(w\right)+2\lambda F^{\prime }\left(w\right)F^{″}\left(w\right)+F^{″}\left(w\right)=0.$$

The second double similarity mapping has the form

$$u={\displaystyle \frac{x^2}{6\lambda }}{\displaystyle \frac{{\tau }^{\prime }\left(t\right)}{\tau \left(t\right)}}+{\displaystyle \frac{1}{x}}F\left(w\right),w={\displaystyle \frac{x^3}{\tau \left(\int \frac{dt}{\tau \left(t\right)}+A\left(y\right)\right)}}$$

that reduces (5) with $c\left(t\right)=b\left(t\right)=\lambda$ to the ordinary differential equation

$$\begin{array}{ccc}& & 27w^2{F}^{\left(iv\right)}\left(w\right)+108w{F}^{‴}\left(w\right)+18\lambda w{F}^{\prime }\left(w\right)F^{″}\left(w\right)-3\lambda F\left(w\right)F^{″}\left(w\right)\hfill \\ & & -w{F}^{″}\left(w\right)+60F^{″}\left(w\right)+6\lambda {\left(F^{\prime }\left(w\right)\right)}^2-2F^{\prime }\left(w\right)=0.\hfill \end{array}$$

We point out that if we set $F_{\eta }=G,F_{\xi }=H$ then the reduced Equation (12) can be written as a system:

$$G_{\eta }+\lambda {\left(GH\right)}_{\xi }+G_{\xi \xi \xi }=0,G_{\xi }=H_{\eta }.$$

5. Non-Lie Reduction Operators

Bluman and Cole [25,26] considered a new approach for constructing reduction operators that are different from Lie operators. In this method, we require the invariance of Equation (5) in conjunction with the invariant surface condition $\tau u_t+\xi u_x+\zeta u_y=\eta$ under the infinitesimal transformations generated by the Lie operator $\mathsf{\Gamma }=\tau {\mathsf{\partial }}_t+\xi {\mathsf{\partial }}_x+\zeta {\mathsf{\partial }}_y+\eta {\mathsf{\partial }}_u$. Here, the task is more difficult since the determining systems consist of non-linear equations. More details and applications of the method can be found, for example, in reference [17].

The analysis for deriving such reduction operators for class (5) is lengthy and very difficult to complete. Here, we present four examples of such non-Lie reduction operators. For arbitrary functions $b\left(t\right)$ and $c\left(t\right)$, Equation (5) admits the operator

$${\mathsf{\partial }}_x+\left[2{\mu }_1(t,y)x+{\mu }_2(t,y)\right]{\mathsf{\partial }}_u,$$

where

$${\mu }_{1ty}+2(b+2c){\mu }_1{\mu }_{1y}=0,{\mu }_{2ty}+2c{\mu }_2{\mu }_{1y}+2(b+c){\mu }_1{\mu }_{2y}=0.$$

We obtain the similarity mapping

$$u={\mu }_1(t,y)x^2+{\mu }_2(t,y)x+F(t,y)$$

that reduces (5) to

$$F_{ty}+2b{\mu }_1{F}_y+c{\mu }_2{\mu }_{2y}=0.$$

In this case, the procedure is equivalent to that of searching for solutions of (5) that are quadratic in x.

If $c\left(t\right)=-b\left(t\right)$ then we have the following two examples of reduction operators admitted by (5):

$${\displaystyle \frac{1}{y}}{\mathsf{\partial }}_x+{\mathsf{\partial }}_y+{\mathrm{e}}^{t-x}{\mathsf{\partial }}_u$$

and

$${\mathsf{\partial }}_y+\varphi \left(t\right){\mathrm{e}}^{\psi \left(t\right)x}{\mathsf{\partial }}_u.$$

The first operator produces the similarity mapping

$$u=xy{e}^{t-x}+F(t,\xi ),\xi =x-ln\left|y\right|$$

that reduces (5) with $c\left(t\right)=-b\left(t\right)$ to

$$F_{t\xi }+F_{\xi \xi \xi \xi }-b\left(t\right)e^{t-\xi }(F_{\xi \xi }+F_{\xi })+e^{2(t-\xi )}b\left(t\right)-3e^{t-\xi }=0.$$

From the second operator, we find the similarity mapping

$$u=y\varphi \left(t\right)e^{\psi \left(t\right)x}+F(t,x)$$

that reduces (5) with $c\left(t\right)=-b\left(t\right)$ to

$$F_{xx}-\psi \left(t\right)F_x=-{\displaystyle \frac{{\varphi }_t+x\varphi {\psi }_t+\varphi {\psi }^3}{b\left(t\right)\varphi }},$$

with the solution

$$F(t,x)={\displaystyle \frac{{\psi }_t{x}^2}{2b\left(t\right)\psi }}+{\displaystyle \frac{{\psi }_{t}x}{b\left(t\right){\psi }^2}}+{\displaystyle \frac{({\varphi }_t+\varphi {\psi }^3)x}{b\left(t\right)\varphi \psi }}+\theta \left(t\right)e^{\psi x}+\eta \left(t\right).$$

Finally, if $c={\displaystyle \frac{1-n}{n}}b,n\ne 0,1$, we find the non-Lie operator

$${\mathsf{\partial }}_y+{\left[\varphi \left(t\right)x+\psi \left(t\right)\right]}^n{\mathsf{\partial }}_u$$

which produces the reduction mapping

$$u={\left[\varphi \left(t\right)x+\psi \left(t\right)\right]}^{n}y+F(t,x)$$

that transforms (5) into

$$F_{xx}-{\displaystyle \frac{(n-1)\varphi \left(t\right)}{\varphi x+\psi }}{F}_x+{\displaystyle \frac{n(x{\varphi }_t+{\psi }_t)}{b(\varphi x+\psi )}}+{\displaystyle \frac{n(n-1)(n-2){\varphi }^3}{b{(\varphi x+\psi )}^3}}=0$$

which can be easily integrated twice with respect to x to give the form of $F(t,x)$.

6. Conclusions

We have performed the enhanced Lie group classification for the class of variable coefficient Boiti–Leon–Manna–Pempinelli equations, which completes the existing results in the literature. These results were achieved with the aid of the equivalence groups admitted by the class. Furthermore, equivalence transformations were used to find those equations from the general class that could be mapped onto constant coefficient equations. The derived Lie symmetries were used to construct similarity reductions. A number of exact solutions were obtained. More solutions could be derived using numerical methods. The method of Bluman and Cole was employed to derive reduction operators that were not equivalent to Lie ones.

The analysis used in the present work can be used for similar classes of differential equations. For example, the class of variable coefficient equations

$$u_{ty}+a\left(t\right)u_{xxxy}+b\left(t\right)u_{xx}{u}_y+c\left(t\right)u_{xy}{u}_x+\alpha \left(t\right)u_{xy}$$

(13)

appeared recently in the literature [27]. From the equivalence group of class (13), we deduce the mapping

$$u\mapsto u+{\displaystyle \frac{\alpha \left(t\right)}{c\left(t\right)}}x$$

which maps it onto class (3). All the properties of (3) can be transformed for class (13), using the above mapping.