Non-local Substitutions for Liouville Equations with Three and Four Independent Variables

Abstract

We obtained the non-local transformations of the Cole—Hopf type, which translate the Liouville equations with three and four independent variables into the Bianchi equations. The solutions with arbitrary functions of these Liouville equations are constructed.

1 On the Group Properties of Bianchi Equations

Consider a homogeneous equation with a dominant partial derivative with variable coefficients (Bianchi equation)

$$\begin{aligned} u_{xyz}+au_{xy}+bu_{yz}+cu_{xz}+du_{x}+eu_{y}+fu_{z}+gu=0. \end{aligned}$$

(1)

In the paper [1] some group properties of this equation have been considered. It is known that the set of equivalence transformations for (1)

$$\begin{aligned} \overline{x}=\alpha (x),\quad \overline{y}=\beta (y),\quad \overline{z}=\gamma (z),\quad u=\omega (x,y,z)\overline{u}. \end{aligned}$$

(2)

Two equations of the form (1) are called equivalent in function [2, p 117], if they pass into each other during transformations (2), in which

$$\begin{aligned} \alpha (x)=x,\quad \beta (y)=y,\quad \gamma (z)=z. \end{aligned}$$

In the paper [3] it was shown that two equations of the form (1) are equivalent in function if and only if the Laplace invariants

$$\begin{aligned} \begin{array}{c} H_1=a_y+ac-d,\, H_2=a_x+ab-e,\,H_3=c_x+bc-f,\\ H_4=b_z+ab-e,\, H_5=b_y+bc-f,\, H_6=c_z+ac-d,\\ H_7=a_{xy}+bd+ce+af-2abc-g,\\ H_8=b_{yz}+bd+ce+af-2abc-g,\\ H_9=c_{xz}+bd+ce+af-2abc-g \end{array} \end{aligned}$$

(3)

are the same for both equations.

If we look for the operator allowed by the Eq. (1)

$$\begin{aligned} \alpha \partial _x +\beta \partial _y +\gamma \partial _z +\tau \partial _u, \end{aligned}$$

then it turns out that part of the system of defining equations will be

$$\begin{aligned} \partial _u \alpha =\partial _u \beta =\partial _u \gamma =0,\quad \partial ^2_u \tau =0. \end{aligned}$$

It is known [2, pp. 99–100] that in this case the Lie algebra of the Eq. (1) there is $L=L^r\oplus L^{\infty }$, where the algebra $L^r$ of dimension r is formed by operators of the form

$$\begin{aligned} X=\xi ^1 (x,y,z)\partial _x +\xi ^2 (x,y,z)\partial _y +\xi ^3 (x,y,z)\partial _z +\sigma (x,y,z)u\partial _u, \end{aligned}$$

(4)

and $L^{\infty }$ is an Abelian subalgebra typical of linear equations with the operator $\omega (x,y,z)\partial _u$, where $\omega$ is the solution of the Eq. (1). It is clear that the operator $u\partial _u$ is allowed by any Eq. (1), therefore, this operator can be included in $L^{\infty }$ and assume that $\sigma (x,y,z)$ is defined in (4) up to a constant summand.

To construct the defining equations we use the third continuation of the operator (4)

$$\begin{aligned} {X_3}=\xi ^1 \partial _x +\xi ^2 \partial _y+\xi ^3 \partial _z +\sigma u\partial _u +\tau ^1 \partial _{u_1}+\tau ^2 \partial _{u_2}+ \tau ^3 \partial _{u_3}+ \end{aligned}$$

$$\begin{aligned} +\tau ^{11} \partial _{u_{11}}+ \tau ^{12} \partial _{u_{12}}+\tau ^{13} \partial _{u_{13}} +\tau ^{22} \partial _{u_{22}}+\tau ^{23} \partial _{u_{23}}+ \tau ^{33} \partial _{u_{33}}+ \end{aligned}$$

$$\begin{aligned} +\tau ^{111} \partial _{u_{111}} +\tau ^{112} \partial _{u_{112}} +\tau ^{113} \partial _{u_{113}} +\tau ^{122} \partial _{u_{122}} +\tau ^{123} \partial _{u_{123}}+ \end{aligned}$$

$$\begin{aligned} +\tau ^{133} \partial _{u_{133}} +\tau ^{222} \partial _{u_{222}} +\tau ^{223} \partial _{u_{223}} +\tau ^{233} \partial _{u_{233}} +\tau ^{333} \partial _{u_{333}}. \end{aligned}$$

The notation used here is $u_1=u_x$, $u_2=u_x$,..., $u_{12}=u_{xy}$,..., $u_{333}=u_{zzz}$. We get

$$\begin{aligned} \begin{array}{c} \tau ^1=\sigma _x u+(\sigma -\xi ^1_x)u_1 -\xi ^2_x u_2 -\xi ^3_x u_3,\\ \tau ^2=\sigma _y u -\xi ^1_y u_1 +(\sigma -\xi ^2_y)u_2 -\xi ^3_y u_3,\\ \tau ^3=\sigma _z u -\xi ^1_z u_1 -\xi ^2_z u_2 +(\sigma -\xi ^3_z)u_3 ,\\ \tau ^{12}=\sigma _{xy} u +(\sigma _y -\xi ^1_{xy})u_1 +(\sigma _x -\xi ^2_{xy})u_2 -\xi ^3_{xy}u_3-\\ -\xi ^1_y u_{11}+(\sigma -\xi ^1_x-\xi ^2_y)u_{12} -\xi ^3_y u_{13}-\xi ^2_x u_{22}-\xi ^3_x u_{23},\\ \tau ^{13}=\sigma _{xz} u +(\sigma _z -\xi ^1_{xz})u_1 -\xi ^2_{xz}u_2+(\sigma _x -\xi ^3_{xz})u_3 -\\ -\xi ^1_z u_{11} -\xi ^2_z u_{12} +(\sigma -\xi ^1_x-\xi ^3_z) u_{13} -\xi ^2_x u_{23}-\xi ^3_x u_{33},\\ \tau ^{23}=\sigma _{yz} u -\xi ^1_{yz}u_1+(\sigma _z -\xi ^2_{yz})u_2 +(\sigma _y -\xi ^3_{yz})u_3 -\\ -\xi ^1_z u_{12} -\xi ^1_y u_{13} -\xi ^2_z u_{22} +(\sigma -\xi ^1_y-\xi ^3_z)u_{23} -\xi ^3_y u_{33},\\ \tau ^{123}=\sigma _{xyz} u +(\sigma _{yz} -\xi ^1_{xyz})u_1 +(\sigma _{xz} -\xi ^2_{xyz})u_2+(\sigma _{xy} -\xi ^3_{xyz})u_3-\\ -\xi ^1_{yz}u_{11}+(\sigma _{z} -\xi ^2_{yz}-\xi ^1_{xz})u_{12} +(\sigma _{y} -\xi ^1_{xy}-\xi ^3_{yz})u_{13}-\\ -\xi ^2_{xz} u_{22} +(\sigma _{x} -\xi ^3_{xz}-\xi ^2_{xy})u_{23} -\xi ^3_{xy} u_{33}-\\ -\xi ^1_{z}u_{112}-\xi ^1_{y}u_{113} -\xi ^2_{z}u_{122} +(\sigma -\xi ^1_{x}-\xi ^2_{y}-\xi ^3_{z})u_{123}-\\ -\xi ^3_{y}u_{133}-\xi ^2_{x}u_{223}-\xi ^3_{x}u_{233}. \end{array} \end{aligned}$$

By applying the operator ${X_3}$ to the Eq. (1), we obtain the defining equations

$$\begin{aligned} \begin{array}{c} \xi ^1_y=\xi ^1_z=\xi ^2_x=\xi ^2_z=\xi ^3_x=\xi ^3_y=0,\\ \sigma _x+(b\xi ^1)_x+b_y \xi ^2 +b_z \xi ^3=0,\\ \sigma _y+c_x \xi ^1+(c\xi ^2)_y +c_z \xi ^3=0,\\ \sigma _z+a_x \xi ^1+a_y \xi ^2+ (a\xi ^3)_z=0,\\ \sigma _{xy}+c\sigma _x+b\sigma _y +(f\xi ^1)_x+(f\xi ^2)_y+f_z \xi ^3=0,\\ \sigma _{xz}+a\sigma _x+b\sigma _z +(e\xi ^1)_x+e_y\xi ^2+(e\xi ^3)_z=0,\\ \sigma _{yz}+a\sigma _y+c\sigma _z +d_x \xi ^1+(d\xi ^2)_y+(d\xi ^3)_z=0,\\ \sigma _{xyz}+a\sigma _{xy}+b\sigma _{yz}+c\sigma _{xz}+d\sigma _x +e\sigma _y+f\sigma _z+\\ +(g\xi ^1)_x+(g\xi ^2)_y+(g\xi ^3)_z=0. \end{array} \end{aligned}$$

(5)

Defining Eq. (5) can be written using Laplace invariants (3) in the form

$$\begin{aligned} \begin{array}{c} \xi ^1_y=\xi ^1_z=\xi ^2_x=\xi ^2_z=\xi ^3_x=\xi ^3_y=0,\\ (\sigma +b\xi ^1+c\xi ^2 +a\xi ^3)_x=(H_3-H_5)\xi ^2+(H_2-H_4)\xi ^3,\\ (\sigma +b\xi ^1+c\xi ^2 +a\xi ^3)_y=(H_5-H_3)\xi ^1+(H_1-H_6)\xi ^3,\\ (\sigma +b\xi ^1+c\xi ^2 +a\xi ^3)_z=(H_4-H_2)\xi ^1+(H_6-H_1)\xi ^2,\\ H_{1x}\xi ^1+(H_1\xi ^2)_y +(H_1\xi ^3)_z=0,\\ H_{6x}\xi ^1+(H_6\xi ^2)_y +(H_6\xi ^3)_z=0,\\ (H_2\xi ^1)_x +H_{2y}\xi ^2+(H_2\xi ^3)_z=0,\\ (H_4\xi ^1)_x +H_{4y}\xi ^2+(H_4\xi ^3)_z=0,\\ (H_3\xi ^1)_x+(H_3\xi ^2)_y +H_{3z}\xi ^3=0,\\ (H_5\xi ^1)_x+(H_5\xi ^2)_y +H_{5z}\xi ^3=0,\\ (H_7\xi ^1)_x+(H_7\xi ^2)_y +(H_{7}\xi ^3)_z=0,\\ (H_8\xi ^1)_x+(H_8\xi ^2)_y +(H_{8}\xi ^3)_z=0,\\ (H_9\xi ^1)_x+(H_9\xi ^2)_y +(H_{9}\xi ^3)_z=0. \end{array} \end{aligned}$$

(6)

The first row in (6) shows that

$$\begin{aligned} \xi ^{i}=\xi ^{i}(x_i), \quad i=\overline{1,3}. \end{aligned}$$

The second, third and fourth rows from (6) are differential equations for determining the function $\sigma$, after $\xi ^1$, $\xi ^2$, $\xi ^3$ have been obtained. The equations starting from the fifth row are responsible for the results of the group classification.

Some consequences can be deduced directly from the defining equations in the form (6). If all $H_i$, $i=\overline{1,9}$, are identically equal to zero, then the Eq. (1) is equivalent to the equation $u_{xyz}=0$ and admits an infinite-dimensional Lie algebra of operators of the form

$$\begin{aligned} \xi ^1(x)\partial _x+\xi ^2(y)\partial _y+\xi ^3(z)\partial _z \end{aligned}$$

with arbitrary $\xi ^1(x)$, $\xi ^2(y)$, $\xi ^3(z)$.

Let’s introduce the relations into consideration

$$\begin{aligned} p_{12}=\frac{H_3}{H_5},\quad p_{13}=\frac{H_2}{H_4},\quad p_{23}=\frac{H_1}{H_6}, \end{aligned}$$

(7)

$$\begin{aligned} \begin{array}{c} q_1=\frac{(\ln H_1)_{yz}}{H_1},\quad q_2=\frac{(\ln H_2)_{xz}}{H_2},\quad q_3=\frac{(\ln H_3)_{xy}}{H_3},\\ q_4=\frac{(\ln H_4)_{xz}}{H_4},\quad q_5=\frac{(\ln H_5)_{xy}}{H_5},\quad q_6=\frac{(\ln H_6)_{yz}}{H_6},\\ q_i=\frac{(\ln H_i)_{xyz}}{H_i},\quad i=7,\, 8,\, 9. \end{array} \end{aligned}$$

(8)

Substitute $H_1=p_{23}H_6$, $H_6\not =0$, in the fifth row (6)

$$\begin{aligned} p_{23}(H_{6x}\xi ^1+(H_6\xi ^2)_y+ (H_6\xi ^3)_z)+p_{23x}H_{6}\xi ^1+p_{23y}H_{6}\xi ^2+p_{23z}H_{6}\xi ^3=0. \end{aligned}$$

Since the term in parentheses vanishes, it follows

$$\begin{aligned} \xi ^1 p_{23x}+\xi ^2 p_{23y}+\xi ^3 p_{23z}=0. \end{aligned}$$

(9)

The identity (9) means that either $p_{23}=const$ or $p_{23}$ is an invariant of the group G with the operator (4).

If $p_{23}=const$, then from the fifth and sixth rows (6) we get

$$\begin{aligned} \xi ^1(\ln {H_{6}})_x+\xi ^2(\ln {H_{6}})_y +\xi ^3(\ln {H_{6}})_z +\xi ^2_y+\xi ^3_z=0. \end{aligned}$$

(10)

Differentiating by y, z we get

$$\begin{aligned} \xi ^1\frac{((\ln {H_{6}})_{yz})_x}{(\ln {H_{6}})_{yz}} +\xi ^2\frac{((\ln {H_{6}})_{yz})_y}{(\ln {H_{6}})_{yz}} +\xi ^3\frac{((\ln {H_{6}})_{yz})_z}{(\ln {H_{6}})_{yz}} +\xi ^2_y+\xi ^3_z=0. \end{aligned}$$

(11)

Subtracting (10) from (11) and then multiplying by $(\ln {H_{6}})_{yz}/H_{6}$, we get

$$\begin{aligned} \xi ^1 q_{6x} +\xi ^2 q_{6y} +\xi ^3 q_{6z}=0. \end{aligned}$$

Thus, again either $q_{6}=const$ or $q_{6}$ is an invariant of the group G with the operator (4).

Then similar identities can be obtained for $p_{12}$, $p_{13}$, $q_i$, $i=\overline{1,5}$.

Similar identities can be obtained for relations

$$\begin{aligned} P_{1}=\frac{H_7}{H_8},\quad P_{2}=\frac{H_7}{H_9},\quad P_{3}=\frac{H_8}{H_9}. \end{aligned}$$

For example, considering the relation $P_{1}$, we come to the identity

$$\begin{aligned} \xi ^1 P_{1x} +\xi ^2 P_{1y} +\xi ^3 P_{1z}=0. \end{aligned}$$

Again, either $P_{1}=const$, or $P_{1}$ is an invariant of the group G with the operator (4). If $P_{1}=const$, then row 12 from (6) gives

$$\begin{aligned} \xi ^1(\ln {H_{8}})_x+\xi ^2(\ln {H_{8}})_y +\xi ^3(\ln {H_{8}})_z +\xi ^1_x+\xi ^2_y+\xi ^3_z=0. \end{aligned}$$

(12)

Differentiating by x, y, z we get

$$\begin{aligned} \xi ^1\frac{((\ln {H_{8}})_{xyz})_x}{(\ln {H_{8}})_{xyz}} +\xi ^2\frac{((\ln {H_{8}})_{xyz})_y}{(\ln {H_{8}})_{xyz}} +\xi ^3\frac{((\ln {H_{8}})_{xyz})_z}{(\ln {H_{8}})_{xyz}} +\xi ^1_x+\xi ^2_y+\xi ^3_z=0. \end{aligned}$$

(13)

Subtracting (12) from (13) and multiplying by $(\ln {H_{8}})_{xyz}/H_{8}$, we get

$$\begin{aligned} \xi ^1 q_{8x} +\xi ^2 q_{8y} +\xi ^3 q_{8z}=0. \end{aligned}$$

Thus, either $q_{8}=const$ or $q_{8}$ is an invariant of the group G with the operator (4).

Based on the above statements, classes of equations of the form (1) admitting Lie algebras of the largest dimensions were listed in the work [1].

In the case when $q_i=const$, $i=\overline{1,6}$, the invariant $H_i$ is a solution of the Liouville equation (this follows from (8)), the formula of the general solution of which is known [2, p 123]. Similarly, if any of the constructions $q_i$, $i=\overline{7,9}$, is constant, then the corresponding invariant $H_i$ is the solution of the equation

$$\begin{aligned} (\ln H_i)_{xyz}=q_i H_i. \end{aligned}$$

In this regard, the task of constructing is of interest exact solutions of the three-dimensional analogue of the Liouville equation

$$\begin{aligned} u_{xyz}=e^u. \end{aligned}$$

(14)

We can propose the following method of constructing an exact solution based on the application of Lie groups of point transformations.

The usual algorithm for calculating the group of point transformations allowed by the Eq. (14) leads to the Lie algebra of operators

$$\begin{aligned} X=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z - (\xi '(x) +\eta '(y) + \zeta '(z))\partial _u, \end{aligned}$$

where $\xi (x)$, $\eta (y)$, $\zeta (z)$ are arbitrary functions.

To determine the invariants of the group allowed by the Eq. (14), we obtain the system

$$\begin{aligned} \frac{dx}{\xi (x)}=\frac{dy}{\eta (y)}= \frac{dz}{\zeta (z)}= \frac{du}{-\xi '(x)-\eta '(y)-\zeta '(z)}. \end{aligned}$$

(15)

The first integrals of the system (15) have the form

$$\begin{aligned} u+\ln |\xi (x)\eta (y)\zeta (z)|=C_1, \end{aligned}$$

$$\begin{aligned} \varphi (x)-\psi (y)=C_2,\quad \varphi (x)-\chi (z)=C_3, \end{aligned}$$

$$\begin{aligned} \varphi '(x)=\frac{1}{\xi (x)},\quad \psi '(y)=\frac{1}{\eta (y)},\quad \chi '(z)=\frac{1}{\zeta (z)}. \end{aligned}$$

Let’s introduce new variables

$$\begin{aligned} v=u+\ln |\xi (x)\eta (y)\zeta (z)|,\quad t=\varphi (x)-\psi (y),\quad \tau =\varphi (x)-\chi (z). \end{aligned}$$

Invariant with respect to the group of point transformations allowed by the Eq. (14) , the solution has the form $v=w(t,\tau )$. As a result, we come to the equation for determining the function w

$$\begin{aligned} w_{tt\tau } +w_{t\tau \tau }=e^w. \end{aligned}$$

(16)

The Eq. (16) has a solution

$$\begin{aligned} w=\ln \frac{-12}{(t+\tau )^3}. \end{aligned}$$

Then (here $\xi (x)\eta (y)\zeta (z)>0$)

$$\begin{aligned} u=-\ln (\xi (x)\eta (y)\zeta (z))+ \ln \frac{-12}{(2\varphi (x)-\psi (y)-\chi (z))^3}= \end{aligned}$$

$$\begin{aligned} =\ln \frac{-12\frac{1}{\xi (x)}\frac{1}{\eta (y)} \frac{1}{\zeta (z)}}{(2\varphi (x)-\psi (y)-\chi (z))^3}. \end{aligned}$$

Denoting $\lambda (x)=2\varphi (x)$, $\mu (y)=-\psi (y)$, $\nu (z)=-\chi (z)$, we obtain an exact solution of the Eq. (14), depending on three arbitrary functions

$$\begin{aligned} u=\ln \frac{-6\lambda '(x)\mu '(y) \nu '(z)}{(\lambda (x)+\mu (y)+\nu (z))^3}. \end{aligned}$$

In [4, 5] some group properties of the fourth-order Bianchi equation were considered. The homogeneous Bianchi equation of the fourth order is

$$\begin{aligned} \begin{array}{c} u_{x_1 x_2 x_3 x_4}+a_{1}u_{x_2 x_3 x_4} +a_{2}u_{x_1 x_3 x_4}+a_{3}u_{x_1 x_2 x_4}+a_{4}u_{x_1 x_2 x_3}+\\ +a_{12}u_{x_3 x_4}+a_{13}u_{x_2 x_4}+a_{14}u_{x_2 x_3} +a_{23}u_{x_1 x_4}+a_{24}u_{x_1 x_3}+ a_{34}u_{x_1 x_2}+\\ +a_{123}u_{x_4}+a_{124}u_{x_3}+a_{134}u_{x_2}+a_{234}u_{x_1}+a_{1234} u=0. \end{array} \end{aligned}$$

(17)

It is implied here that the coefficients are variable.

The Laplace invariants for this equation have the form

$$\begin{aligned} \begin{array}{c} h_{i,j}=a_{ix_j}+a_{i}a_{j}-a_{ij},\\ h_{i,jk}=a_{i x_j x_k}+a_{i}a_{jk}+a_{j}a_{ik}+a_{k}a_{ij}- 2a_{i}a_{j}a_{k}-a_{ijk}, \\ h_{i,jkl}=a_{i x_j x_k x_l}+a_{i}a_{jkl}+a_{j}a_{ikl}+a_{k}a_{ijl}+ a_{l}a_{ijk}+ \\ +a_{ij}a_{kl}+a_{ik}a_{jl}+a_{il}a_{jk}-2a_{i}a_{j}a_{kl} -2a_{i}a_{k}a_{jl}-\\ -2a_{i}a_{l}a_{jk}-2a_{j}a_{k}a_{il}-2a_{j}a_{l}a_{ik} -2a_{k}a_{l}a_{ij}+\\ +6a_{i}a_{j}a_{k}a_{l}-a_{ijkl}, \quad \{ i,j,k,l \}=\{ 1,2,3,4 \},\quad j<k<l. \end{array} \end{aligned}$$

Here we consider coefficients that differ in the order of the indices to be equal (for example, $a_{123}=a_{231}$). There are a total of 28 Laplace invariants for this equation. Two equations of the form (17) are equivalent in function if and only if they have all the corresponding Laplace invariants equal.

Note that if all Laplace invariants are identically zero, then the Eq. (17) is equivalent to the equation $u_{x_1x_2x_3x_4}=0$ and admits an infinite-dimensional Lie algebra of operators of the form

$$\begin{aligned} \xi ^1(x_1)\partial _{x_1}+\xi ^2(x_2)\partial _{x_2} +\xi ^3(x_3)\partial _{x_3}+\xi ^4(x_4)\partial _{x_4} \end{aligned}$$

with arbitrary $\xi ^i(x_i)$.

Similarly to the case of the third-order Bianchi equation, we can introduce into consideration the constructions

$$\begin{aligned} p_{ij}=\frac{h_{j,i}}{h_{i,j}},\quad q_{ij}=\frac{(\ln h_{i,j})_{x_i x_j}}{h_{i,j}},\quad i,\,j=\overline{1,4}; \end{aligned}$$

$$\begin{aligned} p_{ijk}^l=\frac{h_{l,l_1 l_2}}{h_{i,jk}},\quad q_{ijk}=\frac{(\ln h_{i,jk})_{x_i x_j x_k}}{h_{i,jk}}, \quad \{l,l_1,l_2\}=\{i,j,k\}; \end{aligned}$$

$$\begin{aligned} p_{ijkl}^n=\frac{h_{n,n_1 n_2 n_3}}{h_{i,jkl}},\quad q_{ijkl}=\frac{(\ln h_{i,jkl})_{x_1 x_2 x_3 x_4}}{h_{i,jkl}}, \quad \{n,n_1,n_2,n_3\}=\{i,j,k,l\}. \end{aligned}$$

These constructions are used in [5] to obtain classes of fourth-order Bianchi equations with certain group properties.

It is easy to notice that for constants $q_{ij}$, $q_{ijk}$, $q_{ijkl}$ the Laplace invariants are again solutions of the Liouville equation and its three-dimensional and four-dimensional analogues.

2 Three-Dimensional Analogue of the Liouville Equation

Let us consider an approach to the problem of constructing exact solutions to nonlinear equations based on non-local transformations of variables. Equation

$$\begin{aligned} u_{xyz}=\lambda e^u \end{aligned}$$

(18)

is a three-dimensional analogue of the Liouville equation

$$\begin{aligned} u_{xy}=\lambda e^u. \end{aligned}$$

(19)

Equation (19), in particular, plays a key role in the problem of group classification of second-order hyperbolic equations [2, pp. 116–125]

$$\begin{aligned} v_{xy}+a(x,y)v_{x}+b(x,y)v_{y}+c(x,y)v=0. \end{aligned}$$

The general solution of the Eq. (19) is well known and can be constructed in various ways [2, p. 123], [6, pp. 239–240]. As noted earlier, the Eq. (18) is used in the study of the group properties of the third-order Bianchi Eq. (1).

Here a non-local transformation (such as the Cole—Hopf substitution [7]) is constructed, translating the Eq. (18) into the simplest Bianchi equation

$$\begin{aligned} v_{xyz}=0, \end{aligned}$$

(20)

which has a general solution with three arbitrary functions

$$\begin{aligned} v=\alpha (x,y)+\beta (x,z)+\gamma (y,z). \end{aligned}$$

(21)

In this case, an algorithm based on the use of group methods is used [6, pp. 237–241].

Equation (18) admits the Lie algebra of operators

$$\begin{aligned} X=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z - (\dot{\xi }(x) +\dot{\eta }(y) +\dot{\zeta }(z))\partial _u, \end{aligned}$$

where $\xi (x)$, $\eta (y)$, $\zeta (z)$ are arbitrary functions [1].

On the other hand, the Eq. (20) admits the Lie algebra of operators

$$\begin{aligned} X_0=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z , \end{aligned}$$

where $\xi (x)$, $\eta (y)$, $\zeta (z)$ are also arbitrary. In addition, like any linear equation, Eq. (20) admits a stretching operator

$$\begin{aligned} Y=v\partial _v. \end{aligned}$$

In this regard, assume that there is a non-local transformation

$$\begin{aligned} u=\varphi (v,v_x,v_y,v_z) \end{aligned}$$

(22)

such that the system of Eqs. (18), (20), (22) admits the Lie algebra of operators

$$\begin{aligned} \begin{array}{c} X=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z - (\dot{\xi }(x) +\dot{\eta }(y) + \dot{\zeta }(z))\partial _u,\\ Y=v\partial _v. \end{array} \end{aligned}$$

We find the first continuations of operators

$$\begin{aligned} \begin{array}{c} {X}_1=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z - (\dot{\xi }(x) +\dot{\eta }(y) + \dot{\zeta }(z))\partial _u-\\ -(\ddot{\xi }(x)-\dot{\xi }(x)u_x)\partial _{u_x}-(\ddot{\eta }(y)-\dot{\eta }(y)u_y)\partial _{u_y} -(\ddot{\zeta }(z)-\dot{\zeta }(z)u_z)\partial _{u_z}+\\ +\dot{\xi }(x)v_x\partial _{v_x}+\dot{\eta }(y)v_y\partial _{v_y} +\dot{\zeta }(z)v_z\partial _{v_z}, \end{array} \end{aligned}$$

$$\begin{aligned} {Y}_1=v\partial _{v}+v_x\partial _{v_x}+v_y\partial _{v_y}+v_z\partial _{v_z}. \end{aligned}$$

We get relations

$$\begin{aligned} {Y}_1(u-\varphi )\vert _{u=\varphi }=v\varphi _{v}+v_{x}\varphi _{v_x}+v_{y}\varphi _{v_y}+v_{z}\varphi _{v_z}=0, \end{aligned}$$

(23)

$$\begin{aligned} {X}_1(u-\varphi )\vert _{u=\varphi }= -(\dot{\xi }+\dot{\eta }+\dot{\zeta })+\dot{\xi }(x)v_x\varphi _{v_x}+\dot{\eta }(y)v_y\varphi _{v_y} +\dot{\zeta }(z)v_z\varphi _{v_z}=0. \end{aligned}$$

(24)

Since the function v has the form (21), from (23) and (24) we get the system

$$\begin{aligned} \begin{array}{c} (\alpha +\beta +\gamma )\varphi _v+(\alpha _x+\beta _x)\varphi _{v_x}+(\alpha _y+\gamma _y)\varphi _{v_y}+(\beta _z+\gamma _z)\varphi _{v_z}=0,\\ -(\dot{\xi }+\dot{\eta }+\dot{\zeta })+\dot{\xi }(x)(\alpha _x+\beta _x)\varphi _{v_x}+\dot{\eta }(y)(\alpha _y+\gamma _y)\varphi _{v_y} +\dot{\zeta }(z)(\beta _z+\gamma _z)\varphi _{v_z}=0. \end{array} \end{aligned}$$

(25)

The system (25) is satisfied by the relation

$$\begin{aligned} u=\varphi (v,v_x,v_y,v_z)=\ln \frac{cv_x v_y v_z}{v^3}=\ln c +\ln v_x+\ln v_y +\ln v_z -3\ln v. \end{aligned}$$

(26)

Substituting (26) into the Eq. (18) taking into account (21) leads to a formula defining a class of solutions to the Eq. (18) depending on three arbitrary functions

$$\begin{aligned} u=\ln \left( -\frac{6}{\lambda }\frac{f'_1(x)f'_2(y)f'_3(z)}{(f_1(x)+f_2(y)+f_3(z))^3} \right) . \end{aligned}$$

(27)

Here $f_1(x)$, $f_2(y)$, $f_3(z)$—arbitrary continuously differentiable functions.

3 Fourth-Order Analogue of the Liouville Equation

Now consider the equation

$$\begin{aligned} u_{xyzt}=\lambda e^u, \end{aligned}$$

(28)

related to the fourth-order linear Bianchi equation, whose group properties are considered in [4, 5].

Similarly to the case of the Eq. (18), we construct a non-local transformation that translates the Eq. (28) into the equation

$$\begin{aligned} v_{xyzt}=0, \end{aligned}$$

(29)

the general solution of which

$$\begin{aligned} v=\alpha (x,y,z)+\beta (x,y,t)+\gamma (x,z,t)+\delta (y,z,t). \end{aligned}$$

(30)

Equation (28) admits a Lie algebra of operators

$$\begin{aligned} X=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z +\tau (t)\partial _t- (\dot{\xi }(x) +\dot{\eta }(y) + \dot{\zeta }(z)+\dot{\tau }(t))\partial _u, \end{aligned}$$

where $\xi (x)$, $\eta (y)$, $\zeta (z)$, $\tau (t)$ are arbitrary functions [5].

On the other hand, the Eq. (29) admits the Lie algebra of operators

$$\begin{aligned} X_0=\xi (x)\partial _x+\eta (y)\partial _y+\zeta (z)\partial _z +\tau (t)\partial _t, \end{aligned}$$

as well as the stretching operator

$$\begin{aligned} Y=v\partial _v. \end{aligned}$$

Looking for a non-local transformation

$$\begin{aligned} u=\varphi (v,v_x,v_y,v_z,v_t) \end{aligned}$$

(31)

such that the system of Eqs. (28), (29), (31) admits the Lie algebra of operators

$$\begin{aligned} \begin{array}{c} X=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z +\tau (t)\partial _t -(\dot{\xi }(x) +\dot{\eta }(y) + \dot{\zeta }(z)+\dot{\tau }(t))\partial _u,\\ Y=v\partial _v. \end{array} \end{aligned}$$

We calculate the first continuations of operators

$$\begin{aligned} \begin{array}{c} {X}_1=\xi (x)\partial _x +\eta (y)\partial _y +\zeta (z)\partial _z +\tau (t)\partial _t- (\dot{\xi }(x) +\dot{\eta }(y) +\dot{\zeta }(z)+\dot{\tau }(t))\partial _u-\\ -(\ddot{\xi }(x)-\dot{\xi }(x)u_x)\partial _{u_x}-(\ddot{\eta }(y)-\dot{\eta }(y)u_y)\partial _{u_y} -(\ddot{\zeta }(z)-\dot{\zeta }(z)u_z)\partial _{u_z}-\\ -(\ddot{\tau }(t)-\dot{\tau }(t)u_t)\partial _{u_t} +\dot{\xi }(x)v_x\partial _{v_x}+\dot{\eta }(y)v_y\partial _{v_y} +\dot{\zeta }(z)v_z\partial _{v_z}+\dot{\tau }(t)v_t\partial _{v_t}, \end{array} \end{aligned}$$

$$\begin{aligned} {Y}_1=v\partial _{v}+v_x\partial _{v_x}+v_y\partial _{v_y}+v_z\partial _{v_z}+v_t\partial _{v_t} \end{aligned}$$

and we write down the ratios

$$\begin{aligned} {Y}_1(u-\varphi )\vert _{u=\varphi }=v\varphi _{v}+v_{x}\varphi _{v_x}+v_{y}\varphi _{v_y}+v_{z}\varphi _{v_z}+v_{t}\varphi _{v_t}=0, \end{aligned}$$

(32)

$$\begin{aligned} {X}_1(u-\varphi )\vert _{u=\varphi }= -(\dot{\xi }+\dot{\eta }+\dot{\zeta }+\dot{\tau })+\dot{\xi }v_x\varphi _{v_x}+\dot{\eta }v_y\varphi _{v_y} +\dot{\zeta }v_z\varphi _{v_z}+\dot{\tau }v_t\varphi _{v_t}=0. \end{aligned}$$

(33)

The function v has the form (30), therefore from (32)–(33) we get the system

$$\begin{aligned} \begin{array}{c} (\alpha +\beta +\gamma +\delta )\varphi _v+(\alpha _x+\beta _x+\gamma _x)\varphi _{v_x} +(\alpha _y+\beta _y+\delta _y)\varphi _{v_y}+\\ +(\alpha _z+\gamma _z+\delta _z)\varphi _{v_z}+ (\beta _t+\gamma _t+\delta _t)\varphi _{v_t}=0,\\ -(\dot{\xi }+\dot{\eta }+\dot{\zeta }+\dot{\delta })+\dot{\xi }(\alpha _x+\beta _x+\gamma _x)\varphi _{v_x} +\dot{\eta }(\alpha _y+\beta _y+\delta _y)\varphi _{v_y}+\\ +\dot{\zeta }(\alpha _z+\gamma _z+\delta _z)\varphi _{v_z}+\dot{\delta }(\beta _t+\gamma _t+\delta _t)\varphi _{v_t}=0. \end{array} \end{aligned}$$

(34)

The system (34) is satisfied by the relation

$$\begin{aligned} u=\ln \frac{cv_x v_y v_z v_t}{v^4}. \end{aligned}$$

(35)

Substituting (35) into (28) and taking into account (30), we get the solution of the Eq. (28)

$$\begin{aligned} u=\ln \left( \frac{24}{\lambda }\frac{f'_1(x)f'_2(y)f'_3(z)f'_4(t)}{ (f_1(x)+f_2(y)+f_3(z)+f_4(t))^4} \right) , \end{aligned}$$

where $f_1(x)$, $f_2(y)$, $f_3(z)$, $f_4(t)$—arbitrary continuously differentiable functions.

This paper has been supported by the Kazan Federal University Strategic Academic Leadership Program (Priority—2030).