Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein–Gordon equations

Abstract

Essentially generalizing Lie’s results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein–Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.

1 Introduction

Quasilinear second-order hyperbolic equations model many phenomena and processes in physics and mathematics, especially, various kinds of wave propagation, see [14, 16, 17, 47] and references therein. Such equations even with two independent variables are important in a number of areas, including differential geometry, quantum field theory, cosmology, hydro- and gas dynamics, superconductivity, crystal dislocation, waves in ferromagnetic materials, nonlinear optics, low temperature physics, to name a few. This is why these equations have been and are intensively studied in many branches of mathematics, in particular, within the framework of integrability theory and symmetry analysis of differential equations.

In the present paper, we carry out the exhaustive group classification of the class of (1+1)-dimensional generalized nonlinear Klein–Gordon equations of the form

$$\begin{aligned} u_{tx}=f(t,x,u) \quad \text{ with }\quad f_{uu}\ne 0 \end{aligned}$$

(1)

in the light-cone (characteristic) coordinates, which we denote by ${\mathcal {K}}$ and simultaneously refer as the class (1) and the class ${\mathcal {K}}$. Here $u=u(t,x)$ is the unknown function of the independent variables (t, x), and subscripts of functions denote derivatives with respect to the corresponding variables, e.g., $u_{tx}:=\partial ^2 u/\partial t\,\partial x$ and $f_{uu}:=\partial ^2 f/\partial u^2$. The arbitrary element f of the class ${\mathcal {K}}$ runs through the set of smooth functions of (t, x, u) that are not affine in u. The last constraint is imposed on f for excluding linear equations from the class ${\mathcal {K}}$, which is natural in view of several arguments, see Remark 6. The problem of studying the class ${\mathcal {K}}$ within the framework of group analysis of differential equations was posed by Sophus Lie [35].

We discover four important properties of the class ${\mathcal {K}}$, which allows us to obtain stronger results than the ordinary group classification of this class and to simplify all computations. We prove that, firstly, the class ${\mathcal {K}}$ is normalized with respect to its point equivalence group $G^\sim$ and, secondly, the first prolongations of its point admissible transformations exhaust its contact admissible transformations. In view of these two properties, the classification of Lie symmetries of equations from the class ${\mathcal {K}}$ up to the $G^\sim$-equivalence, roughly speaking, coincides with the similar classification up to the general point equivalence and with the classifications of continuous contact symmetries of these equations modulo the equivalences generated by the contact equivalence group and groupoid of ${\mathcal {K}}$. Moreover, this classification problem can be effectively solved by the algebraic method. Thirdly, the dimensions of the maximal Lie invariance algebras of equations from the class ${\mathcal {K}}$ are not greater than four, except the equations that are $G^\sim$-equivalent to the Liouville equation and whose maximal Lie invariance algebras are infinite-dimensional. The fourth property is that each point transformation between any two equations from the class ${\mathcal {K}}$ is projectable both on the space with coordinate t and on the space with coordinate x. As a result, all the point-transformation structures associated with the class ${\mathcal {K}}$, including its equivalence group, its equivalence algebra and the maximal Lie invariance algebras of equations from this class, are also projectable on the same two spaces. This twofold projectability leads to the possibility of extended application of the following classical Lie theorem on realizations of finite-dimensional Lie algebras by vector fields on the line [34, Satz 6, p. 455] in the course of the group classification, see also [43, Theorem 2.70], [51] and references therein.

Theorem 1

(Lie theorem) Inequivalent (up to the local diffeomorphisms of the line) realizations of finite-dimensional Lie algebras by vector fields on the t-line are exhausted by the algebras

$$\begin{aligned} \{0\},\quad \langle \partial _t\rangle ,\quad \langle \partial _t,\,t\partial _t\rangle ,\quad \langle \partial _t,\,t\partial _t,\,t^2\partial _t\rangle . \end{aligned}$$

Previously, the Lie theorem was applied to group classification of classes of evolution and Schrödinger equations [5, 28, 29, 44] and the class of linear ordinary differential equations of an arbitrary fixed order $r\geqslant 2$ [10, Section 3], where there is the similar projectability to the single space coordinatized by the time variable or the independent variable, respectively.

The most important and well-studied subclass ${\mathcal {K}}_9$ of the class ${\mathcal {K}}$ is singled out from ${\mathcal {K}}$ by the additional auxiliary equations $f_t=f_x=0$, i.e., this subclass consists of the nonlinear Klein–Gordon equations, which are of the form $u_{tx}=f(u)$ with $f_{uu}\ne 0$; cf. Remark 14 for justifying the notation. The subclass ${\mathcal {K}}_9$ contains a number of famous equations, which we present in the canonical forms, where the constant parameters are removed by equivalence transformations:

• the Liouville equation $u_{tx}=\mathrm{e}^u$,

• the Tzitzeica equation $u_{tx}=\mathrm{e}^u\pm \mathrm{e}^{-2u}$ called also the Dodd–Bullough–Mikhailov equation,

• the sine-Gordon (or Bonnet) equation $u_{tx}=\sin u$,

• the sinh-Gordon equation $u_{tx}=\sinh u$,

• the double sine-Gordon equation $u_{tx}=\sin u+C\sin 2u$ with $C\ne 0$,

see Section 7.5.1 in [47] and references therein. Contact symmetry transformations of equations from the subclass ${\mathcal {K}}_9$ were described by Sophus Lie himself [35]. In particular, any such transformation was proved to be the first prolongation of a point transformation. Then Lie singled out the Liouville equations, where $f(u)=C\mathrm{e}^{\kappa u}$ with nonzero constants $\kappa$ and C, as the only equations in ${\mathcal {K}}_9$ admitting infinite-dimensional point symmetry groups. The other equations were shown to possess only point symmetry transformations, where each of the t-, x- and u-components depends only on the respective variable, and this dependence is affine. In the introduction of [35], Lie remarked that results obtained therein can be extended to the class ${\mathcal {K}}$. In fact, the present paper essentially generalizes several extensions of Lie’s results to the class ${\mathcal {K}}$.

Specific equations from the subclass ${\mathcal {K}}_9$ and the subclass ${\mathcal {K}}_9$ itself were intensively studied within the framework of symmetry analysis of differential equations. In particular, generalized symmetries of equations from ${\mathcal {K}}_9$ with characteristics not depending on (t, x) were classified over the complex field in [60]. The Liouville equation, the sine-Gordon equation and the Tzitzeica equation were singled out in the subclass ${\mathcal {K}}_9$ as the only equations with infinite-dimensional algebras of such symmetries, see also [23, Section 21.2]. The same equations were also singled out in [13] as the only equations in ${\mathcal {K}}_9$ admitting infinite-dimensional spaces of conservation laws with so-called “polynomial densities” and in [12, 39] as the only ones with the Painlevé property among the equations $u_{tx}=f(u)$, where the function f is a linear combination of exponential functions $\mathrm{e}^{j\alpha u}$ for some fixed nonzero complex constant $\alpha$ and $j\in {\mathbb {Z}}$. Equations from the subclass ${\mathcal {K}}_9$ admitting nonlinear separation of variables in the standard spacetime coordinates to two first-order ordinary differential equations were classified in [58]; see also references therein and [20] for other kinds of nonlinear separation of variables for these equations. The classification of local conservation laws of equations from the subclass ${\mathcal {K}}_9$ over the complex field was begun in [16]. Singular reduction operators [9, 27], i.e., singular nonclassical (or conditional, or Q-conditional) symmetries, of all the equations of the form $u_{tx}=f(u)$ were exhaustively studied in [27]. At the same time, there are still no complete classifications of generalized symmetries, local conservation laws and regular reduction operators of equations from the subclass ${\mathcal {K}}_9$ as well as no exhaustive classifications of such equations admitting nonlinear separation of variables or the Painlevé property, not to mention the entire class ${\mathcal {K}}$. An exception is the general description of regular reduction operators for equations from the class ${\mathcal {K}}$ that was given in [57].

The framework of the algebraic method of group classification originated in Lie’s classification of second-order ordinary differential equations [37] but it became a common tool of group analysis of differential equations considerably later, only since the 1990s, although its applications to problems of complete group classification still involved the normalization property implicitly [2, 18, 19, 22, 32, 33, 38, 59]. When straightforwardly applied to non-normalized classes of differential equations, the algebraic method results in the so-called preliminary group classification of such classes [1, 3, 15, 24]. The algebraic method is usually used to solve group classification problems for classes of differential equations with arbitrary elements depending on several arguments, for which the direct method of group classification, including the method of furcate splitting [4, 41, 45] as its most advanced version, is unproductive. To carry out the group classification of the class ${\mathcal {K}}$, we use the advanced version of the algebraic method, which is based on the normalization of the class of differential equations to be classified [48, 52, 54] and involves the classification of appropriate subalgebras [3, 15] of the corresponding equivalence algebra. This version of the algebraic method was suggested in [52, 54] and was effectively applied to solving group classification problems for various classes of differential equations [3, 5, 10, 28, 29, 44, 50, 52, 54, 55].

The papers [3, 32, 33, 55] are especially relevant in the context of the present paper since they are devoted to group classification of classes of quasilinear hyperbolic second-order equations with two independent variables using the algebraic method. See also references therein for group classifications of other classes of such equations. In particular, the group classification problem for the superclass $\breve{{\mathcal {K}}}$ of ${\mathcal {K}}$ that is constituted by the equations of the form $u_{\check{t}\check{t}}-u_{\check{x}\check{x}}=f(\check{t},\check{x},u,u_{\check{x}})$ in the standard spacetime coordinates $(\check{t},\check{x})=(x+t,x-t)$ was studied in the seminal papers [32, 33]. The superclass $\breve{{\mathcal {K}}}$ was partitioned into four subclasses, which are in fact normalized and are not related by point transformations to each other, and ${\mathcal {K}}$ is one of these subclasses (under using the light-cone coordinates). As a result, the group classification problem for the entire superclass $\breve{{\mathcal {K}}}$ was split into four group classification problems for the subclasses that each was separately studied within the framework of the algebraic method of group classification. Unfortunately, the consideration of the class ${\mathcal {K}}$ had several drawbacks (see the second paragraph of Sect. 7 below), and hence Lie symmetries of equations from this class have needed a more accurate and comprehensive classification, which is done in the present paper. The group classification problems for the non-normalized class of quasilinear hyperbolic and elliptic equations of the form $u_{\check{t}\check{t}}-h(\check{x},u,u_{\check{x}})u_{\check{x}\check{x}}=f(\check{x},u,u_{\check{x}})$ up to the equivalences generated by the corresponding equivalence group and groupoid, respectively, were exhaustively solved in [55] using an original version of the algebraic method of group classification for non-normalized classes of differential equations. When written in the light-cone coordinates, this class nontrivially intersects the class ${\mathcal {K}}$.

The further organization of the present paper is as follows. In Sect. 2, we prove that any contact-transformation structure associated with equations from the class (1) is the first-order prolongation of its point-transformation counterpart, thus justifying the restriction of the further consideration to point-transformation structures. Then we show the normalization of the class (1) (with respect to point transformations) and construct its (point) equivalence group $G^\sim$ and its (point) equivalence algebra ${\mathfrak {g}}^\sim$. Therein, we also single out the equations in the class (1) with infinite-dimensional Lie invariance algebras and derive properties of finite-dimensional appropriate subalgebras of the projection $\varpi _*{\mathfrak {g}}^\sim$ of ${\mathfrak {g}}^\sim$ onto the space with coordinates (t, x, u), which are important in the course of the group classification of this class. The main result of the group classification, which is a complete list of $G^\sim$-inequivalent Lie-symmetry extensions within the class (1), is given by Theorem 13 in Sect. 3. The structure of the partially ordered set of such extensions is represented as a Hasse diagram in Fig. 1. We also discuss relations between Lie-symmetry extensions via limit processes. Section 4 is devoted to the proof of Theorem 13 and its analysis. We find a number of $G^\sim$-invariant integer characteristics of subalgebras of ${\mathfrak {g}}^\sim$ or, equivalently, of $\varpi _*{\mathfrak {g}}^\sim$, which allow us to completely identify $G^\sim$-inequivalent cases of Lie-symmetry extensions within the class (1). In Sect. 5, we use these characteristics to distinguish, modulo the $G^\sim$-equivalence, successive Lie-symmetry extensions among the found ones, thus exhaustively describing the structure of partially ordered set of $G^\sim$-inequivalent Lie-symmetry extensions within the class (1). Possible ways for the group classifications of subclasses of the class (1) are analyzed in Sect. 6. As an example, we use results of [55] to carry out the group classification of the important subclass ${\mathcal {K}}_2$ associated with the constraint $f_x+f_t=0$ up to the equivalence generated by the equivalence group $G^\sim _2$ of this subclass. In Sect. 7, we discuss the obtained results and overview related problems for the further study.

2 Preliminary analysis

Consider the superclass ${\mathcal {K}}_{\mathrm{gen}}$ of all the equation of the general form $u_{tx}=f(t,x,u)$, ${\mathcal {K}}\subset {\mathcal {K}}_{\mathrm{gen}}$. For a fixed value of the arbitrary element f, let $K_f$ denote the equation from the class ${\mathcal {K}}_{\mathrm{gen}}$ with this value of f. We begin with the study of contact admissible transformations within the subclass $\bar{{\mathcal {K}}}$ of ${\mathcal {K}}_{\mathrm{gen}}$ singled out by the constraint $f_u\ne 0$, i.e., we attach to ${\mathcal {K}}$ the linear equations of the form $u_{tx}=f(t,x,u)$ with $f_{uu}=0$ and $f_u\ne 0$. We essentially generalize Lie’s consideration in [35].

Lemma 2

Any contact admissible transformation within the class $\bar{{\mathcal {K}}}$ is the first-order prolongation of a point admissible transformation within this class.

Lemma 2 can be proved by the direct method. It is also a consequence of later results of [49] for the more general class of equations of the form $u_{tx}=f(t,x,u,u_t,u_x)$. In view of Lemma 2, all structures related to contact transformations of equations from the superclass $\bar{{\mathcal {K}}}$ and all its subclasses, including the class ${\mathcal {K}}$, are the first-order prolongations of analogous structures related to point transformations of equations from the same classes. These structures include equivalence groupoids, equivalence groups of these classes and the symmetry groups of equations from them. This is why we restrict the further consideration to point transformations within the class ${\mathcal {K}}$.

The following lemma is an obvious corollary of Lemma 2 and [26, Theorem 4.3c].

Lemma 3

The class (1) is normalized in the usual sense with respect to both point and contact transformations, i.e., its point and contact equivalence groupoids coincide with the action groupoids of its point equivalence group $G^\sim$ and of the first prolongation of this group, respectively. The group $G^\sim$ is generated by the transformations of the form

$$\begin{aligned} {\tilde{t}}=T(t),\quad {\tilde{x}}=X(x),\quad {\tilde{u}}=Cu+U^0(t,x),\quad {\tilde{f}}=\frac{Cf+U^0_{tx}}{T_tX_x} \end{aligned}$$

(2)

and the discrete equivalence transformation ${\mathscr {I}}^0$: ${\tilde{t}}=x$, ${\tilde{x}}=t$, ${\tilde{u}}=u$, ${\tilde{f}}=f$. Here T, X and $U^0$ are arbitrary smooth functions of their arguments with $T_tX_x\ne 0$, and C is an arbitrary nonzero constant.

Corollary 4

A complete list of discrete equivalence transformations of the class (1) that are independent up to combining with each other and with continuous equivalence transformations of this class is exhausted by the (t, x)-permutation ${\mathscr {I}}^0$ and three transformations alternating signs of variables, ${\mathscr {I}}^t:(t,x,u,f)\mapsto (-t,x,u,-f)$, ${\mathscr {I}}^x:(t,x,u,f)\mapsto (t,-x,u,-f)$, ${\mathscr {I}}^u:(t,x,u,f)\mapsto (t,x,-u,-f)$. The quotient group of the equivalence group $G^\sim$ of the class (1) with respect to its identity component is isomorphic to the group ${\mathrm {D}}_4\times {\mathbb {Z}}_2$, where the dihedral group ${\mathrm {D}}_4$ is the symmetry group of a square.

Corollary 5

There is no contact transformation that maps an equation of the form $u_{tx}=f(t,x,u)$ with $f_u\ne 0$ to equations of the same form with $f_u=0$.

Proof

Suppose that there exists such a contact transformation $\Phi$. Repeating the proof of Lemma 2 for $f_u\ne 0$ and ${\tilde{f}}_u=0$, we derive that the transformation $\Phi$ is the first prolongation of a point transformation in the space with the coordinates (t, x, u). Theorem 4.3c from [26] implies that the (point) equivalence group of the superclass ${\mathcal {K}}_{\mathrm{gen}}$ of all the equation of the form $u_{tx}=f(t,x,u)$ coincides with the group $G^\sim$, and any point admissible transformation within ${\mathcal {K}}_{\mathrm{gen}}$ is generated by an element of $G^\sim$. The conditions $f_u\ne 0$ and $f_u=0$ are $G^\sim$-invariant, which contradicts the supposition. $\square$

Analyzing results of [35, 36], one can deduce that assertions like Lemmas 2 and 3 and Corollary 5 may have been known to Sophus Lie. In general, similar assertions are typical for the theory of contact equivalence of Monge–Ampère equations, see, e.g., Lemma 1 in [31, p. 205] and references therein. In particular, the above Corollary 5 follows from Corollary 1 in [30, p. 238].

Remark 6

The above assertions imply that the class ${\mathcal {K}}_{\mathrm{gen}}$ and its subclasses ${\mathcal {K}}_{\mathrm{gen}}{\setminus }\bar{{\mathcal {K}}}$, $\bar{{\mathcal {K}}}$, $\bar{{\mathcal {K}}}{\setminus }{\mathcal {K}}$ and ${\mathcal {K}}$, which are singled out by the auxiliary constraints $f_u=0$, $f_u\ne 0$, $f_u\ne 0\wedge f_{uu}=0$ and $f_{uu}\ne 0$, respectively, have the same point equivalence group $G^\sim$ and are normalized in the point sense. The subclasses $\bar{{\mathcal {K}}}$, $\bar{{\mathcal {K}}}{\setminus }{\mathcal {K}}$ and ${\mathcal {K}}$ are also normalized in the contact sense. Equations from the class ${\mathcal {K}}$ are not mapped by contact transformations to equations from the class ${\mathcal {K}}_{\mathrm{gen}}{\setminus }{\mathcal {K}}$. Although the group classification of the class ${\mathcal {K}}_{\mathrm{gen}}{\setminus }{\mathcal {K}}$ up to the $G^\sim$-equivalence has not be carried out in the literature, Lie’s solution of the group classification problem for the wider class of all linear hyperbolic equations with two independent variables is well known [36]. Moreover, symmetry properties of the linear and the nonlinear equations from ${\mathcal {K}}_{\mathrm{gen}}$, which constitute the classes ${\mathcal {K}}_{\mathrm{gen}}{\setminus }{\mathcal {K}}$ and ${\mathcal {K}}$, respectively, are quite different. The last three facts justify the exclusion of linear equations from the further consideration.

Lemma 3 implies that the transformations of the form (2) constitute a subgroup H of $G^\sim$. Any such transformation ${\mathscr {T}}$ can be represented as a composition

$$\begin{aligned} {\mathscr {T}} = {\mathscr {D}}^t(T)\circ {\mathscr {D}}^x(X)\circ {\mathscr {Z}}(U^0)\circ {\mathscr {D}}^u(C) \end{aligned}$$

of the elementary equivalence transformations

which are an arbitrary transformation in t, an arbitrary transformation in x, a scaling of u and a shift of u with arbitrary functions of (t, x), respectively. The transformation parameters are described in Lemma 3, and their values are the same as in the form (2). The families of elementary transformations $\{{\mathscr {D}}^t(T)\}$, $\{{\mathscr {D}}^x(X)\}$, $\{{\mathscr {D}}^u(C)\}$ and $\{{\mathscr {Z}}(U^0)\}$, where the corresponding constant or functional parameter varies, are subgroups of $G^\sim$. One more elementary equivalence transformation of the class (1) is ${\mathscr {I}}^0$. Each transformation ${\mathscr {T}}$ from $G^\sim {\setminus } H$ can be decomposed as

$$\begin{aligned} {\mathscr {T}} = {\mathscr {I}}^0\circ {\mathscr {D}}^t(T)\circ {\mathscr {D}}^x(X)\circ {\mathscr {Z}}(U^0)\circ {\mathscr {D}}^u(C). \end{aligned}$$

Corollary 7

The equivalence algebra of the class (1) is ${\mathfrak {g}}^\sim :=\big \langle {\hat{D}}^t(\tau ),\!{\hat{D}}^x(\xi ),\!{\hat{I}},\!{\hat{Z}}(\eta ^0)\big \rangle ,$ where the parameter functions $\tau =\tau (t)$, $\xi =\xi (x)$ and $\eta ^0=\eta ^0(t,x)$ run through the sets of smooth functions of their arguments, and

$$\begin{aligned}&{\hat{D}}^t(\tau ):=\tau (t)\partial _t-\tau _t(t)f\partial _f,\quad {\hat{D}}^x(\xi ):=\xi (x)\partial _x-\xi _x(x)f\partial _f,\\&{\hat{I}}:=u\partial _u+f\partial _f,\quad {\hat{Z}}(\eta ^0):=\eta ^0(t,x)\partial _u+\eta ^0_{tx}(t,x)\partial _f. \end{aligned}$$

Using the infinitesimal invariance criterion [6, 7, 42], we prove the following assertion.

Proposition 8

The maximal Lie invariance algebra ${\mathfrak {g}}_f$ of an equation $K_f$ from the class (1) consists of the vector fields of the form $\tau (t)\partial _t+\xi (x)\partial _x+\big (\eta ^1u+\eta ^0(t,x)\big )\partial _u$, where the parameter functions $\tau =\tau (t)$, $\xi =\xi (x)$ and $\eta ^0=\eta ^0(t,x)$ and the constant $\eta ^1$ satisfy the classifying equation

$$\begin{aligned} \tau f_t + \xi f_x + \big (\eta ^1u + \eta ^0\big )f_u=\big (\eta ^1-\tau _t-\xi _x\big )f+\eta ^0_{tx}. \end{aligned}$$

(3)

Consider the linear span ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ of all the maximal Lie invariance algebras of equations from the class (1),

$$\begin{aligned} \begin{aligned} {\mathfrak {g}}_{\tiny {\langle \,\rangle }}:={}&\textstyle \sum _f{\mathfrak {g}}_f=\big \{Q=\tau (t)\partial _t+\xi (x)\partial _x+(\eta ^1u +\eta ^0(t,x))\partial _u\big \}\\ ={}&\textstyle \big \langle D^t(\tau ),\,D^x(\xi ),\,I,\,Z(\eta ^0)\big \rangle \ne \bigcup _f {\mathfrak {g}}_f, \end{aligned} \end{aligned}$$

(4)

where the parameters $\tau$, $\xi$ and $\eta ^0$ run through the sets of smooth functions of their arguments, $\eta ^1$ is an arbitrary constant, and

$$\begin{aligned} D^t(\tau ):=\tau (t)\partial _t,\quad D^x(\xi ):=\xi (x)\partial _x,\quad I:=u\partial _u,\quad Z(\eta ^0):=\eta ^0(t,x)\partial _u. \end{aligned}$$

It is obvious that any vector field Q of the above form with $(\tau ,\xi )\ne (0,0)$ belongs to ${\mathfrak {g}}_f$ for any f satisfying the classifying equation with the components of Q, and such a value of the arbitrary element f necessarily exists. The last inequality in (4) holds since the vector fields from $\langle I,Z(\eta ^0)\rangle$ do not belong to ${\mathfrak {g}}_f$ for any f in view of the auxiliary inequality $f_{uu}\ne 0$ for the arbitrary element f within the class (1). At the same time, we can represent such vector fields as linear combinations of vector fields from ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ with $(\tau ,\xi )\ne (0,0)$. This is why the second equality in (4) holds as well, and thus the algebra ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ coincides with the projection $\varpi _*{\mathfrak {g}}^\sim$ of ${\mathfrak {g}}^\sim$. Moreover, the span ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ is a Lie algebra, since it is closed with respect to the Lie bracket of vector fields.

Here and in what follows  $\varpi$ denotes the projection of the space with coordinates (t, x, u, f) onto the space with coordinates (t, x, u). We also use the notation $\pi ^{t,x}$, $\pi ^t$ and $\pi ^x$ for the projections of the space with coordinates (t, x, u) onto the spaces with coordinates (t, x), t and x, respectively.

The nonidentity actions of elementary equivalence transformations on the above vector fields spanning ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ are the following:

$$\begin{aligned}&\big (\varpi _*{\mathscr {D}}^t(T)\big )_*D^t(\tau )=D^t\big (\tau ({\hat{T}})/{\hat{T}}_t\big ),\qquad \quad \big (\varpi _*\mathscr {D}^t(T)\big )_*Z(\eta ^0)=Z\big (\eta ^0(\hat{T},x)\big ),\\&\big (\varpi _*{\mathscr {D}}^x(X)\big )_*D^x(\xi )=D^x\big (\xi ({\hat{X}})/{\hat{X}}_x\big ),\qquad \quad \big (\varpi _*\mathscr {D}^x(X)\big )_*Z(\eta ^0)=Z\big (\eta ^0(t,\hat{X})\big ), \\&\big (\varpi _*{\mathscr {Z}}(U^0)\big )_*D^t(\tau )=D^t(\tau )+Z(\tau U^0_t),\qquad \quad \big (\varpi _*{\mathscr {D}}^u(C)\big )_*Z(\eta ^0)=Z(C\eta ^0), \\&\big (\varpi _*{\mathscr {Z}}(U^0)\big )_*D^x(\xi )=D^x(\xi )+Z(\xi U^0_x),\qquad \quad \big (\varpi _*{\mathscr {Z}}(U^0)\big )_*I =I-Z(U^0), \end{aligned}$$

where ${\hat{T}}={\hat{T}}(t)$ and ${\hat{X}}={\hat{X}}(x)$ are the inverses of the functions T and X, respectively.

Definition 9

A subalgebra ${\mathfrak {s}}$ of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ is said to be appropriate if there exists a value of the arbitrary element f such that ${\mathfrak {s}}={\mathfrak {g}}_f$.

In view of Lemma 3, the group classification of the class (1) reduces to the classification of appropriate subalgebras of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}= \varpi _* {\mathfrak {g}}^\sim$ up to the $\varpi _*G^\sim$-equivalence.

Splitting the classifying equation (3) with respect to the arbitrary element f and its derivations, we obtain the trivial system $\tau =0$, $\xi =0$, $\eta ^1=0$ and $\eta ^0=0$. This means that the following assertion holds.

Lemma 10

The kernel Lie invariance algebra of the equations from the class (1) is ${\mathfrak {g}}^\cap =\{0\}$.

Since the kernel Lie invariance algebra ${\mathfrak {g}}^\cap$ is zero, the condition of the necessary inclusion of it into each appropriate algebra makes no constraint for such algebras. Analyzing the classifying equation (3) deeper, we derive really essential constraints for such algebras.

Lemma 11

(i) ${\mathfrak {g}}_f\cap \big \langle I,\,Z(\eta ^0)\big \rangle =\{0\}$ for any $f=f(t,x,u)$ with $f_{uu}\ne 0$, and therefore $\dim {\mathfrak {g}}_f=\dim \pi ^{t,x}_*{\mathfrak {g}}_f$. Here $\eta ^0$ runs through the set of smooth functions depending on (t, x).

(ii) $\dim {\mathfrak {g}}_f=\infty$ if and only if $f=\mathrm{e}^u$ $(\!{}\bmod G^\sim )$.

(iii) If $f\ne \mathrm{e}^u$ $(\!{}\bmod G^\sim )$, then $\dim {\mathfrak {g}}_f\leqslant 4$.

Proof

Suppose that for a value of the arbitrary element f, the algebra ${\mathfrak {g}}_f$ contains a vector field $\eta ^1I+Z(\eta ^0)$ with $(\eta ^1,\eta ^0)\ne (0,0)$. The classifying equation (3) implies that the function f satisfies the equation $(\eta ^1u+\eta ^0)f_u=\eta ^1f+\eta ^0_{tx}$. Considering the cases $\eta ^1\ne 0$ and $\eta ^1=0$ separately, we easily show that in both cases f is affine in u, which contradicts the auxiliary inequality $f_{uu}\ne 0$ for the class (1). This proves item (i) of the lemma.

We differentiate the classifying equation (3) with respect to u and, in view of the auxiliary inequality $f_{uu}\ne 0$, divide the result of differentiation by $f_{uu}$. Then we differentiate the obtained equation once more with respect to u, which gives

$$\begin{aligned} \tau \left( \frac{f_{ut}}{f_{uu}}\right) _u+\xi \left( \frac{f_{ux}}{f_{uu}}\right) _u+\eta ^1=-(\tau _t+\xi _x)\left( \frac{f_u}{f_{uu}}\right) _u. \end{aligned}$$

(5)

We need to consider two cases depending on whether or not the expression $(f_u/f_{uu})_u$ vanishes.

Upon the condition $(f_u/f_{uu})_u\ne 0$, we can rewrite the equation (5) as

$$\begin{aligned} \tau _t+\xi _x=-\tau \frac{(f_{ut}/f_{uu})_u}{({f_u}/{f_{uu}})_u} -\xi \frac{(f_{ux}/f_{uu})_u}{({f_u}/{f_{uu}})_u}-\eta ^1\frac{1}{({f_u}/{f_{uu}})_u}. \end{aligned}$$

After fixing a value $u=u_0$, the last equation takes the form $\tau _t+\xi _x=A(t,x)\tau +B(t,x)\xi +C(t,x)$, where the coefficients A, B and C are obviously expressed via derivatives of f. After additionally fixing a value $t=t_0$, we derive the first-order inhomogeneous linear ordinary differential equation

$$\begin{aligned}\xi _x=B(t_0,x)\xi -\tau _t(t_0)+A(t_0,x)\tau (t_0)+\eta ^1C(t_0,x)\end{aligned}$$

with respect to $\xi$, and its inhomogeneity involves three constant parameters $\tau (t_0)$, $\tau _t(t_0)$ and $\eta ^1$. The general solution of this equation can be represented in the form

$$\begin{aligned}\xi =C_1\xi ^1(x)+\tau (t_0)\xi ^2(x)+\tau _t(t_0)\xi ^3(x)+\eta ^1\xi ^4(x),\end{aligned}$$

where $\xi ^k(x)$, $k=1,\dots ,4$, are fixed smooth functions of x, and hence it is linearly parameterized at most four independent arbitrary constants. In other words, $\dim \pi ^x_*{\mathfrak {g}}_f\leqslant 4$. Similarly, after taking into account the derived expression for $\xi$, we fix a value $x_0$ for x instead of t and obtain the first-order inhomogeneous linear ordinary differential equation

$$\begin{aligned} \begin{aligned} \tau _t={}&A(t,x_0)\tau +B(t,x_0)\big (C_1\xi ^1(x_0)+\tau (t_0)\xi ^2(x_0)+\tau _t(t_0)\xi ^3(x_0)+\eta ^1\xi ^4(x_0)\big )\\&{}-C^1\xi ^1_x(x_0)-\tau (t_0)\xi ^2_x(x_0)-\tau _t(t_0)\xi ^3_x(x_0)-\eta ^1\xi ^4_x(x_0)+C(t,x_0) \end{aligned} \end{aligned}$$

with respect to $\tau$, where the inhomogeneity involves the four constant parameters $\tau (t_0)$, $\tau _t(t_0)$, $\eta ^1$ and $C_1$. Since the value of $\tau$ in the fixed point $t=t_0$ is among these parameters, the general solution of this equation merely involves these very parameters, i.e., $\dim \pi ^t_*{\mathfrak {g}}_f\leqslant 4$. Therefore, $\dim \pi ^{t,x}_*{\mathfrak {g}}_f\leqslant 4$ and thus, in view of item (i) of the lemma, $\dim {\mathfrak {g}}_f\leqslant 4$.

Now we study the second case $(f_u/f_{uu})_u=0$, i.e., $f=\gamma (t,x)\mathrm{e}^{\alpha (t,x)u}+\beta (t,x)$, where $\alpha \gamma \ne 0$ in view of the auxiliary inequality $f_{uu}\ne 0$ for the class (1). Hence $\gamma =1$ $(\!{}\bmod G^\sim )$. Substituting the expression for f into the classifying equation (3), we collect the coefficients of the linearly independent functions $u\mathrm{e}^{\alpha u}$, $\mathrm{e}^{\alpha u}$ and 1, treated as functions of u, which leads to the system

$$\begin{aligned} \begin{aligned}&\tau \alpha _t+\xi \alpha _x+\eta ^1\alpha \!=\!0,\quad \alpha \eta ^0\!=\!\eta ^1-\tau _t-\xi _x, \\&\tau \beta _t\!+\!\xi \beta _x\!=\!\big (\eta ^1-\tau _t-\xi _x\big )\beta +\eta ^0_{tx}. \end{aligned} \end{aligned}$$

(6)

If $\alpha _x\ne 0$, then $\xi =-\big ((\tau \alpha _t+\eta ^1\alpha )/\alpha _x\big )\big |_{t=t_0}$, i.e., the component $\xi$ involves at most two varying constants, $\tau (t_0)$ and $\eta ^1$. Similarly, if $\alpha _t\ne 0$, then $\tau =\big ((\xi \alpha _x+\eta ^1\alpha )/\alpha _t\big )\big |_{x=x_0}$, i.e., the component $\tau$ also involves at most two varying constants, $\xi (x_0)$ and $\eta ^1$. Therefore, in the case $\alpha _t\alpha _x\ne 0$, the components $\tau$ and $\xi$ in total involve at most three different varying constants, and hence $\dim {\mathfrak {g}}_f\leqslant 3$. If exactly one of the derivatives $\alpha _t$ and $\alpha _x$ is nonzero, then up to the equivalence transformation ${\mathscr {I}}^0$, we can assume that $\alpha _x\ne 0$ and $\alpha _t=0$. Then the substitution of the expressions for $\xi$ and $\eta ^0$ implied by the first two equations of (6), $\xi =-\eta ^1\alpha /\alpha _x$ and $\eta ^0=(\eta ^1-\tau _t-\xi _x)/\alpha$, into the last equation of (6) leads to the second-order inhomogeneous linear ordinary differential equation $(1/\alpha )_x\tau _{tt}-(\beta \tau )_t=\eta ^1(\alpha \beta /\alpha _x)_x-\eta ^1\beta$ with respect to $\tau$, where the leading coefficient $(1/\alpha )_x$ does not vanish, and the inhomogeneity involves the single constant parameter $\eta ^1$. Analogously to the above consideration, we obtain $\dim {\mathfrak {g}}_f\leqslant 3$. Otherwise, $\alpha _t=\alpha _x=0$, and hence $\alpha =\mathrm{const}$. Since $\alpha \ne 0$, we can set $\alpha =1$ by scalings of u and ${\mathscr {I}}^u$. The above system reduces to $\eta ^1=0$, $\eta ^0=-\tau _t-\xi _x$, $\tau \beta _t+\xi \beta _x=-(\tau _t+\xi _x)\beta$. For $\beta \ne 0$, we treat the last equation in the same way as the equation (5) and conclude that $\dim {\mathfrak {g}}_f\leqslant 3$ in this case. If $\beta =0$, then the equation $K_f$ coincides with the Liouville equation $u_{tx}=\mathrm{e}^u$, whose maximal Lie invariance algebra is ${\mathfrak {g}}_f=\langle \tau (t)\partial _t+\xi (x)\partial _x-(\tau _t(t)+\xi _x(x))\partial _u\rangle$, where the components $\tau$ and $\xi$ run through the sets of smooth functions of t or x, respectively. Therefore, $\dim {\mathfrak {g}}_f=\infty$, and up to the $G^\sim$-inequivalence, this is the only case with the infinite dimension of $\dim {\mathfrak {g}}_f$, which proves item (ii) of the lemma.

For all the other equations from the class (1), we derive that the dimensions of the corresponding maximal Lie invariance algebras do not exceed four, which implies item (iii) of the lemma. $\square$

Corollary 12

If $f\ne \mathrm{e}^u$ $(\!{}\bmod G^\sim )$, then $\dim \pi ^t_*{\mathfrak {g}}_f\leqslant 3$ and $\dim \pi ^x_*{\mathfrak {g}}_f\leqslant 3$.

Proof

In view of item (iii) of Lemma 11, we have $\dim {\mathfrak {g}}_f\leqslant 4$ if $f\ne \mathrm{e}^u$ $(\!{}\bmod G^\sim )$. Then $\dim \pi ^t_*{\mathfrak {g}}_f\leqslant \dim {\mathfrak {g}}_f\leqslant 4$ and $\dim \pi ^x_*{\mathfrak {g}}_f\leqslant \dim {\mathfrak {g}}_f\leqslant 4$, i.e., $\pi ^t_*{\mathfrak {g}}_f$ and $\pi ^x_*{\mathfrak {g}}_f$ are finite-dimensional Lie algebras of vector fields on the t- and the x-lines, respectively. Then the required inequalities directly follow from the Lie theorem on such algebras.$\square$

3 Result of group classification

The main result of the paper is the following theorem.

Theorem 13

A complete list of $G^\sim$-inequivalent (maximal) Lie-symmetry extensions in the class (1) is exhausted by the following cases:

• 0. General case $f=f(t,x,u)$: $\{0\}$;

• 1. $f={\hat{f}}(x,u)$: $\langle \partial _t\rangle$;

• 2. $f={\hat{f}}(x-t,u)$: $\langle \partial _t+\partial _x\rangle$;

• 3. $f=\mathrm{e}^t{\hat{f}}(x,\mathrm{e}^{-t}u)$: $\langle \partial _t+u\partial _u\rangle$;

• 4. $f=\mathrm{e}^{x+t}{\hat{f}}(x-t,\mathrm{e}^{-x-t}u)$: $\langle \partial _t+\partial _x+2u\partial _u\rangle$;

• 5. $f=\mathrm{e}^t{\hat{f}}(\mathrm{e}^{-t}u)$: $\langle \partial _t+u\partial _u,\,\partial _x\rangle$;

• 6. $f=\mathrm{e}^{x+t}{\hat{f}}(\mathrm{e}^{-x-t}u)$: $\langle \partial _t+u\partial _u,\,\partial _x+u\partial _u\rangle$;

• 7. $f=|x-t|^{-q-2}{\hat{f}}(|x-t|^qu)$, $q\ne 0$: $\langle \partial _t+\partial _x,\,t\partial _t+x\partial _x-qu\partial _u\rangle$;

• 8. $f=|x|^{-q-2}{\hat{f}}(|x|^qu)$, $q\ne 0$: $\langle \partial _t,\,t\partial _t+x\partial _x-qu\partial _u\rangle$;

• 9. $f={\hat{f}}(u)$: $\langle \partial _t,\,\partial _x,\,t\partial _t-x\partial _x\rangle$;

• 10. $f=(x-t)^{-2}{\hat{f}}(u)$: $\langle \partial _t+\partial _x,\,t\partial _t+x\partial _x,\,t^2\partial _t+x^2\partial _x\rangle$;

• 11. $f=\mathrm{e}^{u/x}$: $\langle \partial _t,\,t\partial _t-x\partial _u,\,x\partial _x+u\partial _u\rangle$;

• 12. $f=|u|^p u$, $p\ne -1,0$: $\langle \partial _t,\,\partial _x,\,t\partial _t-x\partial _x,\,-pt\partial _t+u\partial _u\rangle$;

• 13. $f=\mathrm{e}^u$: $\langle \tau (t)\partial _t+\xi (x)\partial _x-(\tau _t(t)+\xi _x(x))\partial _u\rangle$.

Here ${\hat{f}}$ is an arbitrary smooth function of its arguments whose second derivative with respect to the argument involving u is nonzero, and q and p are arbitrary constants that satisfy the conditions indicated in the corresponding cases. In Case 13, the components $\tau$ and $\xi$ run through the sets of smooth functions of t or x, respectively.

The proof of Theorem 13 is presented in the next Section 4.

Remark 14

There are two ways of interpreting the classification cases listed in Theorem 13, in a weak sense and in a strong sense. Within the framework of the weak group classification, we consider the entire subclass ${\mathcal {K}}_N$ of the equations from the class ${\mathcal {K}}$ with the form of f presented in Case N,

$$\begin{aligned} N\in \Gamma :=\{0,\dots ,6,7_q,8_q,9, 10,11,12_p,13\mid q\ne 0,\, p\ne -1,0\}, \end{aligned}$$

and then the corresponding algebra is the kernel Lie invariance algebra ${\mathfrak {g}}^\cap _N$ of the equations from the subclass ${\mathcal {K}}_N$. Here we use the notation $7_q:=(7,q)$, $8_q:=(8,q)$ and $12_p:=(12,p)$. We refer to Cases 7, 8 and 12 as to collections of Cases $7_q$, $8_q$ and $12_p$ with fixed values of q or p, respectively. It is obvious that ${\mathcal {K}}_0={\mathcal {K}}$. Under the strong group classification, Case N includes only the equations from the subclass ${\mathcal {K}}_N$ for which ${\mathfrak {g}}_f={\mathfrak {g}}^\cap _N$. Thus, the discussion after the equation (4) implies that ${\mathfrak {g}}_f=\{0\}$ and hence $K_f$ belongs to strong Case 0 if and only if f does not satisfy the classifying equation (3) for any constant $\eta ^1$ and any smooth functions $\tau =\tau (t)$, $\xi =\xi (x)$ and $\eta ^0=\eta ^0(t,x)$ with $(\tau ,\xi )\ne (0,0)$. For Cases 1–9 to merely collect maximal Lie-symmetry extensions, the parameter function ${\hat{f}}$ should take only values for which the associated values of the arbitrary element f are not $G^\sim$-equivalent to ones from the other listed cases with maximal Lie invariance algebras of greater dimensions. In other words, a value of the parameter function ${\hat{f}}$ leads to a maximal Lie-symmetry extension if and only if it satisfies no equation among those associated with the corresponding case in Proposition 25 or 26 below. Case 10 is special since ${\mathfrak {g}}_f={\mathfrak {g}}^\cap _{10}$ for any $K_f\in {\mathcal {K}}_{10}$, and thus it does not depend on interpreting the group classification. We will mostly omit the attributes “weak” and “strong”, explicitly indicating all places where the weak interpretation is used.

Remark 15

Cases 3–6 and 8 can be replaced by $G^\sim$-equivalent cases, for each of which the arbitrary element f just runs through the set of arbitrary smooth functions of either one or two arguments without an additional multiplier:

$$\begin{aligned}&3'.\ f={\hat{f}}\big (x,t^{-1}u\big ):\ \ {\mathfrak {g}}_{f}=\langle t\partial _t+u\partial _u\rangle ;\\&4'.\ f= {\hat{f}}\big (t^{-1}x,(tx)^{-1}u\big ):\ \ {\mathfrak {g}}_f=\langle t\partial _t+x\partial _x+2u\partial _u\rangle ;\\&5'.\ f= {\hat{f}}\big (t^{-1}u\big ):\ \ {\mathfrak {g}}_f=\langle t\partial _t+u\partial _u,\,\partial _x\rangle ;\\&6'.\ f= {\hat{f}}\big ((tx)^{-1}u\big ):\ \ {\mathfrak {g}}_f=\langle t\partial _t+u\partial _u,\,x\partial _x+u\partial _u\rangle ;\\&8'a.\ f= {\hat{f}}\big (|x|^{q'}u\big ), \ q'\ne 0,-1:\ \ {\mathfrak {g}}_f=\langle \partial _t,\,(q'+1)t\partial _t-x\partial _x+q'u\partial _u\rangle ;\\&8'b.\ f= {\hat{f}}\big (\mathrm{e}^{-x}u\big ):\ \ {\mathfrak {g}}_f=\langle \partial _t,\,t\partial _t+\partial _x+u\partial _u\rangle . \end{aligned}$$

Here Case 8 splits into two subcases, 8$'$a and 8$'$b, respectively associated with $q\ne 0,-1$ and $q=-1$.

Remark 16

The group classification of the class (1) can be easily mapped via the point transformation $\check{t}=x+t$, $\check{x}=x-t$, $\check{u}=u$, $\check{f}=f$ to the group classification of nonlinear Klein–Gordon equations in the standard spacetime variables, $\check{u}_{\check{t}\check{t}}-\check{u}_{\check{x}\check{x}}=\check{f}(\check{t},\check{x},\check{u})$.

Remark 17

We found eight triples of $G^\sim$-invariant integer characteristics of subalgebras ${\mathfrak {s}}$ of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$, which suffice for distinguishing the $G^\sim$-inequivalent cases of Lie-symmetry extensions from each other, see Remark 24 below. The most significant among these eight triples is $(r_3,j_1,r_2)$, where

$$\begin{aligned} r_3=r_3({\mathfrak {s}}):= & {} 3-\min \big \{\dim \big \langle D^t(\tau ),D^x(\xi )\big \rangle \mid \\&\exists \,\eta ^0:D^t(\tau )+D^x(\xi )+I+Z(\eta ^0)\in {\mathfrak {s}}\big \},\\ j_1=j_1({\mathfrak {s}}):= & {} \max (\dim {\mathfrak {s}}^1,\dim {\mathfrak {s}}^2),\\ r_2=r_2({\mathfrak {s}}):= & {} \min (\dim \pi ^t_*{\mathfrak {s}}^{12},\dim \pi ^x_*{\mathfrak {s}}^{12}) \end{aligned}$$

with ${\mathfrak {s}}^1 :={\mathfrak {s}}\cap \big \langle D^t(\tau ),Z(\eta ^0)\big \rangle$, ${\mathfrak {s}}^2 :={\mathfrak {s}}\cap \big \langle D^x(\xi ),Z(\eta ^0)\big \rangle$ and ${\mathfrak {s}}^{12}:={\mathfrak {s}}\cap \big \langle D^t(\tau ),D^x(\xi ),Z(\eta ^0)\big \rangle$.

In the next four remarks, we discuss weak Lie-symmetry extensions, omitting the attribute “weak”.

Remark 18

We can partially order the collection of Lie-symmetry extensions within the class (1). Here $\text{ Case } N\prec \text{ Case } {\bar{N}}$ with $N,{\bar{N}}\in \Gamma$ means that Case ${\bar{N}}$ is a further Lie-symmetry extension of Case N modulo the $G^\sim$-equivalence, i.e., there exists ${\mathscr {T}}\in G^\sim$ such that ${\mathfrak {s}}\varsubsetneq (\varpi _*{\mathscr {T}})\bar{{\mathfrak {s}}}$, where ${\mathfrak {s}}$ and $\bar{{\mathfrak {s}}}$ are subalgebras of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ associated with Cases N and ${\bar{N}}$, respectively. For the corresponding subclasses ${\mathcal {K}}_N$ and ${\mathcal {K}}_{{\bar{N}}}$, we have the inverse inclusion, ${\mathcal {K}}_N\varsupsetneq (\varpi _*{\mathscr {T}}){\mathcal {K}}_{{\bar{N}}}$ with the same ${\mathscr {T}}\in G^\sim$.

Remark 19

We assign two more $G^\sim$-invariant integer characteristics of subalgebras ${\mathfrak {s}}$ of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ to those from Remark 17,

$$\begin{aligned} n=n({\mathfrak {s}}):=\dim {\mathfrak {s}},\quad k=k({\mathfrak {s}}):=\min (\dim \pi ^t_*{\mathfrak {s}},\dim \pi ^x_*{\mathfrak {s}}), \end{aligned}$$

see Remark 24 below. Thus, we consider tuples of the form $(n,r_3,r_2,j_1,k)$, where the entries are ordered according to their importance, see the beginning of Sect. 5. We partially order the set of these tuples, assuming $(n,r_3,r_2,j_1,k)<({\bar{n}},{\bar{r}}_3,{\bar{r}}_2,{\bar{j}}_1,{\bar{k}})$ if $n<{\bar{n}}$, $r_3\leqslant {\bar{r}}_3$, $r_2\leqslant {\bar{r}}_2$, $j_1\leqslant {\bar{j}}_1$ and $k\leqslant {\bar{k}}$. Let $(n,r_3,r_2,j_1,k)$ and $({\bar{n}},{\bar{r}}_3,{\bar{r}}_2,{\bar{j}}_1,{\bar{k}})$ correspond to the algebras ${\mathfrak {s}}$ and $\bar{{\mathfrak {s}}}$ of Cases N and ${\bar{N}}$, respectively. Then the relation $\text{ Case } N\prec \text{ Case } {\bar{N}}$ is equivalent to the relation $(n,r_3,r_2,j_1,k)<({\bar{n}},{\bar{r}}_3,{\bar{r}}_2,{\bar{j}}_1,{\bar{k}})$.

Remark 20

The Hasse diagram for the partially ordered set of $G^\sim$-inequivalent Lie-symmetry extensions within the class (1) is presented in Fig. 1. According to the rule of constructing Hasse diagrams, the arrows in Fig. 1 depict only the direct Lie-symmetry extensions. We say that the pair $(\text{ Case } N,\text{ Case } {\bar{N}})$ of classification cases with $\text{ Case } N\prec \text{ Case } {\bar{N}}$ presents a direct Lie-symmetry extension if there does not exist $\check{N}\in \Gamma$ such that $\text{ Case } N\prec \text{ Case } \check{N}\prec \text{ Case } {\bar{N}}.$

Fig. 1

The Hasse diagram of Lie-symmetry extensions within the class (1)

Remark 21

There are many pairs among $\{(\text{ Case } N,\text{ Case } {\bar{N}}),\,N,{\bar{N}}\in \Gamma \}$, where the pair components are related to each other via limit processes supplemented, if necessary, with preliminary or subsequent equivalence transformations.¹ All these limits lead to contractions ${\mathfrak {g}}^\cap _N\rightarrow {\mathfrak {h}}\subseteq {\mathfrak {g}}^\cap _{{\bar{N}}}$ as contractions of realizations of abstract Lie algebras by vector fields, where ${\mathfrak {h}}$ is a subalgebra of ${\mathfrak {g}}^\cap _{{\bar{N}}}$, which often coincides with the entire ${\mathfrak {g}}^\cap _{{\bar{N}}}$. The most obvious limit process is

$$\begin{aligned} \text{ Case } \text{7 }_q\rightarrow \text{ Case } \text{10 }\quad \text{ as }\quad q\rightarrow 0 \quad \text{ with }\quad {\mathfrak {g}}^\cap _{7,1}\simeq {\mathfrak {g}}^\cap _{7,q}\rightarrow \langle \partial _t+\partial _x,\,t\partial _t+x\partial _x\rangle \subsetneq {\mathfrak {g}}^\cap _{10}. \end{aligned}$$

An example with a necessary subsequent equivalence transformation is

$$\begin{aligned}&\text{ Case } \text{8 }_q\rightarrow {\mathscr {D}}^x(-x^{-1})_*(\text{ Case } \text{9 })\quad \text{ as }\quad q\rightarrow 0 \\&\qquad \text{ with }\quad {\mathfrak {g}}^\cap _{8,1}\simeq {\mathfrak {g}}^\cap _{8,q}\rightarrow \langle \partial _t,\,t\partial _t+x\partial _x\rangle \subsetneq \big (\varpi _*D^x(-x^{-1})\big )_*{\mathfrak {g}}^\cap _9. \end{aligned}$$

The limit processes

$$\begin{aligned}&\text{ Case } \text{2 }\rightarrow \text{ Case } \text{1 },\quad \text{ Case } \text{4 }\rightarrow \text{ Case } \text{2 },\quad \text{ Case } \text{6 }\rightarrow \text{ Case } \text{5 },\quad \text{ Case } \text{7 }\rightarrow \text{ Case } \text{8 },\\&\text{ Case } \text{10 }\rightarrow {\mathscr {D}}^x(-x^{-1})_*(\text{ Case } \text{9 }) \end{aligned}$$

as $q\rightarrow 0$ with contractions between the entire corresponding Lie algebras are realized via preliminarily introducing the parameter q using a scaling equivalence transformation. For instance, for each value of the parameter function ${\hat{f}}$ in Case 2, we take the family $f^q=q^{-1}{\hat{f}}(x-t,u)$ with $q\ne 0$ and act on each equation $K_{f^q}$ by the equivalence transformation ${\mathscr {D}}^t(q^{-1}t)$. For each $q\ne 0$, we obtain the equation $K_{{\tilde{f}}^q}$ with ${\tilde{f}}^q={\hat{f}}({\tilde{x}}-q{\tilde{t}},{\tilde{u}})$, which is invariant with respect to the algebra $\langle \partial _{{\tilde{t}}}+q\partial _{{\tilde{x}}}\rangle$. The limit as $q\rightarrow 0$ then gives Case 1. One more set of limit processes

$$\begin{aligned} \text{ Case } \text{7 }\rightarrow \text{ Case } \text{6 },\quad \text{ Case } \text{8 }\rightarrow ({\mathscr {I}}^t\circ {\mathscr {I}}^0)_*(\text{ Case } \text{5 }),\quad \text{ Case } \text{12 }\rightarrow \text{ Case } \text{13 } \end{aligned}$$

is based on the remarkable limit $(1+q^{-1})^q\rightarrow \mathrm{e}$ as $q\rightarrow +\infty$. The first two limit processes again give the contractions between the entire corresponding Lie algebras, whereas the last limit process results in a contraction to a proper subalgebra. We describe in detail the second limit process. For each value of the parameter function ${\hat{f}}$ in Case 8, we act on the equation $K_{f^q}$ with $f^q=q|x|^{-q-2}{\hat{f}}(|x|^qu)$, $q\in {\mathbb {R}}_{\ne 0}$, by the equivalence transformation ${\mathscr {D}}^x\big (q(x-1)\big )$, which leads to the equation $K_{{\tilde{f}}^q}$ with ${\tilde{f}}^q=|1+{\tilde{x}}/q|^{-q-2}{\hat{f}}(|1+{\tilde{x}}/q|^q{\tilde{u}})$. The Lie invariance algebra of $K_{{\tilde{f}}^q}$ is $\langle q^{-1}{\tilde{t}}\partial _{{\tilde{t}}}+(1+q^{-1}{\tilde{x}})\partial _{{\tilde{x}}}-{\tilde{u}}\partial _{{\tilde{u}}}\rangle$. Proceeding to the limit as $q\rightarrow +\infty$, we get $({\mathscr {I}}^t\circ {\mathscr {I}}^0)_*(\text{ Case } \text{5 })$. A less standard limit process is $\text{ Case } \text{11 }\rightarrow \text{ Case } \text{13 }$, where we introduce the parameter q via the transformation ${\mathscr {D}}^t(qt)\circ {\mathscr {D}}^x\big (q^{-1}(x-1)\big )$ and take the limit as $q\rightarrow 0$. The contracted algebra $\langle \partial _t,\,t\partial _t-\partial _u,\,\partial _x\rangle$ is a (proper) subalgebra of the infinite-dimensional algebra ${\mathfrak {g}}^\cap _{13}$.

4 Proof of group classification

In view of Lemma 11(ii), we can exclude Case 13, which corresponds to the Liouville equation, from the further consideration. For convenience, denote by ${\mathcal {L}}$ the subclass of ${\mathcal {K}}$ consisting of the equations that are $G^\sim$-equivalent to the Liouville equation. The equations from the class ${\mathcal {K}}$ with finite-dimensional Lie invariance algebras, for which $f\ne \mathrm{e}^u$ $(\!{}\bmod G^\sim )$, constitute the complement ${\mathcal {K}}{\setminus }{\mathcal {L}}$. In this notation, we need to classify only Lie symmetries of equations from the class ${\mathcal {K}}{\setminus }{\mathcal {L}}$. The classification of such equations splits into different cases depending on the following three $G^\sim$-invariant integer values

$$\begin{aligned} m:=\max (\dim \pi ^t_*{\mathfrak {g}}_f,\dim \pi ^x_*{\mathfrak {g}}_f),\quad n:=\dim {\mathfrak {g}}_f,\quad k:=\min (\dim \pi ^t_*{\mathfrak {g}}_f,\dim \pi ^x_*{\mathfrak {g}}_f), \end{aligned}$$

which are defined by f and are listed in accordance with their influence on the classification splitting. In view of their definition, they satisfy the inequality $0\leqslant n-m\leqslant k\leqslant m\leqslant n$. According to Lemma 11(iii), we can suppose from the very beginning that $n\leqslant 4$, whereas Corollary 12 implies that $m\leqslant 3$. Moreover, it follows from Lemma 11(i) that $m>0$ for any equation in the class (1) with Lie-symmetry extension. In other words, looking for $G^\sim$-inequivalent equations $K_f$ from ${\mathcal {K}}{\setminus }{\mathcal {L}}$ with nonzero maximal Lie invariance algebra ${\mathfrak {g}}_f$, we select the appropriate triples (m, n, k) among those that satisfy the condition

$$\begin{aligned} m\in \{1,2,3\},\quad n\in \{1,2,3,4\},\quad 0\leqslant n-m\leqslant k\leqslant m\leqslant n. \end{aligned}$$

Since the class (1) possesses the discrete equivalence transformation ${\mathscr {I}}^0$, which permutes the variables t and x, without loss of generality we can assume that $m=\dim \pi ^t_*{\mathfrak {g}}_f$. We choose an initial basis of ${\mathfrak {g}}_f$ consisting of vector fields

$$\begin{aligned} Q^i=\tau ^i(t)\partial _t+\xi ^i(x)\partial _x+\big (\eta ^{1i}u +\eta ^{0i}(t,x)\big )\partial _u, \quad i=1,\dots ,n \quad \text{ with }\quad n\leqslant 4, \end{aligned}$$

see Proposition 8. In addition to Corollary 12, the application of the Lie theorem to the group classification of the class (1) is based on the fact that $\pi ^t_*G^\sim$ and $\pi ^x_*G^\sim$ coincide with the diffeomorphism groups on the t- and the x-lines, respectively.

We separately consider the cases with different values of $m\in \{1,2,3\}$ in descending order.

$\varvec{m=3.}$ Therefore, $n\in \{3,4\}$. We change the basis $(Q^1,\dots ,Q^n)$ in such a way that $\tau ^1$, $\tau ^2$ and $\tau ^3$ are linearly independent and, if $n=4$, $\tau ^4=0$. Then $\pi ^t_*{\mathfrak {g}}_f=\langle \tau ^1\partial _t,\tau ^2\partial _t,\tau ^3\partial _t\rangle$ is a faithful realization of the algebra $\mathfrak {sl}(2,{\mathbb {R}})$ on the line. This fact has several implications. In particular, the algebra ${\mathfrak {f}}:=\langle Q^1,Q^2,Q^3\rangle$ is, up to combining $Q^1$, $Q^2$ and $Q^3$ with $Q^4$ if $n=4$, a faithful realization of $\mathfrak {sl}(2,{\mathbb {R}})$ as well, and $\pi ^x_*{\mathfrak {f}}=\langle \xi ^1\partial _x,\xi ^2\partial _x,\xi ^3\partial _x\rangle$ is also a (not necessarily faithful) realization of $\mathfrak {sl}(2,{\mathbb {R}})$.² The last realization should be either faithful or zero since the kernel of any homomorphism of a Lie algebra ${\mathfrak {g}}$ to a Lie algebra is an ideal of ${\mathfrak {g}}$, and the algebra $\mathfrak {sl}(2,{\mathbb {R}})$ has no proper ideals. In other words, either $\xi ^1=\xi ^2=\xi ^3=0$ or $\xi ^1\xi ^2\xi ^3\ne 0$. In view of the Lie theorem, up to linearly combining $Q^1$, $Q^2$ and $Q^3$ with each other, we can set $(\tau ^1,\tau ^2,\tau ^3)=(1,t,t^2)$ $(\!{}\bmod \pi ^t_*G^\sim )$. Moreover, the derived algebra $[{\mathfrak {f}},{\mathfrak {f}}]$ of the algebra ${\mathfrak {f}}$ coincides with the algebra ${\mathfrak {f}}$ itself, and thus $\eta ^{1i}=0$, $i=1,2,3$.

In the case $\xi ^1=\xi ^2=\xi ^3=0$, we have $Q^i=t^{i-1}\partial _t+\eta ^{0i}\partial _u$, $i=1,2,3$, with $\eta ^{01}=0$ $(\!{}\bmod G^\sim )$ and, if $n=4$, $\tau ^4=0$. The commutator relations of ${\mathfrak {f}}$,

$$\begin{aligned}{}[Q^1,Q^2]= & {} \partial _t+\eta _t^{02}\partial _u=Q^1, \\ {[}Q^1,Q^3{]}= & {} 2t\partial _t+\eta _t^{03}\partial _u=2Q^2,\\ {[}Q^2,Q^3{]}= & {} t^2\partial _t+(t\eta _t^{03}-t^2\eta ^{02}_t)\partial _u=Q^3, \end{aligned}$$

imply the system $\eta _t^{02}=0$, $\eta ^{03}_t=2\eta ^{02}$, $t\eta ^{03}_t=\eta ^{03}$, and thus $Q^1=\partial _t$, $Q^2=t\partial _t-\mu (x)\partial _u$, $Q^3=t^2\partial _t-2t\mu (x)\partial _u$, where $\mu$ is a smooth function of x. Successively substituting the obtained vector fields $Q^1$, $Q^2$ and $Q^3$ into the classifying equation (3), we derive the following system with respect to the arbitrary element f: $f_t=0$, $\mu f_u=f$, $2\mu tf_u=2tf+\mu _x$. An obvious consequence of this system is $\mu _x=0$ and, since $f\ne 0$, we also get $\mu \ne 0$ and $f=\nu (x)\mathrm{e}^{u/\mu }$. Therefore, the equation $K_f$ is $G^\sim$-equivalent to the Liouville equation, cf. the proof of Lemma 11, i.e., $\dim {\mathfrak {g}}_f=\infty$, which contradicts the supposition $\dim {\mathfrak {g}}_f\leqslant 4$.

Now we consider the case $\xi ^1\xi ^2\xi ^3\ne 0$ and, up to the $G^\sim$-equivalence, set $\xi ^1=1$ and $\eta ^{01}=0$. After expanding the commutator relations of ${\mathfrak {f}}$ and collecting the components of vector fields,

$$\begin{aligned}{}[Q^1, Q^2]= & {} \partial _t+\xi ^2_x\partial _x+(\eta ^{02}_t+\eta _x^{02})\partial _u=Q^1, \\ {[}Q^1, Q^3{]}= & {} 2t\partial _t+\xi ^3_x\partial _x+(\eta ^{03}_t+\eta ^{03}_x)\partial _u=2Q^2, \\ {[}Q^2, Q^3{]}= & {} t^2\partial _t+(\xi ^2\xi ^3_x-\xi ^3\xi ^2_x)\partial _x +(t\eta ^{03}_t+\xi ^2\eta ^{03}_x-t^2\eta ^{02}_t-\xi ^3\eta ^{02}_x)\partial _u=Q^3, \end{aligned}$$

we obtain the system

$$\begin{aligned} \xi _x^2=1,\quad \eta _t^{02}+\eta _x^{02}=0,\quad \xi ^3_x=2\xi ^2,\quad \eta _t^{03}+\eta _x^{03}=2\eta ^{02},\\ \xi ^2\xi ^3_x-\xi ^3\xi ^2_x=\xi ^3,\quad t\eta ^{03}_t+\xi ^2\eta ^{03}_x-t^2\eta ^{02}_t-\xi ^3\eta ^{02}_x=\eta ^{03}. \end{aligned}$$

From the first, the third and the fifth equations of this system, up to the equivalence transformations of shifts with respect to x, we derive $\xi ^2=x$ and $\xi ^3=x^2$. From the second and the fourth equations, we find $\eta ^{02}=\rho (\omega ^-)$ and $\eta ^{03}=\rho (\omega ^-)\omega ^+ +\theta (\omega ^-)$ with $\omega ^-:=x-t$ and $\omega ^+:=x+t$. Then the sixth equation of the system reduces to $\omega ^-\theta _{\omega ^-}=\theta$, which integrates to $\theta =\lambda \omega ^-$, where $\lambda$ is an arbitrary constant. Thus, $Q^1=\partial _t+\partial _x$, $Q^2=t\partial _t+x\partial _x+\rho (\omega ^-)\partial _u$, $Q^3=t^2\partial _t +x^2\partial _x+\big ((x+t)\rho (\omega ^-)+\lambda (x-t)\big )\partial _u$, where we can set the parameter function $\rho =\rho (\omega ^-)$ to zero by the transformation ${\tilde{t}}=t$, ${\tilde{x}}=x$, ${\tilde{u}}=u-\int (\omega ^-)^{-1}\rho (\omega ^-)\,\mathrm{d}\omega ^-$, which is the projection of an equivalence transformation. We successively substitute the components of the vector fields $Q^1$, $Q^2$ and $Q^3$ into the classifying equation (3). The derived system with respect to the arbitrary element f,

$$\begin{aligned} f_t+f_x=0,\quad tf_t+xf_x=-2f,\quad t^2f_t+x^2f_x+\lambda (x-t)f_u=-2(x+t)f, \end{aligned}$$

is consistent under the inequality $f_u\ne 0$ if and only if $\lambda =0$.

Suppose that $n=4$, $\tau ^4=0$ and $\xi ^4\ne 0$. Since $\dim \pi ^x_*{\mathfrak {g}}_f\leqslant 3$, the component $\xi ^4$ should belong to $\langle \xi ^1,\xi ^2,\xi ^3\rangle =\langle 1,x,x^2\rangle$. Moreover, $[{\mathfrak {f}},\langle Q^4\rangle ]\subseteq \langle Q^4\rangle$, which implies $[\pi ^x_*{\mathfrak {f}},\langle \xi ^4\partial _x\rangle ]\subseteq \langle \xi ^4\partial _x\rangle$, i.e., $\langle \xi ^4\partial _x \rangle$ is an ideal of $\pi ^x_*{\mathfrak {f}}\simeq \mathfrak {sl}(2,{\mathbb {R}})$. The algebra $\mathfrak {sl}(2,{\mathbb {R}})$ has no proper ideals. Hence $\xi ^4=0$, which is a contradiction. This gives Case 10 of Theorem 13. Moreover, Case 10 admits no further Lie-symmetry extensions via specifying ${\hat{f}}$; cf. the beginning of Sect. 5.

$\varvec{m=2.}$ We change the basis $(Q^1,\dots ,Q^n)$ in such a way that $\tau ^1$, $\tau ^2$ are linearly independent and $\tau ^i=0$, $i=3,\dots ,n$. Then $\pi ^t_*{\mathfrak {g}}_f=\langle \tau ^1\partial _t,\tau ^2\partial _t\rangle$ is a faithful realization of a two-dimensional Lie algebra on the t-line, and according to the Lie theorem, up to linearly combining $Q^1$ and $Q^2$, we can set $(\tau ^1,\tau ^2)=(1,t)$ $(\!{}\bmod \pi ^t_*G^\sim )$. The possible values of $n=\dim {\mathfrak {g}}_f$ are 2, 3 and 4.

Suppose that $n=4$. Then $\dim \pi ^x_*{\mathfrak {g}}_f=\dim \langle \xi ^3\partial _x,\xi ^4\partial _x\rangle =2$ in view of Lemma 11(i). Up to linearly combining $Q^3$ and $Q^4$, we can set $(\xi ^3,\xi ^4)=(1,x)$ $(\!{}\bmod \pi ^x_*G^\sim )$. Since $\xi ^1,\xi ^2\in \langle \xi ^3,\xi ^4\rangle$, we can further linearly combine $Q^1$ and $Q^2$ with $Q^3$ and $Q^4$ to annihilate $\xi ^1$ and $\xi ^2$. We have

$$\begin{aligned} Q^1=\partial _t+(\eta ^{11}u+\eta ^{01})\partial _u,\quad Q^2=t\partial _t+(\eta ^{12}u+\eta ^{02})\partial _u,\\ Q^3=\partial _x+(\eta ^{13}u+\eta ^{03})\partial _u,\quad Q^4=x\partial _x+(\eta ^{14}u+\eta ^{04})\partial _u, \end{aligned}$$

where $\eta ^{11}$, ..., $\eta ^{14}$ are constants and $\eta ^{01}$,..., $\eta ^{04}$ are smooth functions of (t, x). The commutation relations $[Q^1, Q^2]=Q^1$ and $[Q^3, Q^4]=Q^3$ imply $\eta ^{11}=\eta ^{13}=0$. Acting by a transformation $\varpi _*{\mathscr {Z}}(U^0)$, we can set $\eta ^{01}=0$. From the commutation relation $[Q^1, Q^3]=\eta ^{03}_t\partial _u=0$, we derive $\eta ^{03}_t=0$. All transformations $\varpi _*{\mathscr {Z}}(U^0)$ with $U^0=U^0(x)$ preserve the reduced form of $Q^1=\partial _t$, and among them there is a transformation allowing us to set $\eta ^{03}=0$. We still need to take into account the following commutation relations:

$$\begin{aligned}{}[Q^1, Q^2]=\partial _t+\eta ^{02}_t\partial _u=Q^1, \quad {[}Q^3, Q^2{]}=\eta ^{02}_x\partial _u=0, \\ {[}Q^3, Q^4{]}=\partial _x+\eta ^{04}_x\partial _u=Q^3,\quad {[}Q^1, Q^4{]}=\eta ^{04}_t\partial _u=0, \end{aligned}$$

which leads to $\eta ^{02}_t=\eta ^{02}_x=\eta ^{04}_t=\eta ^{04}_x=0$, i.e., $\eta ^{02}$ and $\eta ^{04}$ are constants. Substituting the components of the vector fields $Q^1$, $Q^2$, $Q^3$ and $Q^4$ into the classifying equation (3), we obtain a system with respect to the arbitrary element f,

$$\begin{aligned} f_t=f_x=0,\quad (\eta ^{12}u+\eta ^{02})f_u=f,\quad (\eta ^{14}u+\eta ^{04})f_u=f. \end{aligned}$$

This system is consistent if and only if $\eta ^{12}=\eta ^{14}$ and $\eta ^{02}=\eta ^{04}$. Moreover, $\eta ^{12}=\eta ^{14}\ne 0$ since otherwise we have the Liouville equation. In view of the derived conditions, we can set $\eta ^{02}=\eta ^{04}=0$ using a shift of u by a constant, which preserves all the posed constraints of the components of $Q^1$, ..., $Q^4$. Denoting $p:=-1/\eta ^{12}$, we derive Case 12, where the constant multiplier ${\hat{C}}$ of f can be removed by the transformation ${\mathscr {D}}^t({\hat{C}} t)$.

Let $n=3$. Hence $\xi ^3\ne 0$ and $k:=\dim \pi ^x_*{\mathfrak {g}}_f=\dim \langle \xi ^1,\xi ^2,\xi ^3\rangle \in \{1,2\}$.

If $k=2$, then the Lie theorem implies that $\xi ^i=a_ix+b_i$ with some constants $a_i$ and $b_i$, $i=1,2,3$. Therefore, we need to examine the cases with respect to coefficients of $\xi ^i$. For $a_3\ne 0$, we can set $a_3=1$ and $b_3=0$ by rescaling the vector field $Q^3$ and by shifting x, respectively. We can further linearly combine $Q^1$ and $Q^2$ with $Q^3$ to make $a_1=a_2=0$. Thus, the basis elements $Q^1$, $Q^2$ and $Q^3$ satisfy the commutation relations $[Q^1,Q^2]=Q^1$, $[Q^1,Q^3]=0$ and $[Q^2,Q^3]=0$. The last two commutation relations imply $b_1=b_2=0$, which means $k=1$, contradicting the supposed condition $k=2$. For $a_3=0$, the condition $\xi ^3\ne 0$ is equivalent to $b_3\ne 0$. Rescaling $Q^3$ and linearly combining $Q^1$ and $Q^2$ with $Q^3$, we make $b_3=1$ and $b_1=b_2=0$. The commutation relation $[Q^1,Q^2]=Q^1$ leads to $a_1=0$ and $\eta ^{11}=0$, and thus $a_2\ne 0$ since $k=2$. Acting by a transformation $\varpi _*{\mathscr {Z}}(U^0)$, we can set $\eta ^{01}=0$. The basis elements take the form

$$\begin{aligned} Q^1=\partial _t,\quad Q^2=t\partial _t+a_2x\partial _x+(\eta ^{12}u+\eta ^{02})\partial _u, \quad Q^3=\partial _x+(\eta ^{13}u+\eta ^{03})\partial _u, \end{aligned}$$

where $\eta ^{1i}$ is a constant and $\eta ^{0i}$ is a smooth function of (t, x), $i=2,3$. From the commutation relation $[Q^1, Q^3]=\eta ^{03}_t\partial _u=0$, we derive $\eta ^{03}_t=0$. Since all the pushforwards $\varpi _*{\mathscr {Z}}(U^0)$ with $U^0=U^0(x)$ preserve the reduced form of $Q^1=\partial _t$, in view of the equation $\eta ^{03}_t=0$ we can assume that $\eta ^{03}=0$ up to these pushforwards. More conditions on the basis elements follow from the commutation relations

$$\begin{aligned}{}[Q^1, Q^2]=\partial _t+\eta _t^{02}\partial _u=Q^1,\quad [Q^3, Q^2]= a_2\partial _x+(\eta _x^{02}-\eta ^{13}\eta ^{02})\partial _u=a_2Q^3, \end{aligned}$$

which are $\eta ^{13}=0$, $\eta _t^{02}=0$, $\eta _x^{02}=0$, i.e., $\eta ^{02}=\mathrm{const}$. Substituting the components of the vector fields $Q^1$, $Q^2$ and $Q^3$ into the classifying equation (3), we obtain a system with respect to the arbitrary element f,

$$\begin{aligned} f_t=f_x=0,\quad (\eta ^{12}u+\eta ^{02})f_u=(\eta ^{12}-a_2-1)f. \end{aligned}$$

If $\eta ^{12}=\eta ^{02}=0$, then $a_2=-1$ since $f\ne 0$, and we get Case 9. For $\eta ^{12}=0$ and $\eta ^{02}\ne 0$, the inequality $f_u\ne 0$ implies $a_2\ne -1$, which leads to the Liouville equation. If $\eta ^{12}\ne 0$, then the corresponding values of the arbitrary element f are, up to the shifts of u, of the form as in Case 12, where $n=4$.

Consider the case $k=1$. Then $\xi ^3\ne 0$, and we can assume $\xi ^3=1$, $\xi ^1=\xi ^2=0$. In view of the commutation relations $[Q^1,Q^2]=Q^1$ and $[Q^1,Q^3]=0$, analogously to previous cases, we have $\eta ^{11}=0$ and we can set $\eta ^{01}=\eta ^{03}=0$ acting by the transformation ${\mathscr {Z}}(U^0)$. Further expanding commutation relations, $[Q^1, Q^2]=\partial _t+\eta ^{02}_t\partial _u=Q^1$, $[Q^3, Q^2]=(\eta ^{02}_x-\eta ^{13}\eta ^{02})\partial _u=0$, we derive $\eta ^{02}_t=0$, $\eta ^{02}_x=\eta ^{13}\eta ^{02}$. From the classifying equation (3), we obtain the system on the arbitrary element f,

$$\begin{aligned} f_t=0,\quad f_x+\eta ^{13}uf_u=\eta ^{13}f,\quad (\eta ^{12}u+\eta ^{02})f_u=(\eta ^{12}-1)f. \end{aligned}$$

If $\eta ^{13}=0$, then $f_t=f_x=0$, and thus $\langle t\partial _t-x\partial _x\rangle \in {\mathfrak {g}}_f$, which contradicts the condition $\xi ^2=0$. Hence $\eta ^{13}\ne 0$, and we can set $\eta ^{13}=1$ by scaling of x, and thus $\eta ^{02}=C\mathrm{e}^x$ for some constant C. If additionally $\eta ^{12}\ne 0$, then we can make $\eta ^{02}=0$ acting by the transformation ${\mathscr {Z}}(-pC\mathrm{e}^x)$ with $p:=-1/\eta ^{12}\ne 0$, which preserves $Q^1$ and $Q^3$. The general solution of the system for f with the auxiliary inequality $f_{uu}\ne 0$ is $f={\hat{C}}\mathrm{e}^{-px}|u|^pu$, where ${\hat{C}}$ is an arbitrary nonzero constant. The family of equivalence transformations ${\tilde{t}}={\hat{C}}t$, ${\tilde{x}}=-\mathrm{e}^{-px}/p$, ${\tilde{u}}=u$, ${\tilde{f}}=\mathrm{e}^{px}f/{\hat{C}}$ maps such values of f to that from Case 12, where $n=4$. Otherwise, $\eta ^{12}=0$, and thus $C\ne 0$ since $f\ne 0$. Scaling u, we can set $C=1$. From the above system for f, we derive $f={\hat{C}}\mathrm{e}^{x-\mathrm{e}^{-x}u}$. The equivalence transformation ${\tilde{t}}=-{\hat{C}}t$, ${\tilde{x}}=\mathrm{e}^x$, ${\tilde{u}}=-u$, ${\tilde{f}}=\mathrm{e}^{-x}f/{\hat{C}}$ leads to Case 11.

If $n=2$, then $k\in \{0,1,2\}$.

Suppose that $k=2$. From the commutation relation $[Q^1, Q^2]=Q^1$, we derive $\eta ^{11}=0$ and, modulo the transformations $\varpi _*{\mathscr {D}}^x(X)$ and $\varpi _*{\mathscr {Z}}(U^0)$, $\xi ^1=1$, $\xi ^2=x$ and $\eta ^{01}=0$, and then $\eta ^{02}_t+\eta ^{02}_x=0$. Hence $\eta ^{02}$ is a function of $x-t$, and additionally acting by a transformation $\varpi _*{\mathscr {Z}}(\theta )$, where $\theta$ is also a function of $x-t$, we can set $\eta ^{02}=0$. The basis vector fields take the form $Q^1=\partial _t+\partial _x$, $Q^2=t\partial _t+x\partial _x+\eta ^{12}u\partial _u$. Substituting the components of $Q^1$ and $Q^2$ into the classifying equation (3), we obtain the system

$$\begin{aligned} f_t+f_x=0, \quad tf_t+xf_x+\eta ^{12}uf_u=(\eta ^{12}-2)f. \end{aligned}$$

This system implies that $\eta ^{12}\ne 0$ since otherwise we get Case 10 with $n=3$. The condition $\eta ^{12}\ne 0$ singles out Case 7, where $q:=-\eta ^{12}$.

In the case $k=1$, the commutation relation $[Q^1, Q^2]=Q^1$ implies $\xi ^1=0$ and $\eta ^{11}=0$. It is possible and convenient to make $\xi ^2=x$ using a transformation $\varpi _*{\mathscr {D}}^x(X)$. Similarly to the previous cases, we can simultaneously set $\eta ^{01}=0$ and $\eta ^{02}=0$ via acting by a transformation $\varpi _*{\mathscr {Z}}(U^0)$. The system $f_t=0$, $xf_x+\eta ^{12}uf_u=(\eta ^{12}-2)f$ following from the classifying equation (3) implies that $\eta ^{12}\ne 0$ since otherwise we again get Case 9 with $n=3$. As a result, we have Case 8, where $q:=-\eta ^{12}$.

Remark 22

The (nonzero) parameter q in Cases 7 and 8 cannot be further gauged by equivalence transformations. We show this only for Case 8 since the argumentation for Case 7 is similar. For each value of f from Case 8, the derived algebra of the corresponding algebra ${\mathfrak {g}}_f$ is spanned by the vector field $Q^1=\partial _t$. Hence the projection $\varpi _*{\mathscr {T}}$ of any element ${\mathscr {T}}$ of $G^\sim$ that maps the equation $K_f$ to an equation from the same Case 8 should preserve the span of $Q^1$, i.e., $(\varpi _*{\mathscr {T}})_*\partial _t\in \langle \partial _t\rangle$. This implies that the t-component of ${\mathscr {T}}$ is affine in t. Such a transformation cannot change the ratio of the coefficients of $t\partial _t$ and of $u\partial _u$ in the vector field $Q^2$.

For $k=0$, $\xi ^1=\xi ^2=0$, and the further consideration is similar to the case $k=1$. Modulo the $G^\sim$-equivalence, from the commutation relation $[Q^1,Q^2]=Q^1$ we derive that $\eta ^{11}=0$ and $\eta ^{01}=\eta ^{02}=0$. The associated system for f is $f_t=0$, $\eta ^{12}uf_u=(\eta ^{12}-1)f$, where $\eta ^{12}\ne 0,1$, and hence such a value of f can be reduced by a transformation ${\mathscr {D}}^x(X)$ to the form from Case 12, where $n=4$.

$\varvec{m=1.}$ Then $n\leqslant 2$.

If $n=2$, then linearly combining basis elements and pushing forward the algebra ${\mathfrak {g}}_f$ by $\varpi _*{\mathscr {D}}^t(T)$ and $\varpi _*{\mathscr {Z}}(U^0)$, we make $\tau ^1=1$, $\tau ^2=0$ and $\eta ^{01}=0$. Hence $\xi ^2\ne 0$, i.e., $\xi ^2=1$ $(\!{}\bmod G^\sim )$, and we additionally set $\xi ^1=0$ by linearly combining the basis elements and repairing the gauge $\eta ^{01}=0$ using a transformation $\varpi _*{\mathscr {Z}}(U^0)$. In view of the commutation relation $[Q^1, Q^2]=(\eta ^{02}_t-\eta ^{11}\eta ^{02})\partial _u=0$, we can conclude that $\eta ^{02}=\mathrm{e}^{\eta ^{11}t}\zeta (x)$ for some smooth function $\zeta$ of x. Therefore, we can set $\eta ^{02}=0$ by the transformation $\varpi _*{\mathscr {Z}}\big (\mathrm{e}^{\eta ^{11}t}\theta (x)\big )$, where $\theta$ is a solution of the first-order linear ordinary differential equation $\theta _x-\eta ^{12}\theta +\zeta =0$; the vector field $Q^1$ is preserved by this transformation. The basis elements take the form $Q^1=\partial _t+\eta ^{11}u\partial _u$, $Q^2=\partial _x+\eta ^{12}u\partial _u$. Here $(\eta ^{11},\eta ^{12})\ne (0,0)$ since otherwise we have Case 9 with $n=3$. If both the coefficients $\eta ^{11}$ and $\eta ^{12}$ are nonzero, then by scaling t and x, we can set $\eta ^{11}=\eta ^{12}=1$, and the associated system for the arbitrary element f is $f_t+uf_u=f$, $f_x+uf_u=f$, which leads to Case 6. If one of the parameters $\eta ^{11}$ or $\eta ^{12}$ is nonzero, then up to the discrete equivalence transformation ${\mathscr {I}}^0$, which permutes the variables t and x, we can assume that $\eta ^{11}\ne 0$, and we can set $\eta ^{11}=1$ by scaling t. From the classifying equation (3), we obtain the system $f_t+uf_u=f$, $f_x=0$, whose integration gives Case 5.

In the case $n=1$, since $\tau ^1\ne 0$, we can set $\tau ^1=1$, $\xi ^1=\delta$, $\eta ^{01}=0$ and $\eta ^{11}=(1+\delta )~\delta '$ with $\delta ,\delta '\in \{0,1\}$ modulo the transformations $\varpi _*{\mathscr {D}}^t(T)$, $\varpi _*{\mathscr {D}}^x(X)$ and $\varpi _*{\mathscr {Z}}(U^0)$ and of a simultaneous scaling of (t, x), respectively. As a result, we obtain the following $G^\sim$-inequivalent cases for $Q^1$: $\partial _t$, $\partial _t+\partial _x$, $\partial _t+u\partial _u$, and $\partial _t+\partial _x+2u\partial _u$, which correspond to Cases 1–4.

Remark 23

We have split the group classification of equations from the class (1) with finite-dimensional maximal Lie invariance algebras into different cases depending on values of the triple (m, n, k) of $G^\sim$-invariant integers, which are defined as

$$\begin{aligned}&m=m(f):=\max (\dim \pi ^t_*{\mathfrak {g}}_f,\dim \pi ^x_*{\mathfrak {g}}_f),\\&n=n(f):=\dim {\mathfrak {g}}_f,\\&k=k(f):=\min (\dim \pi ^t_*{\mathfrak {g}}_f,\dim \pi ^x_*{\mathfrak {g}}_f). \end{aligned}$$

It is clear that most of the values in ${\mathbb {Z}}^3$ are inappropriate for (m, n, k). As a preliminary step of the classification, we have significantly narrowed down the set of candidates to be looked through for such triples. According to Lemma 11, Corollary 12 and the definition of (m, n, k), we have the constraints $m\in \{1,2,3\}$, $n\in \{1,2,3,4\}$, and $0\leqslant n-m\leqslant k\leqslant m\leqslant n$ but even the (quite restricted) set S of triples satisfying these constraints contains many elements that are not realized for equations from the class (1). The appropriate values for (m, n, k) are exhausted by (1, 1, 0), (1, 1, 1), (1, 2, 1), (2, 2, 2), (2, 2, 1), (2, 3, 2), (3, 3, 3), (2, 3, 1), (2, 4, 2), which are associated with pairs of Cases 1 and 3, 2 and 4, 5 and 6, and single Cases 7, 8, 9, 10, 11, 12 of Theorem 13, respectively. Therefore, the inappropriate triples in S are (3, 4, k) with $k=1,2,3$, (3, 3, k) with $k=0,1,2$ and (2, 2, 0), and their number is significant but less than the number of appropriate triples. The only appropriate triple with $m=3$ is (3, 3, 3), i.e., the value $m=3$ uniquely defines the possible values for n and k. This interesting observation may be related to the fact that for $m=3$ both the projections $\pi ^t_*{\mathfrak {g}}_f$ and $\pi ^x_*{\mathfrak {g}}_f$ as well as the algebra ${\mathfrak {g}}_f$ itself are necessarily isomorphic to $\mathfrak {sl}(2,{\mathbb {R}})$ in view of the Lie theorem and the simplicity of $\mathfrak {sl}(2,{\mathbb {R}})$. The separation of the appropriate values from the inappropriate ones in the set S cannot be implemented in the course of preliminary analysis of Lie symmetries of equations from the class (1) since it is an integral part of the group classification procedure for this class.

Remark 24

To check algebraically that the cases given in Theorem 13 are $G^\sim$-inequivalent to each other, we need more values associated with the maximal Lie invariance algebras of equations from the class (1), since there are pairs of $G^\sim$-inequivalent cases with the same triples (m, n, k). These are Cases 1 and 3 with $(m,n,k)=(1,1,0)$, Cases 2 and 4 with $(m,n,k)=(1,1,1)$ and Cases 5 and 6 with $(m,n,k)=(1,2,1)$. To introduce additional $G^\sim$-invariant values for complete identification of classification cases, we represent the span ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ as a direct sum of its subspaces,

$$\begin{aligned}&{\mathfrak {g}}_{\tiny {\langle \,\rangle }}=\big \langle D^t(\tau )\big \rangle \dotplus \big \langle D^x(\xi )\big \rangle \dotplus \big \langle I\big \rangle \dotplus \big \langle Z(\eta ^0)\big \rangle , \end{aligned}$$

where the parameter functions $\tau =\tau (t)$, $\xi =\xi (x)$ and $\eta ^0=\eta ^0(t,x)$ run through the sets of smooth functions of their arguments. By ${\mathfrak {P}}_i$ we denote the projection from ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ onto the ith summand of the above representation for ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$, $i=1,\dots ,4$. Although $\dim {\mathfrak {P}}_1{\mathfrak {g}}_f=\dim \pi ^t_*{\mathfrak {g}}_f$ and $\dim {\mathfrak {P}}_2{\mathfrak {g}}_f=\dim \pi ^x_*{\mathfrak {g}}_f$, we can define the new $G^\sim$-invariant integer values

$$\begin{aligned}&l=l(f):=\dim {\mathfrak {P}}_3{\mathfrak {g}}_f,\\&j_1=j_1(f):=\max (\dim {\mathfrak {g}}_f^1,\dim {\mathfrak {g}}_f^2),\\&j_2=j_2(f):=\min (\dim {\mathfrak {g}}_f^1,\dim {\mathfrak {g}}_f^2),\\&j_{12}=j_{12}(f):=\dim {\mathfrak {g}}_f^{12},\\&j_{13}=j_{13}(f):=\max (\dim {\mathfrak {g}}_f^{13},\dim {\mathfrak {g}}_f^{23}),\\&j_{23}=j_{23}(f):=\min (\dim {\mathfrak {g}}_f^{13},\dim {\mathfrak {g}}_f^{23}),\\&r_1=r_1(f):=\max (\dim \pi ^t_*{\mathfrak {g}}_f^{12},\dim \pi ^x_*{\mathfrak {g}}_f^{12}),\\&r_2=r_2(f):=\min (\dim \pi ^t_*{\mathfrak {g}}_f^{12},\dim \pi ^x_*{\mathfrak {g}}_f^{12}),\\&r_3=r_3(f):=3-\min \big \{\dim \big \langle D^t(\tau ),D^x(\xi )\big \rangle \mid \\&\qquad \qquad \qquad \quad \,\,\, \exists \,\eta ^0:D^t(\tau )+D^x(\xi )+I+Z(\eta ^0)\in {\mathfrak {g}}_f\big \}, \end{aligned}$$

where $r_3:=0$ if the set in its definition is empty,

$$\begin{aligned}&{\mathfrak {g}}_f^1:={\mathfrak {g}}_f\cap \big \langle D^t(\tau ),Z(\eta ^0)\big \rangle ,\quad {\mathfrak {g}}_f^2:={\mathfrak {g}}_f\cap \big \langle D^x(\xi ),Z(\eta ^0)\big \rangle ,\\&{\mathfrak {g}}_f^{12}:={\mathfrak {g}}_f\cap \big \langle D^t(\tau ),D^x(\xi ),Z(\eta ^0)\big \rangle ,\\&{\mathfrak {g}}_f^{13}:={\mathfrak {g}}_f\cap \big \langle D^t(\tau ),I,Z(\eta ^0)\big \rangle ,\quad {\mathfrak {g}}_f^{23}:={\mathfrak {g}}_f\cap \big \langle D^x(\xi ),I,Z(\eta ^0)\big \rangle . \end{aligned}$$

The values of $(m,n,k,l,j_1,j_2,j_{12},j_{13},j_{23},r_1,r_2,r_3)$ differ from each other for different cases of Theorem 13,

$$\begin{aligned} 1&.\ (1,1,0,0,1,0,1,1,0,1,0,0);&\!\!\!\!\!\!\!\!2.\ (1,1,1,0,0,0,1,0,0,1,1,0);\\ 3&.\ (1,1,0,1,0,0,0,1,0,0,0,2);\!&\!\!\!\!\!\!\!\!4.\ (1,1,1,1,0,0,0,0,0,0,0,1);\\ 5&.\ (1,2,1,1,1,0,1,1,1,1,0,2);\!&\!\!\!\!\!\!\!\!6.\ (1,2,1,1,0,0,1,1,1,1,1,2);\\ 7&.\ (2,2,2,1,0,0,1,0,0,1,1,1);\!&\!\!\!\!\!\!\!\!8.\ (2,2,1,1,1,0,1,1,0,1,0,1);\\ 9&.\ (2,3,2,0,1,1,3,1,1,2,2,0);\!&\!\!\!\!\!\!\!\!10.\ (3,3,3,0,0,0,3,0,0,3,3,0);\\ 11&.\ (2,3,1,1,2,0,2,2,1,2,0,2);\!&\!\!\!\!\!\!\!\!12.\ (2,4,2,1,1,1,3,2,1,2,2,2);\\ 13&.\ (\infty ,\infty ,\infty ,0,\infty ,\infty ,\infty ,\infty ,\infty ,\infty ,\infty ,0). \end{aligned}$$

This is why these cases are $G^\sim$-inequivalent. At the same time, this integer tuple is redundant for distinguishing the classification cases from each other. The most remarkable minimal sufficient tuple is the triple $(r_3,j_1,r_2)$, where we order the characteristics according to their importance. The other sufficient triples are $(r_3,j_1,n)$, $(r_3,j_1,k)$, $(r_3,j_1,j_{12})$, $(r_3,j_1,r_1)$, $(r_3,r_2,n)$, $(r_3,r_2,j_{12})$, $(r_3,r_2,r_1)$. Nevertheless, in the course of the study of successive extension in the next section, we need to extend triples with other values among the above ones, although the values $r_3$, $j_1$ and $r_2$ jointly with n are still of primary importance.

5 Successive Lie-symmetry extensions

Throughout this section, the classification cases listed in Theorem 13 are interpreted in the weak sense. We intend to identify all the pairs $(\text{ Case } N,\text{ Case } {\bar{N}})$ of $G^\sim$-equivalent Lie-symmetry extensions with $\text{ Case } N\prec \text{ Case } {\bar{N}}$, i.e., where Case ${\bar{N}}$ is an additional Lie-symmetry extension of Case N modulo the $G^\sim$-equivalence, see Remark 18. For this purpose, we use a technique similar to that for the classification of contractions of low-dimensional Lie algebras, see, e.g., [11, 21, 40] and references therein. Let ${\mathfrak {s}}$ and $\bar{{\mathfrak {s}}}$ be subalgebras of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ associated with Cases N and ${\bar{N}}$, and

$$\begin{aligned} (m,n,k,l,j_1,j_2,j_{12},j_{13},j_{23},r_1,r_2,r_3) \end{aligned}$$

and

$$\begin{aligned} ({\bar{m}},{\bar{n}},{\bar{k}},{\bar{l}},{\bar{j}}_1,{\bar{j}}_2,{\bar{j}}_{12},{\bar{j}}_{13},{\bar{j}}_{23},{\bar{r}}_1,{\bar{r}}_2,{\bar{r}}_3) \end{aligned}$$

are the tuples of their $G^\sim$-invariant characteristics that are defined in Remarks 23 and 24. It is obvious that the relation $\text{ Case } N\prec \text{ Case } {\bar{N}}$ implies

$$\begin{aligned}&n<{\bar{n}},\quad m\leqslant {\bar{m}},\quad k\leqslant {\bar{k}},\quad l\leqslant {\bar{l}},\quad j_1\leqslant {\bar{j}}_1,\quad j_2\leqslant {\bar{j}}_2,\\&j_{12}\leqslant {\bar{j}}_{12},\quad j_{13}\leqslant {\bar{j}}_{13},\quad j_{23}\leqslant {\bar{j}}_{23},\quad r_1\leqslant {\bar{r}}_1,\quad r_2\leqslant {\bar{r}}_2,\quad r_3\leqslant {\bar{r}}_3. \end{aligned}$$

In other words, if at least one of the above inequalities does not hold, then $\text{ Case } N\nprec \text{ Case } {\bar{N}}$. Examining all the pairs of the cases listed in Theorem 13, we exclude the pairs $(\text{ Case } N,\text{ Case } {\bar{N}})$ with $\text{ Case } N\nprec \text{ Case } {\bar{N}}$. It turns out that for this exclusion it suffices to use only a tuple of five of the above $G^\sim$-invariant integer values, e.g., $(n,r_3,r_2,j_1,k)$, which is minimally sufficient. The other minimally sufficient tuples of five characteristics are obtained by replacing k by m or $r_1$. We order the characteristics according to their importance in the elimination procedure. The principal characteristic is the dimension n of the entire general Lie invariance algebra of the corresponding case, and the inequality between n and ${\bar{n}}$ should only be strict for $\text{ Case } N$ and $\text{ Case } {\bar{N}}$ to be ordered. The characteristics $r_3$, $r_2$, $j_1$, k, m and $r_1$ detect the following cases of disordering with $n<{\bar{n}}$:

$$\begin{aligned} r_3:\&\text{ Case } 3\nprec \text{ Cases } 7,8,9,10,13,\quad \text{ Cases } 4,5,6,7,8\nprec \text{ Cases } 9,10,13,\\&\text{ Cases } 11,12\nprec \text{ Case } 13,\\ r_2:\&\text{ Case } 2\nprec \text{ Cases } 5,8,11,\quad \text{ Cases } 6,7\nprec \text{ Case } 11,\\ j_1:\&\text{ Case } 1 \nprec \text{ Cases } 6,7,10,\quad \text{ Cases } 5,8\nprec \text{ Case } 10,\quad \text{ Case } 11\nprec \text{ Case } 12,\\ k:\&\text{ Case } 7\nprec \text{ Case } \text{11 },\quad \text{ Case } 10\nprec \text{ Case } 12,\quad \text{ or }\quad m,r_1:\ \text{ Case } 10\nprec \text{ Case } 12. \end{aligned}$$

The direct inspection shows that the remaining pairs $(\text{ Case } N,\text{ Case } {\bar{N}})$ with $n<{\bar{n}}$ are necessarily ordered, except the pairs $(\text{ Case } \text{7 },\text{ Case } \text{10 })$ and $(\text{ Case } \text{8 },\text{ Case } \text{9 })$, which are related to limit processes for Cases 7 and 8 as $q\rightarrow 0$. Therefore, the Hasse diagram in Fig. 1 represents the structure of the partially ordered set of Lie-symmetry extensions within the class (1), cf. Remark 20. Note that there are two characteristics, $j_2$ and $j_{12}$, that detect no cases of disordering with $n<{\bar{n}}$. Each of the characteristics l, $j_{13}$ and $j_{23}$ detects only cases of disordering with $n<{\bar{n}}$, that are detected by other characteristics. For example, the characteristic l detects $\text{ Cases } 3,4,5,6,7,8\nprec \text{ Cases } 9,10,13$ and $\text{ Cases } 11,12\nprec \text{ Case } 13$, which is completely covered by the characteristic $r_3$.

We derive the necessary and sufficient conditions for the parameter function ${\hat{f}}$ under which equations from Cases 1–9 have wider Lie invariance algebras than equations with general values of ${\hat{f}}$. Here we omit Case 10 since we have shown in Sect. 4 that this case admits no further Lie-symmetry extension. We first consider Cases 1–4, for each of which the parameter function ${\hat{f}}$ depends on two arguments and the corresponding common Lie invariance algebra is one-dimensional. By default, we assume that the second derivative of ${\hat{f}}$ with respect to the argument involving u is nonzero.

Case 1 possesses, modulo the $G^\sim$-equivalence, three families of further Lie-symmetry extensions, which are given by Cases 5, 8 and 9. Analyzing them, we conclude that for any further Lie-symmetry extension of Case 1, the corresponding invariance algebra contains $Q^2\in {\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ with $\pi ^x_*Q^2\ne 0$ and $[Q^1,Q^2]\in \langle Q^1,Q^2\rangle$. Therefore, $[Q^1,Q^2]\in \langle Q^1\rangle$. Up to rescaling of $Q^2$, we can assume $[Q^1,Q^2]=\delta Q^1$ with $\delta \in \{0,1\}$. We split the last commutation relation componentwise and integrate the obtained equations for the components of $Q^2$. Linearly combining $Q^2$ with $Q^1$ if necessary, we derive the representation $Q^2=\delta t\partial _t +\xi (x)\partial _x+\big (\eta ^1u+\eta ^0(x)\big )\partial _u$, where $\xi$ and $\eta ^0$ are arbitrary smooth functions of x with $\xi \ne 0$, and $\eta ^1$ is an arbitrary constant. The substitution of this representation into the classifying equation (3) leads to the equation

$$\begin{aligned} \xi {\hat{f}}_x +\big (\eta ^1u+\eta ^0\big ){\hat{f}}_u=\big (\eta ^1-\delta -\xi _x\big ){\hat{f}}. \end{aligned}$$

(7)

For any value of the parameter function ${\hat{f}}$ satisfying the last equation, we indeed have a further Lie-symmetry extension of Case 1, which belongs, up to the $G^\sim$-equivalence, to Case 5 if $\eta ^1\ne 0$ and $\delta =0$, to Case 8 if $\eta ^1\ne 0$ and $\delta =1$, or to Case 9 if $\eta ^1=0$.

Case 3 is considered similarly to Case 1. The further Lie-symmetry extensions of Case 3 are exhausted, modulo the $G^\sim$-equivalence, by Cases 5 and 6. The additional Lie-symmetry vector field $Q^2\in {\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ satisfies the conditions $\pi ^x_*Q^2\ne 0$ and $[Q^1,Q^2]=0$. This is why we can assume without loss of generality that, up to linearly combining $Q^2$ with $Q^1$ and rescaling  $Q^2$, $Q^2=\xi (x)\partial _x+(\delta u+\eta ^0(x))\partial _u$, where $\xi$ and $\eta ^0$ are arbitrary smooth functions of x with $\xi \ne 0$, and $\delta \in \{0,1\}$. Substituting such $Q^2$ into the classifying equation (3) and successively splitting with respect to t under assuming x and $\omega :=\mathrm{e}^{-t}u$ as the other independent variables, we derive one more constraint $\eta ^0=0$ for components of $Q^2$ and the equation

$$\begin{aligned} \xi {\hat{f}}_x+\delta \omega {\hat{f}}_\omega =\big (\delta -\xi _x\big ){\hat{f}}. \end{aligned}$$

(8)

The last equation defines, up to the $G^\sim$-equivalence, further Lie-symmetry extensions to Case 5 or Case 6 if $\delta =0$ or $\delta =1$, respectively.

Up to the $G^\sim$-equivalence, Case 2 has further Lie-symmetry extensions to Cases 6, 7, 9 and 10. For the additional Lie-symmetry vector field $Q^2\in {\mathfrak {g}}_{\tiny {\langle \,\rangle }}$, we have $[Q^1,Q^2]=\delta Q^1+\kappa Q^2$ for some constants $\delta$ and $\kappa$. If $\kappa =0$, then, up to rescaling of $Q^2$ and linearly recombining $Q^2$ with $Q^1$, we can set $Q^2=(\delta t+\kappa ')\partial _t+\delta x\partial _x+\big (\eta ^1u+{\hat{\eta }}^0(\omega )\big )\partial _u$, where ${\hat{\eta }}^0$ is an arbitrary smooth function of $\omega :=x-t$, $\eta ^1$ is an arbitrary constant, $\delta \in \{0,1\}$, $\kappa '$ is an arbitrary constant if $\delta =1$, and $\kappa '=1$ if $\delta =0$. Analogously to the previous cases, we substitute $Q^2$ into the classifying equation (3) and obtain the equation

$$\begin{aligned} (\delta \omega -\kappa '){\hat{f}}_\omega + \big (\eta ^1u+{\hat{\eta }}^0(\omega )\big ){\hat{f}}_u=\big (\eta ^1-2\delta \big ){\hat{f}}-{\hat{\eta }}^0_{\omega \omega }. \end{aligned}$$

(9)

In view of this equation, up to the $G^\sim$-equivalence, we have further Lie-symmetry extensions to Case 6 if $\delta =0$ and $\eta ^1\ne 0$, to Case 7 if $\delta =1$ and $\eta ^1\ne 0$, to Case 9 if $\delta =\eta ^1=0$, and to Case 10 if $\delta =1$ and $\eta ^1=0$. If $\kappa \ne 0$, then $\eta ^1=0$. Linearly recombining $Q^2$ with $Q^1$, we can set $\delta =0$. Hence $Q^2=C_1\mathrm{e}^{\kappa t}\partial _t+C_2\mathrm{e}^{\kappa x}\partial _x+\mathrm{e}^{\kappa t}{\hat{\eta }}^0(\omega )\partial _u$, where ${\hat{\eta }}^0$ is again an arbitrary smooth function of $\omega :=x-t$, and $C_1$ and $C_2$ are arbitrary constants with $(C_1,C_2)\ne (0,0)$. The classifying equation (3) with such $Q^2$ results in the equation

$$\begin{aligned} (C_2\mathrm{e}^{\kappa \omega } -C_1){\hat{f}}_\omega +{\hat{\eta }}^0{\hat{f}}_u=-\kappa (C_1+C_2\mathrm{e}^{\kappa \omega }){\hat{f}}+\kappa {\hat{\eta }}^0_\omega -{\hat{\eta }}^0_{\omega \omega }. \end{aligned}$$

(10)

Here the conditions $C_1C_2=0$ and $C_1C_2\ne 0$ are associated with further Lie-symmetry extensions to Cases 9 and 10, respectively.

All the classification cases with $n>1$ and $l>0$ are, up to the $G^\sim$-equivalence, further Lie-symmetry extensions of Case 4. Its direct Lie-symmetry extensions are exhausted, modulo the $G^\sim$-equivalence, by Cases 5, 6, 7 and 8. In view of the form of $Q^1$, we have the following commutation relation of $Q^1$ with the additional Lie-symmetry vector field $Q^2\in {\mathfrak {g}}_{\tiny {\langle \,\rangle }}$: $[Q^1,Q^2]=\kappa Q^2$ for some constant $\kappa$. The commutation relation implies the representation $Q^2=C_1\mathrm{e}^{\kappa t}\partial _t+C_2\mathrm{e}^{\kappa x}\partial _x+\mathrm{e}^{(\kappa +2)t}{\hat{\eta }}^0(\omega _1)\partial _u$, where ${\hat{\eta }}^0$ is again an arbitrary smooth function of $\omega _1:=x-t$, and $C_1$ and $C_2$ are arbitrary constants with $(C_1,C_2)\ne (0,0)$ and, if $\kappa =0$, additionally $C_1\ne C_2$. Substituting this representation into the classifying equation (3), we obtain the equation

$$\begin{aligned} \begin{aligned}&\big (C_2\mathrm{e}^{\kappa \omega _1}-C_1\big ){\hat{f}}_{\omega _1}+ \big (\mathrm{e}^{-\omega _1}{\hat{\eta }}^0(\omega _1)- C_2\omega _2\mathrm{e}^{\kappa \omega _1}-C_1\omega _2\big ){\hat{f}}_{\omega _2}\\&\qquad = -(\kappa +1) \big (C_1 +C_2\mathrm{e}^{\kappa \omega _1}\big ){\hat{f}}+ (\kappa +2)\mathrm{e}^{-\omega _1} {\hat{\eta }}^0_{\omega _1}- \mathrm{e}^{-\omega _1} {\hat{\eta }}^0_{\omega _1\omega _1}, \end{aligned} \end{aligned}$$

(11)

where $\omega _2:=\mathrm{e}^{-x-t}u$. Modulo the $G^\sim$-equivalence, we obtain extensions to Case 5 if $\kappa =0$ and $C_1C_2=0$, to Case 6 if $\kappa =0$ and $C_1C_2\ne 0$, to Case 7 if $\kappa \ne 0$ and $C_1C_2\ne 0$, and to Case 8 if $\kappa \ne 0$ and $C_1C_2=0$.

We summarize the above consideration in the following proposition.

Proposition 25

A generalized nonlinear Klein–Gordon equation from Cases 1–4 admits an additional Lie-symmetry extension if and only if the corresponding value of the parameter function ${\hat{f}}$ satisfies an equation (7) in Case 1, an equation (8) in Case 3, an equation (9) or (10) in Case 2, or an equation (11) in Case 4.

Now we derive the conditions on the parameter function ${\hat{f}}$ for the Lie invariance algebras presented in Cases 5–9 of Theorem 13 to be maximal for the corresponding equations from the class (1). In each of these cases, the arbitrary element f takes a value of the form $f=\alpha (t,x){\hat{f}}(\omega )$, where $\omega :=\beta (t,x)u$, $\alpha$ and $\beta$ are nonzero known functions of (t, x), and ${\hat{f}}_{\omega \omega }\ne 0$ since $f_{uu}\ne 0$. Substituting this form for f into the classifying equation (3), we obtain the classifying equation in terms of ${\hat{f}}$,

$$\begin{aligned}&\bigg (\bigg (\frac{\beta _t}{\beta }\tau +\frac{\beta _x}{\beta }\xi +\eta ^1\bigg )\omega +\beta \eta ^0\bigg ){\hat{f}}_\omega \nonumber \\&\quad +\bigg (\tau _t+\xi _x+\frac{\alpha _t}{\alpha }\tau +\frac{\alpha _x}{\alpha }\xi -\eta ^1\bigg ){\hat{f}}-\frac{\eta ^0_{tx}}{\alpha }=0. \end{aligned}$$

(12)

We apply the method of furcate splitting, see [4, 41, 45] and references therein. Fixing values of the variables t and x gives the template form of equations for values of ${\hat{f}}$, for which the equation $K_f$ possesses an additional Lie-symmetry extension,

$$\begin{aligned} (a\omega +b){\hat{f}}_\omega +c{\hat{f}}-d=0, \end{aligned}$$

(13)

where a, b, c and d are constants with $(a,b)\ne (0,0)$. Additionally, in view of ${\hat{f}}_{\omega \omega }\ne 0$ we have $c\ne -a$ if $a\ne 0$ and $c\ne 0$ if $a=0$. Moreover, the number of equations of the form (13) with linearly independent tuples (a, b, c, d) cannot exceed one since otherwise ${\hat{f}}_{\omega \omega }=0$. In other words, we have exactly one independent equation of the form (13) if the equation $K_f$ possesses an additional Lie-symmetry extension. This means that the left-hand side of (12) is proportional to that of (13) with nonvanishing multiplier $\lambda$ depending on (t, x),

$$\begin{aligned}&\bigg (\bigg (\frac{\beta _t}{\beta }\tau +\frac{\beta _x}{\beta }\xi +\eta ^1\bigg )\omega +\beta \eta ^0\bigg ){\hat{f}}_\omega +\bigg (\tau _t+\xi _x+\frac{\alpha _t}{\alpha }\tau +\frac{\alpha _x}{\alpha }\xi -\eta ^1\bigg ){\hat{f}}-\frac{\eta ^0_{tx}}{\alpha }\\&\qquad {}=\lambda \big ((a\omega +b){\hat{f}}_\omega +c{\hat{f}}-d\big ). \end{aligned}$$

The last equation can be split with respect to ${\hat{f}}$ and ${\hat{f}}_\omega$ into the system

$$\begin{aligned} \begin{aligned}&\frac{\beta _t}{\beta }\tau +\frac{\beta _x}{\beta }\xi +\eta ^1=a\lambda ,\quad \tau _t+\xi _x+\frac{\alpha _t}{\alpha }\tau +\frac{\alpha _x}{\alpha }\xi -\eta ^1=c\lambda ,\\&\quad \beta \eta ^0=b\lambda ,\qquad \eta ^0_{tx}=d\alpha \lambda . \end{aligned} \end{aligned}$$

(14)

If $a\ne 0$, then we can make $a=1$ by rescaling the template-form equation (13), and thus $c\ne -1$. The first two equations of the system (14) are combined to

$$\begin{aligned}&\lambda =\frac{\beta _t}{\beta }\tau +\frac{\beta _x}{\beta }\xi +\eta ^1,\nonumber \\&\tau _t+\left( \frac{\alpha _t}{\alpha }-c\frac{\beta _t}{\beta }\right) \tau +\xi _x+\left( \frac{\alpha _x}{\alpha }-c\frac{\beta _x}{\beta }\right) \xi =(c+1)\eta ^1, \end{aligned}$$

(15)

and only the last equation plays the role of a classifying condition. The third and fourth equations of the system (14) merely establish a relation between the constant parameters b and d. Indeed, in each of Cases 5–9, we have $(1/\beta )_{tx}$ is proportional to $\alpha$, $(1/\beta )_{tx}=C\alpha$ for some constant C. Hence we can always set $b=0$ by the equivalence transformation ${\mathscr {Z}}(b/\beta )$, which adds bC to ${\hat{f}}$ and also makes $\eta ^0=0$ and $d=0$. In the general case $a\ne 0$, the relation between b and d is $ad=bcC$. For specific values of the parameter functions $\alpha$ and $\beta$ in Cases 5–9, we obtain the following values of the constant parameter C and the following forms of the reduced classifying equation (15):

$$\begin{aligned}&\text{ Case } \text{5 }:\quad C=0, \quad \tau _t+c\tau +\xi _x=(c+1)\eta ^1; \\&\text{ Case } \text{6 }:\quad C=1, \quad \tau _t+c\tau +\xi _x+c\xi =(c+1)\eta ^1;\\&\text{ Case } \text{7 }:\quad C=q(q+1),\\&\quad \qquad \qquad \,\, (x-t)(\tau _t+\xi _x)+(q+2+cq)(\tau -\xi )=(c+1)(x-t)\eta ^1;\\&\text{ Case } \text{8 }:\quad C=0, \quad \tau _t+\xi _x-(q+2+cq)x^{-1}\xi =(c+1)\eta ^1;\\&\text{ Case } \text{9 }:\quad C=0, \quad \tau _t+\xi _x=(c+1)\eta ^1. \end{aligned}$$

In each of Cases 5, 6, 8 and 9, the corresponding classifying equation implies that there is a further Lie-symmetry extension if and only if the associated value of the parameter function ${\hat{f}}$ satisfies the equation (13) with arbitrary $a\ne 0$, c and b and with $d=bcC/a$. Here the extension is given by Case 12 with an arbitrary nonzero constant p. Case 7 is similar but the constant c is related to a according to $c=-(1+2/q)a$, $d=-(q+1)(q+2)b$, and the extension for a fixed q is given by Case 12$_p$ with $p=2/q$.

If $a=0$, then $bc\ne 0$, and we rescale the template-form equation (13) for making $c=1$. Then

$$\begin{aligned} \begin{aligned}&\frac{\beta _t}{\beta }\tau +\frac{\beta _x}{\beta }\xi +\eta ^1=0,\quad \lambda =\tau _t+\xi _x+\frac{\alpha _t}{\alpha }\tau +\frac{\alpha _x}{\alpha }\xi -\eta ^1,\quad \eta ^0=b\frac{\lambda }{\beta },\quad \\&\quad \left( \frac{\lambda }{\beta }\right) _{tx}=\frac{d}{b}\alpha \lambda . \end{aligned} \end{aligned}$$

(16)

In Cases 5–9, the first two equations of (16) respectively reduce to

$$\begin{aligned}&\text{ Case } \text{5 }:\ \tau =\eta ^1,\ \ \lambda =\xi _x;\\&\text{ Case } \text{6 }:\ \tau +\xi -\eta ^1=0,\ \ \lambda =\tau _t+\xi _x;\\&\text{ Case } \text{7 }:\ q\frac{\tau -\xi }{x-t}-\eta ^1=0,\ \lambda =\tau _t+\xi _x+2\frac{\tau -\xi }{x-t};\\&\text{ Case } \text{8 }:\ q\xi +\eta ^1x=0,\ \ \lambda =\tau _t-2\frac{\eta ^1}{q};\quad \text{ Case } \text{9 }:\ \eta ^1=0,\ \ \lambda =\tau _t+\xi _x. \end{aligned}$$

In Case 5, we have no additional constraints on the parameters b, c and d. The last two equations of (16) in view of the first ones just define $\eta ^0$ and $\xi$, $\eta ^0=b\xi _x\mathrm{e}^t$, $b\xi _{xx}=d\xi _x$.

In Cases 6 and 7, the above equations imply $\lambda =0$, which contradicts the inequality $\lambda \ne 0$. This means that these cases possess no further Lie-symmetry extensions with $a=0$.

The system (16) implies $\tau _{tt}=0$ and $d=0$ in Case 8 and $d=0$ in Case 9, which correspond to the Lie-symmetry extensions to Cases 11 and 13, respectively.

Merging the conditions derived separately for $a\ne 0$ and $a=0$, we obtain the following proposition.

Proposition 26

A generalized nonlinear Klein–Gordon equation from Cases 5–9 admits an additional Lie-symmetry extension if and only if the corresponding value of the parameter function ${\hat{f}}$ satisfies an equation (13) with $(a,b)\ne (0,0)$, where $ad=0$ in Case 5, $ad=bc$ in Case 6, $c=-(1+2/q)a$, $d=-(q+1)(q+2)b$ in Case 7$_q$, and $d=0$ in Cases 8 and 9.

Remark 27

In view of the infinitesimal counterpart of [54, Proposition 10], all the subclasses ${\mathcal {K}}'_N$, $N\in \Gamma$, of ${\mathcal {K}}$ that are associated with strong Cases 0–13 are normalized, see Remark 14 for notation and definitions. At the same time, this is not the case for most of the subclasses ${\mathcal {K}}_N$, $N\in \Gamma$. More specifically, the subclasses ${\mathcal {K}}_1$, ..., ${\mathcal {K}}_6$, ${\mathcal {K}}_{7_q}$, ${\mathcal {K}}_{8_q}$ and ${\mathcal {K}}_9$ are not normalized in view of the following arguments:

• sep=1ex

• ${\mathcal {K}}_1\supset {\mathcal {K}}_9$ but $G^\sim _1\nsupseteq G^\sim _9$ since ${\mathscr {I}}^0\in G^\sim _9{\setminus } G^\sim _1$;

• ${\mathcal {K}}_2\supset {\mathcal {K}}_9$ but $G^\sim _2\nsupseteq G^\sim _9$ since ${\mathscr {I}}^t\in G^\sim _9{\setminus } G^\sim _2$;

• ${\mathcal {K}}_3\ni K_f$ with $f=\mathrm{e}^{-t}u^2$ but $\varpi _*G^\sim _3\nsupseteq G_f$ since $\varpi _*\big ({\mathscr {D}}^t(-\ln t)\circ {\mathscr {I}}^0\circ {\mathscr {D}}^t(\mathrm{e}^{-t})\big )\in G_{f}{\setminus }\varpi _*G^\sim _3$;

• ${\mathcal {K}}_4\ni K_f$ with $f=\mathrm{e}^{-p(t+x)}|u|^pu$ for any $p\ne 0$ but $\varpi _*{\mathfrak {g}}^\sim _4\nsupseteq {\mathfrak {g}}_f$ since $D^t(\mathrm{e}^{pt})\in {\mathfrak {g}}_f{\setminus }\varpi _*{\mathfrak {g}}^\sim _4$;

• ${\mathcal {K}}_5\ni K_f$ with $f=\mathrm{e}^{-t}u^2$ but $\varpi _*G^\sim _5\nsupseteq G_{f}$ since $\varpi _*\big ({\mathscr {D}}^t(-\ln t)\circ {\mathscr {I}}^0\circ {\mathscr {D}}^t(\mathrm{e}^{-t})\big )\in G_{f}{\setminus }\varpi _*G^\sim _5$;

• ${\mathcal {K}}_6\ni K_f$ with $f=\mathrm{e}^{-p(t+x)}|u|^pu$ for any $p\ne 0$ but $\varpi _*{\mathfrak {g}}^\sim _6\nsupseteq {\mathfrak {g}}_f$ since $D^t(\mathrm{e}^{pt})\in {\mathfrak {g}}_f{\setminus }\varpi _*{\mathfrak {g}}^\sim _6$;

• ${\mathcal {K}}_{7,q}\supset {\mathcal {K}}_{12,p}$ with $p=2/q$ but $G^\sim _{7,q}\nsupseteq G^\sim _{12,p}$ since ${\mathscr {D}}^t(2t)\circ {\mathscr {D}}^x(x/2)\in G^\sim _{12,p}{\setminus } G^\sim _{7,q}$;

• ${\mathcal {K}}_{8,q}\supset {\mathcal {K}}_{12,p}$ with $p=2/q$ but $G^\sim _{8,q}\nsupseteq G^\sim _{12,p}$ since ${\mathscr {I}}^0\in G^\sim _{12,p}{\setminus } G^\sim _{8,q}$;

• ${\mathcal {K}}_9\supset {\mathcal {K}}_{13}=\{K_{\mathrm{e}^u}\}$ but $G^\sim _9\nsupseteq G^\sim _{13}$ since ${\mathscr {D}}^t(-t^{-1})\circ {\mathscr {Z}}(2\ln |t|)\in G^\sim _{13}{\setminus } G^\sim _9$.

Here $G_f$ denotes the point symmetry group of the equation $K_f$. The subclasses ${\mathcal {K}}_0$, ${\mathcal {K}}_{10}$, ${\mathcal {K}}_{11}$, ${\mathcal {K}}_{12,p}$ and ${\mathcal {K}}_{13}$ are normalized since they coincide with ${\mathcal {K}}$, ${\mathcal {K}}'_{10}$ and the singletons $\{K_{\mathrm{e}^{u/x}}\}$, $\{K_{|u|^pu}\}$ and $\{K_{\mathrm{e}^u}\}$, respectively. (It is obvious that a class consisting of a single system of differential equations is normalized.)

6 On group classification of subclasses

Although we have exhaustively solved the group classification problem for the class ${\mathcal {K}}$, which consists of the equations of the form (1), this does not directly lead to the solution of the group classification problem for each of the subclasses of ${\mathcal {K}}$. Given a subclass $\hat{{\mathcal {K}}}$ of ${\mathcal {K}}$, Theorem 13 is used for the group classification of the subclass $\hat{{\mathcal {K}}}$ with respect to its equivalence group $G^\sim _{\hat{{\mathcal {K}}}}$ as follows.

• Recalling the normalization of ${\mathcal {K}}$, construct the equivalence group $G^\sim _{\hat{{\mathcal {K}}}}$ as the subgroup of $G^\sim$ that consists of the elements of $G^\sim$ preserving the subclass $\hat{{\mathcal {K}}}$.

• For each $N\in \Gamma$, intersect the subclass $\hat{{\mathcal {K}}}$ with the $G^\sim$-orbit $G^\sim _*{\mathcal {K}}'_N$ the subclass ${\mathcal {K}}'_N$ (resp. with the $G^\sim$-orbit $G^\sim _*{\mathcal {K}}_N$ the subclass ${\mathcal {K}}_N$). This is realized via selecting those values of the arbitrary element f for equations from the orbit that satisfy the additional auxiliary constraint singling out the subclass $\hat{{\mathcal {K}}}$ from the class ${\mathcal {K}}$. The collection of the intersections presents a complete list of Lie symmetry extensions within the subclass $\hat{{\mathcal {K}}}$.

• In the selected values of f, gauge parameters by transformations from $G^\sim _{\hat{{\mathcal {K}}}}$.

Since the subclass $\hat{{\mathcal {K}}}$ is in general not normalized, the above procedure looks easier than directly solving the group classification problem for the subclass $\hat{{\mathcal {K}}}$ although its computational complexity is quite high.

The subclasses ${\mathcal {K}}_N$, $N\in \Gamma$, of ${\mathcal {K}}$ that are associated with weak Cases 1–9, cf. Remark 14, are specific in regard to the above procedure. The group classification of each of these subclasses up to the equivalence generated by the corresponding equivalence groupoid can be easily derived via analyzing the Hasse diagram in Fig. 1, which depicts the structure of the partially ordered set of these cases. Nevertheless, this is not the case for the group classification up to the equivalence generated by the corresponding equivalence group since most of the subclasses ${\mathcal {K}}_N$, $N\in \Gamma$, are not normalized, see Remark 27. Note that both the group classifications of the subclasses ${\mathcal {K}}_{10}$, ${\mathcal {K}}_{11}$, ${\mathcal {K}}_{12,p}$ and ${\mathcal {K}}_{13}$ are trivial since the kernel Lie invariance algebra of equations from each of these subclasses is the maximal Lie invariance algebra for every such equation.

Consider in detail the subclass ${\mathcal {K}}_2$ of the class (1), which is related to weak Case 2 of Theorem 13 and thus consists of the equations of the form

$$\begin{aligned} u_{tx}=f(\omega ,u), \quad \text{ where }\quad \omega :=x-t,\quad f_{uu}\ne 0, \end{aligned}$$

or, in the variables $(\check{t},\check{x},\check{u})=(x+t,x-t,u)$ with $\check{f}(\check{x},\check{u})=f(\omega ,u)$,

$$\begin{aligned} \check{u}_{\check{t}\check{t}}-\check{u}_{\check{x}\check{x}}=\check{f}(\check{x},\check{u}),\quad \check{f}_{\check{u}\check{u}}\ne 0. \end{aligned}$$

Lemma 28

The equivalence group $G^\sim _2$ of the class ${\mathcal {K}}_2$ consists of the transformations of the form

$$\begin{aligned} {\tilde{t}}=c_1t+c_2,\quad {\tilde{x}}=c_1x+c_3,\quad {\tilde{u}}=c_4u+U^0(\omega ), \quad {\tilde{f}}=c_1^{-2}(c_4f-U^0_{\omega \omega }) \end{aligned}$$

(17)

and the discrete equivalence transformation ${\mathscr {I}}^0$: ${\tilde{t}}= x$, ${\tilde{x}}=t$, ${\tilde{u}}=u$, ${\tilde{f}}=f$. Here $c_1$, ..., $c_4$ are arbitrary constants with $c_1c_4\ne 0$, and $U^0$ is an arbitrary smooth function of $\omega :=x-t$.

Proof

Since the class (1) is normalized, the equivalence group of any subclass of (1) is the subgroup of $G^\sim$ that consists of the elements of $G^\sim$ preserving this subclass. It is obvious that the transformation ${\mathscr {I}}^0$ belongs to $G^\sim _2$. Any transformation of the form (2) that is contained by the group $G^\sim _2$ satisfies the equation

$$\begin{aligned} T_tX_x{\tilde{f}}(X-T,U)=Cf(x-t,u)+U^0_{tx}. \end{aligned}$$

(18)

We act on (18) by the operator $\partial _t+\partial _x$, obtaining

$$\begin{aligned}&(X_x-T_t)\,{\tilde{f}}_{{\tilde{\omega }}}(X-T,U)+(U^0_t+U^0_x)\,{\tilde{f}}_{{\tilde{u}}}(X-T,U)\\&\quad +\frac{T_{tt}X_x+T_tX_{xx}}{T_tX_x}\,{\tilde{f}}(X-T,U)=\frac{U^0_{ttx}+U^0_{txx}}{T_tX_x}. \end{aligned}$$

Since ${\tilde{f}}$ is an unconstrained value of the arbitrary element of the class ${\mathcal {K}}_2$, for computing $G^\sim _2$, we can split the last equation with respect to ${\tilde{f}}$ and its derivatives. As a result, we derive the equations $T_t=X_x$ and $U^0_t+U^0_x=0$ on the parameters involved in the form (2). The integration of these equations implies the form (17). $\square$

The appropriate subalgebra ${\mathfrak {s}}=\langle \partial _t+\partial _x\rangle$ of ${\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ is the kernel Lie invariance algebra of the equations from the class ${\mathcal {K}}_2$. In other words, Case 2 is the general case with no Lie-symmetry extensions within this class. It is not normalized since the action groupoid of $G^\sim _2$ is properly contained in the equivalence groupoid ${\mathcal {G}}^\sim _2$. Indeed, many equations in ${\mathcal {K}}_2$, e.g., the Liouville equation, possess points symmetries that are not related to equivalence transformations of ${\mathcal {K}}_2$. Moreover, Theorem 30 below implies that the class $\mathcal K_2$ is not semi-normalized as well. This is why it is natural that the group classifications of the class ${\mathcal {K}}_2$ up to the ${\mathcal {G}}^\sim _2$- and the $G^\sim _2$-equivalences are different. We easily see from the Hasse diagram in Fig. 1 that ${\mathcal {G}}^\sim _2$-inequivalent cases of Lie-symmetry extensions within the class ${\mathcal {K}}_2$ are exhausted by Cases 6, 7, 9, 10, 12 and 13, which gives the complete group classification of this class up to the ${\mathcal {G}}^\sim _2$-equivalence. (In Case 6, we should additionally alternate the sign of x.)

The complete group classification of the class ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence is more delicate. It can be derived from the group classification of the superclass ${\mathcal {W}}$ of ${\mathcal {K}}_2$, which consists of the equations of the following form in the variables $(\check{t},\check{x},\check{u})$:

$$\begin{aligned} \check{u}_{\check{t}\check{t}}-\check{g}(\check{x},\check{u})\check{u}_{\check{x}\check{x}}=\check{f}(\check{x},\check{u}),\quad (\check{g}_{\check{u}},\check{f}_{\check{u}\check{u}})\ne (0,0). \end{aligned}$$

It is obvious that the class ${\mathcal {K}}_2$ is singled out from the superclass ${\mathcal {W}}$ by the constraint $\check{g}=1$. The comprehensive group analysis of the class ${\mathcal {W}}$ was carried out in [55], where a different notation of the arbitrary elements $\check{g}$ and $\check{f}$ was used, $\check{g}\rightsquigarrow f$ and $\check{f}\rightsquigarrow g$. The equivalence group $G^\sim _{{\mathcal {W}}}$ and the equivalence groupoid ${\mathcal {G}}^\sim _{{\mathcal {W}}}$ of ${\mathcal {W}}$ were described in [55, Theorem 6] and in [55, Theorem 9], respectively. The action groupoid of $G^\sim _{{\mathcal {W}}}$ is a proper subgroupoid of the groupoid ${\mathcal {G}}^\sim _{{\mathcal {W}}}$, i.e., the superclass ${\mathcal {W}}$ is not normalized. The restriction of the action groupoid of $G^\sim _{{\mathcal {W}}}$ to the subclass ${\mathcal {K}}_2$ of ${\mathcal {W}}$ coincides with the action groupoid of $G^\sim _2$. This is why the complete group classification of the class ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence can be singled out from the complete group classification of the superclass ${\mathcal {W}}$ up to the $G^\sim _{{\mathcal {W}}}$-equivalence, which was presented in [55, Theorem 8]. Since the gauge $\check{g}=1$ modulo the $G^\sim _{{\mathcal {W}}}$-equivalence was used for representatives of Lie-symmetry extensions whenever it was possible, to classify Lie symmetries of equations from ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence it suffices to select all the cases of [55, Table 1] with $\check{g}=1$, i.e., $f=1$ in the notation of [55], write them in the variables (t, x, u), and supplement the result with Cases 6 and 7 of Theorem 13 of the present paper, which are the counterparts of the appropriate portions of Cases 1 and 2 of [55, Table 1]. As a result, we prove the following theorem.

Theorem 29

A complete list of $G^\sim _2$-inequivalent cases of Lie-symmetry extensions of the kernel Lie invariance algebra ${\mathfrak {g}}^\cap =\langle \partial _t+\partial _x\rangle$ in the class ${\mathcal {K}}_2$ are exhausted by the following cases:

$$\begin{aligned}&2.\ \ \text{ General } \text{ case } f={\hat{f}}(x-t,u):\ {\mathfrak {g}}_f=\langle \partial _t+\partial _x\rangle ;\\&6.\ \ f=\mathrm{e}^{-x+t}{\hat{f}}(\mathrm{e}^{x-t}u):\ {\mathfrak {g}}_f=\langle \partial _t+u\partial _u,\,\partial _x-u\partial _u\rangle ;\\&7.\ \ f=|x-t|^{-q-2}{\hat{f}}(|x-t|^qu), \ q\ne 0:\ \\&\qquad {\mathfrak {g}}_f=\langle \partial _t+\partial _x,\,t\partial _t+x\partial _x-qu\partial _u\rangle ;\\&9\mathrm{a}.\ \ f={\hat{f}}(u):\ {\mathfrak {g}}_f=\langle \partial _t,\,\partial _x,\,t\partial _t-x\partial _x\rangle ;\\&9\mathrm{b}.\ \ f={\hat{f}}(u)\mathrm{e}^{x-t}:\ {\mathfrak {g}}_f=\langle \mathrm{e}^t\partial _t,\,\mathrm{e}^{-x}\partial _x,\,\partial _t+\partial _x\rangle ;\\&10\mathrm{a}.\ \ f={\hat{f}}(u)(x-t)^{-2}:\ {\mathfrak {g}}_f=\langle \partial _t+\partial _x,\,t\partial _t+x\partial _x,\,t^2\partial _t+x^2\partial _x\rangle ;\\&10\mathrm{b}.\ \ f={\hat{f}}(u)\cos ^{-2}(x-t):\ \\&\qquad \quad {\mathfrak {g}}_f=\langle \partial _t+\partial _x,\,\cos 2t\,\partial _t-\cos 2x\,\partial _x,\,\sin 2t\,\partial _t-\sin 2x\,\partial _x\rangle ;\\&10\mathrm{c}.\ \ f={\hat{f}}(u)\cosh ^{-2}(x-t):\ \\&\qquad \quad {\mathfrak {g}}_f=\langle \partial _t+\partial _x,\,\mathrm{e}^{2t}\partial _t-\mathrm{e}^{2x}\partial _x,\,\mathrm{e}^{-2t}\partial _t-\mathrm{e}^{-2x}\partial _x\rangle ;\\&10\mathrm{d}.\ \ f={\hat{f}}(u)\sinh ^{-2}(x-t):\ \\&\qquad \quad {\mathfrak {g}}_f=\langle \partial _t+\partial _x,\,\mathrm{e}^{2t}\partial _t+\mathrm{e}^{2x}\partial _x,\,\mathrm{e}^{-2t}\partial _t+\mathrm{e}^{-2x}\partial _x\rangle ;\\&12\mathrm{a}.\ \ f=|u|^p u, \ p\ne -1,0:\ {\mathfrak {g}}_f=\langle \partial _t,\,\partial _x,\,t\partial _t-x\partial _x,\,-pt\partial _t+u\partial _u\rangle ;\\&12\mathrm{b}.\ \ f=|u|^p u\mathrm{e}^{x-t}, \ p\ne -1,0:\ {\mathfrak {g}}_f=\langle \mathrm{e}^t\partial _t,\,\mathrm{e}^{-x}\partial _x,\,\partial _t+\partial _x,\,p\partial _t+u\partial _u\rangle ;\\&13.\ \ f=\mathrm{e}^u:\ {\mathfrak {g}}_f=\langle \tau (t)\partial _t+\xi (x)\partial _x-(\tau _t(t)+\xi _x(x))\partial _u\rangle . \end{aligned}$$

Here ${\hat{f}}$ is an arbitrary smooth function of its arguments whose second derivative with respect to the argument involving u is nonzero, q and p are arbitrary constants that satisfy the conditions indicated in the corresponding cases. In Case 13, the components $\tau$ and $\xi$ run through the sets of smooth functions of t or x, respectively.

We use the two-level numeration for the classification cases listed in Theorem 29 for indicating the presence of additional equivalences between these cases. Namely, numbers with the same Arabic numerals and different Roman letters correspond to cases that are $G^\sim _2$-inequivalent but ${\mathcal {G}}^\sim _2$-equivalent and hence $G^\sim$-equivalent as Lie-symmetry extensions within the class ${\mathcal {K}}$. Related cases in Theorems 13 and 29 have numbers with the same Arabic numerals.

To find all additional equivalence transformations among $G^\sim _2$-inequivalent classification cases for ${\mathcal {K}}_2$ and thus to relate the group classification of ${\mathcal {K}}_2$ modulo the $G^\sim _2$-equivalence to that modulo the ${\mathcal {G}}^\sim _2$-equivalence, we need to classify admissible transformations within the class ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence. The description of the equivalence groupoid ${\mathcal {G}}^\sim _2$ of the class ${\mathcal {K}}_2$ can be derived from [55, Theorem 9] analogously to the above derivation of the group classification of ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence. See necessary notions in [55, Section 2].

Theorem 30

A generating (up to the $G^\sim _2$-equivalence) set of admissible transformations for the class ${\mathcal {K}}_2$, which is minimal and self-consistent with respect to the $G^\sim _2$-equivalence, is the union of the following families of admissible transformations $(f,\Phi ,{\tilde{f}})$:

$$\begin{aligned}&\mathrm{T1 }.\ \ f= {\hat{f}}(u),\ \ {\tilde{f}}=-f,\quad \Phi :\ \ {\tilde{t}}=-t,\ \ {\tilde{x}}=x,\ \ {\tilde{u}}=u;\\&\mathrm{T2 }.\ \ f= {\hat{f}}(u),\ \ {\tilde{f}}=f,\quad \Phi :\ \ {\tilde{t}}=t\mathrm{e}^\gamma ,\ \ {\tilde{x}}=x\mathrm{e}^{-\gamma },\ \ {\tilde{u}}=u,\ \ \gamma \in {\mathbb {R}}_{\ne 0};\\&\mathrm{T3 }.\ \ f={\hat{f}}(u)\mathrm{e}^{x-t},\ \ {\tilde{f}}={\hat{f}}({\tilde{u}}),\quad \Phi :\ \ {\tilde{t}}=-\mathrm{e}^{-t},\ \ {\tilde{x}}=\mathrm{e}^x,\ \ {\tilde{u}}=u;\\&\mathrm{T4a}.\ \ f= {\hat{f}}(u)(x-t)^{-2},\ \ {\tilde{f}}={\hat{f}}({\tilde{u}})({\tilde{x}}-{\tilde{t}})^{-2},\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \!\Phi :\ \ {\tilde{t}}=t^{-1},\ \ {\tilde{x}}=x^{-1}, \ {\tilde{u}}=u;\\&\mathrm{T4b}.\ \ f= -{\hat{f}}(u)\cos ^{-2}(x-t),\ \ {\tilde{f}}={\hat{f}}({\tilde{u}})({\tilde{x}}-{\tilde{t}})^{-2},\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \!\Phi :\ \ {\tilde{t}}= \tan t, {\tilde{x}}=\cot x,\ \ {\tilde{u}}=u;\\&\mathrm{T4c}.\ \ f=-{\hat{f}}(u)\cosh ^{-2}(x-t),\ \ {\tilde{f}}={\hat{f}}({\tilde{u}})({\tilde{x}}-{\tilde{t}})^{-2},\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \!\Phi :\ \ {\tilde{t}}=-\tfrac{1}{2}\mathrm{e}^{2t}, \ {\tilde{x}}=\tfrac{1}{2}\mathrm{e}^{2x},\ \ {\tilde{u}}=u;\\&\mathrm{T4d}.\ \ f= {\hat{f}}(u)\sinh ^{-2}(x-t),\ \ {\tilde{f}}={\hat{f}}({\tilde{u}})({\tilde{x}}-{\tilde{t}})^{-2},\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \!\Phi :\ \ {\tilde{t}}=\tfrac{1}{2}\mathrm{e}^{2t},\ \ {\tilde{x}}=\tfrac{1}{2}\mathrm{e}^{2x}, \ {\tilde{u}}=u;\\&\mathrm{T5 }.\ \ f=\mathrm{e}^u,\ \ {\tilde{f}}=\mathrm{e}^{{\tilde{u}}},\quad \Phi :\ \ {\tilde{t}}=T(t),\ \ {\tilde{x}}=X(x),\ \ {\tilde{u}}=u-\ln (T_tX_x), \end{aligned}$$

where (T, X) runs through a complete set of representatives of cosets of (T, X) with $T_tX_x>0$ and $(T_{tt},X_{xx})\ne (0,0)$ with respect to the action of the group constituted by the transformations of the form ${\hat{t}}=c_1t+c_2$, ${\hat{x}}=c_1x+c_3$, ${\hat{T}}={\tilde{c}}_1T+{\tilde{c}}_2$, ${\hat{X}}={\tilde{c}}_1X+{\tilde{c}}_3$, where $c_1$, $c_2$, $c_3$, ${\tilde{c}}_1$, ${\tilde{c}}_2$ and ${\tilde{c}}_3$ are arbitrary constants with $c_1{\tilde{c}}_1\ne 0$.

Having the classification of admissible transformations within the class ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence, we can directly find all independent additional equivalence transformations among classification cases listed in Theorem 29. These transformations are

$$\begin{aligned}&\mathrm{T1 } :\ \text{ Case } \text{9a }_{{\hat{f}}}\rightarrow \text{ Case } \text{9a }_{-{\hat{f}}},\ \ \\&\mathrm{T3 } :\ \text{ Case } \text{9b } \rightarrow \text{ Case } \text{9a, } \text{ Case } \text{12b } \rightarrow \text{ Case12a },\\&\mathrm{T4b} :\ \text{ Case } \text{10b }\rightarrow \text{ Case } \text{10a },\ \ \\&\mathrm{T4c} :\ \text{ Case } \text{10c }\rightarrow \text{ Case } \text{10a },\ \ \\&\mathrm{T4d} :\ \text{ Case } \text{10d }\rightarrow \text{ Case } \text{10a }. \end{aligned}$$

7 Conclusion

In the present paper, we have carried out the complete (contact) group classification of the class (1) of (1+1)-dimensional generalized nonlinear Klein–Gordon equations up to the $G^\sim$-equivalence. This has substantially enhanced the results on Lie symmetries of such equations that were obtained in the seminal papers [32, 33]. At first, extending results of Lie’s paper [35], we have shown in Lemma 2 that any contact admissible transformation within the class (1) is the first-order prolongation of a point admissible transformation within this class. In other words, the study of contract-transformation structures related to equations from the class (1) reduces to the study of their point-transformation counterparts. We have proved in Lemma 3 that the class (1) is normalized. Therefore, applying the algebraic method, we have reduced the group classification of (1) to classifying the appropriate subalgebras of the projection $\varpi _*{\mathfrak {g}}^\sim ={\mathfrak {g}}_{\tiny {\langle \,\rangle }}$ of the equivalence algebra ${\mathfrak {g}}^\sim$. In addition to this, we have employed the specific structure of ${\mathfrak {g}}^\sim$ for twofold involving the classical Lie theorem on realizations of Lie algebras by vector fields on the line [34] into the classification procedure. Moreover, the normalization of the class (1) means that the action groupoid [55] of the equivalence group $G^\sim$ coincides with the entire equivalence groupoid ${\mathcal {G}}^\sim$ of the class (1). Hence the complete group classification of the class (1) up to the $G^\sim$-equivalence coincides with its complete group classification up to the ${\mathcal {G}}^\sim$-equivalence, which is just the general point equivalence within this class. In other words, we have no additional point equivalences between $G^\sim$-inequivalent classification cases.

Lie symmetries of equations from the class (1) were considered in Section 6 of [33], and cases with two-, three- and four-dimensional Lie invariance algebras were listed in Table 1 therein, see also Section V and Table I in [32]. Cases 1–6, 8 and 9 of Table 1 and the equation (5.4) in [33] correspond to Cases 5, 6, 7$_{q=1}$, 8$_{q=-1}$, 10, 11, 9, 12 and 13 of Theorem 13 in the present paper. Case 7 of Table 1 in [33] should be excluded from the classification since it is equivalent to Case 9 therein, see the discussion of the case $(m,n,k)=(2,3,1)$ in Sect. 4. Cases 7$_{q\ne 1}$ and 8$_{q\ne -1}$ were missed in [33] owing to superfluously constraining the parameter q, see Remark 22.

We have additionally enhanced the results of [32, 33] by explicitly singling out the equations from the class (1) with infinite-dimensional maximal Lie invariance algebras in Lemma 11. It turns out that any such equation is $G^\sim$-equivalent to the Liouville equation. For the other equations from the class (1), whose maximal Lie invariance algebras are finite-dimensional, we have found the least upper bound of dimensions of these algebras, which is equal to four. One more tool for arranging the classification is to assign a value of the triple (m, n, k) of $G^\sim$-invariant integers to each case of Lie-symmetry extension in the class (1). We have strongly restricted the set of candidates for appropriate values of the triple at the stage of preliminary analysis using Lemma 11 and the Lie theorem. The final selection of the appropriate values has been done in the course of the group classification. It was important for simplifying the computations on all classification stages that for equations from the class (1), in contrast to evolution equations, the Lie theorem can be applied to both the t- and the x-projections of Lie-symmetry vector fields.

Although the characteristic triple (m, n, k) has a simple interpretation and is principal for the proof of Theorem 13, it does not suffice for completely distinguishing $G^\sim$-inequivalent classification cases. This is why we attempted to find as many $G^\sim$-invariant integer characteristics of the classification cases as possible, and have found even twelve of them in total, m, n, k, l, $j_1$, $j_2$, $j_{12}$, $j_{13}$, $j_{23}$, $r_1$, $r_2$ and $r_3$. The comprehensive analysis has shown that the complete tuple of these twelve characteristics is redundant. As found out in Remark 24, these characteristics can constitute no pairs and exactly eight triples that suffice for distinguishing $G^\sim$-inequivalent classification cases, and the most remarkable triple is $(r_3,j_1,r_2)$. The same triple is, simultaneously with n, of primary importance for identifying the pairs of $G^\sim$-inequivalent weak classification cases that do not represent successive Lie-symmetry extensions, i.e., the pairs $(\text{ Case } N,\text{ Case } {\bar{N}})$ with $\text{ Case } N\nprec \text{ Case } {\bar{N}}$, see Sect. 5. To be sufficient for this task, the tuple $(n,r_3,r_2,j_1)$ should be extended with one of the characteristics k, m or $r_1$, and the thus obtained three tuples exhaust the set of such sufficient tuples of minimum size, which is equal to five. In the same section, we have directly checked that all the other pairs of $G^\sim$-inequivalent classification cases are indeed associated with successive Lie-symmetry extensions. The consideration has been summarized in the Hasse diagram in Fig. 1, which represents the structure of the partially ordered set of $G^\sim$-inequivalent Lie-symmetry extensions within the class (1). Analyzing the Hasse diagram allows one to easily solve the group classification problems up to the general point equivalence for the subclasses ${\mathcal {K}}_N$, $N\in \Gamma$, of ${\mathcal {K}}$, which correspond, under the interpretation in the weak sense, to the classification cases that have been listed in Theorem 13; see Remark 14. Since the subclasses ${\mathcal {K}}_1$, ..., ${\mathcal {K}}_6$, ${\mathcal {K}}_{7,q}$, ${\mathcal {K}}_{8,q}$ and ${\mathcal {K}}_9$ are not normalized, the group classification of any such ${\mathcal {K}}_N$ up to the $G^\sim _N$-equivalence is not so easy. The last claim has been illustrated in Sect. 6 by carrying out the group classification of the subclass ${\mathcal {K}}_2$ up to the $G^\sim _2$-equivalence. Therein, we have also discussed a procedure of using Theorem 13 for the group classification of any subclass of the class ${\mathcal {K}}$ with respect to the equivalence group of this subclass.

The classification of Lie symmetries is the first necessary step for extended symmetry analysis of equations from the class (1). It can be used for the classification of Lie reductions and further finding exact invariant solutions of these equations. Since the general solution of the Liouville equation is well known, Lie reductions should be carried out only for equations from the class (1) with finite-dimensional maximal Lie invariance algebras. In view of Lemma 11(iii), which states upper bound four for the dimensions of such algebras, the classification of subalgebras of three- and four-dimensional Lie algebras in [46] is extremely relevant here. As an example, consider Case 10. It is the only case among those with finite-dimensional maximal Lie invariance algebras, where the algebra ${\mathfrak {g}}_f$ is not solvable. More precisely, it is isomorphic to the algebra $\mathfrak {sl}(2,{\mathbb {R}})$, and its inequivalent one-dimensional subalgebras and the associated Lie reductions to ordinary differential equations are the following:

$$\begin{aligned}&1.\ \langle \partial _t+\partial _x\rangle :\ \ u=\varphi (\omega ),\ \ \omega =x-t,\ \ \varphi _{\omega \omega }=-{\hat{f}}(\varphi )\omega ^{-2};\\&2.\ \langle t\partial _t+x\partial _x\rangle :\ \ u=\varphi (\omega ),\ \ \omega =\tfrac{1}{2}\ln |x|-\tfrac{1}{2}\ln |t|,\ \ \varphi _{\omega \omega }=-{\hat{f}}(\varphi )\sinh ^{-2}\omega ;\\&3.\ \langle (1+t^2)\partial _t+(1+x^2)\partial _x\rangle :\ \ u=\varphi (\omega ),\ \ \omega =\arctan x-\arctan t,\\&\quad \ \ \varphi _{\omega \omega }=-{\hat{f}}(\varphi )\sin ^{-2}\omega . \end{aligned}$$

The knowledge of Lie symmetries of equations from the class (1) is also needed for classification of reduction operators of these equations. In the course of classifying reduction operators, it is natural to exclude those of them that are induced by Lie symmetries, i.e., to look only for non-Lie reduction operators. Unfortunately, the general description of regular reduction operators for equations from the class ${\mathcal {K}}$ as a whole from [57] cannot be used in a reasonable way for describing reduction operators of particular equations from this class or for classifying reduction operators of equations constituting its proper subclasses. Recall that regarding singular reduction operators of equations from the class (1), a systematic study has been done in the literature only for equations with $f={\hat{f}}(u)$, which constitute the subclass ${\mathcal {K}}_9$ associated with Case 9 of Theorem 13, see [27, Section 6].

The consideration of the presented paper can be extended to the much wider superclass of generalized nonlinear Klein–Gordon equations of the form

$$\begin{aligned} u_{tx}=f(t,x,u,u_t,u_x). \end{aligned}$$

(19)

A preliminary study has shown that Lemma 2 may be generalized to this superclass, and the Lie theorem should be relevant for its group classification in the same way as for that of the class (1). The principal precondition for applying the Lie theorem to the group classification of the superclass (19) is to prove an analogue of Lemma 11 for this superclass, which seems a much more difficult problem than proving Lemma 11. In particular, one needs to single out, within the superclass (19), the equations with infinite-dimensional maximal Lie invariance algebras.