# Lie symmetry classification for the 1+1 and 1+2 generalized Zoomeron
equations

Andronikos Paliathanasis1,2 and P.G.L Leach1
1Institute of Systems Science, Durban University of Technology
PO Box 1334, Durban 4000, Republic of South Africa
2Departamento de Matemáticas, Universidad Católica del Norte,
Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile Email:
<EMAIL_ADDRESS>
###### Abstract

We present a complete algebraic classification of the Lie symmetries for
generalized Zoomeron equations. For the generalized 1+1 and 2+1 Zoomeron
equations we solve the Lie symmetry conditions in order to constrain the free
functions of the equations. We find that the differential equations of our
consideration admit the same number of Lie symmetries with the non-generalized
equations. The admitted Lie symmetries form the Lie algebras $2A_{1}$,
$A_{3,3}$ for the 1+1 generalized Zoomeron equation, and the $A_{4,5}^{ab}\,$,
$3A_{1}\otimes_{s}2A_{1}$ in the case of the 2+1 generalized Zoomeron
equation. The one-dimensional optimal system is constructed for the two
equations and similarity solutions are derived. The similarity transformation
lead to the derivation of kink solutions. Indeed, the similarity exact
solutions determined in this work are asymptotic solutions near to the
singular behaviour of the kink behaviour.

Keywords: Lie symmetries; Invariant functions; Zoomeron; nonlinear partial
differential equations

MSC2020: 35A09 ; 35C05 ; 35C07

##  1 Introduction

Nonlinear differential equations play a significant role in applied science
because they are the main approach to the mathematical description of real
world phenomena [1]. The Korteweg-de Vries (KdV) equation is probably the most
fully understood and well studied 1+1 nonlinear partial differential equation
which arises in various physical phenomena. Indeed, the KdV equation provides
exact solutions which are used for the mathematical description of waves in
various media, such as surface gravity waves, internal waves in the ocean,
acoustic waves, optical waves and many others [2, 3, 4].

With the study of KdV the concept of solitons was introduced. Solitons are
solitary wave solutions the shape and speed of which are invariant after the
collision, that is, the solitary waves have particle-like properties. The
first known soliton was found in the Fermi-Pasta-Ulam lattice [5]. There are
many studies in the literature where the KdV is modified so as to describe
physical phenomena. For more details we refer the reader to [6]. Moreover, the
concept of topological solitons has been also introduced [7]. Topological
solitons are exact solutions in classical field theory and they describe
particle-like objects such as the Yang-Mills instatons or the Skyrmions [8, 9,
10].

In the attempt to generalize the KdV equation there has been introduced an 1+1
nonlinear partial differential equation known as the Zoomeron equation [11]

$\left(\frac{u_{xt}}{u}\right)_{tt}-\left(\frac{u_{xt}}{u}\right)_{xx}+2\left(u^{2}\right)_{xt}=0.$
(1)

The dynamics, the analytic solutions and the algebraic properties of equation
(1) have been widely studied in the literature, see for instance [12, 13] and
references therein.

The novelty of equation (1) is that describes solitons which move with a
variable speed and there exist nontrivial polarization effects. These solitons
are categorized in two families; the accelerated solitons which are coming in
the past from one side and boomeranging back to that side with the same speed
in the remote future, the second family of solutions have the characteristic
to describe trapped oscillating in a spatial domain changing direction
multiple times around a fixed point; that is,. a potential well. The latter
solutions are known as boomeron and trappon solutions [11].

The 2+1 Zoomeron equation,

$\left(\frac{u_{xy}}{u}\right)_{tt}-\left(\frac{u_{xy}}{u}\right)_{xx}+2\left(u^{2}\right)_{xt}=0,$
(2)

was introduced in [14] in which the
$\left(\frac{G^{\prime}}{G}\right)-$expansion method was applied to find exact
solutions. Furthermore, travelling wave solutions for equation (2) were found
by using the MSE method and the Exp-function method in [15, 16]. The Lie
symmetries of equation (2) were the subject of study in [17]. It was found
that equation (2) admits a five-dimensional Lie algebra and the Lie symmetries
were applied to determine new similarity solutions.

Equation (2) is of special interest because it describes specific solitons
with important features related to the nonlinear optics and fluid mechanics
[18]. In [19] the physical properties of the solitons provided by equation (2)
investigated in details. The authors determined solutions described by
rational, trigonometric and hyperbolic functions; consequently, kink, singular
solitons, periodic solitons and lump-like solitons can be described by
equation (2). The authors discussed how equation (2) can be used to describe
foam drainage phenomena [20]. Solitary wave solutions derived in [21] with the
application of the tanh-coth approach, while new periodic solutions determined
in [22] by the use of the sine-cosine method. For more

A generalization of equation (2) was introduced in [23]

$\left(\frac{u_{xy}}{u}\right)_{tt}-k^{2}\left(\frac{u_{xy}}{u}\right)_{xx}+2\alpha\left(u^{2n}\right)_{xt}=0.$
(3)

For the latter 2+1 generalized equation it was found that it admits a five-
dimensional Lie algebra which forms the $3A_{1}\otimes_{s}2A_{1}$ Lie algebra
for$~{}n>0$ [24]. With the use of Lie symmetry analysis exact solutions were
found. Specifically, equation (3) under the application of similarity
transformations can be written in the form of the Airy equation, the Lane-
Emden equation or the Bernoulli equation. Consequently, trappon and boomeron
solutions are provided by equation (3).

Lie symmetry analysis is a systematic approach for the study of nonlinear
differential equations [25, 26, 27, 28]. The novelty of the Lie symmetry
approach is that though a systematic approach the existence of invariant
transformations can be determined. The latter can be used to simplify the
given differential equation with the use of similarity transformations. Under
the application of similarity transformations in a given differential equation
the number of independent variables can be reduced or the order of the
differential equation can be reduced. Furthermore, conservation laws can be
constructed which are essential for the study of the properties for the given
differential equation [25]. Lie symmetries have been applied for the study of
nonlinear differential equations in all areas of applied mathematics [29, 30,
31, 32, 33, 34]. In [35], the Lie symmetries were used for the study of the
nonlinear Schrödinger equation and for the construction of exact solutions
which describe solitons. Furthermore, solitons of the 2+1 dimensional Nizhnik-
Novikov-Vesselov equation were determined by using the Lie symmetry analysis
in [36].

The Lie symmetries for the 2+1 generalized Hirota-Satsuma-Ito equation
investigated in [37] where new conservation laws and new soliton solutions
were determined. Applications of the symmetry analysis in higher-dimensional
partial differential equations or in higher-order equations are presented for
instance in [38, 39, 40, 41, 42] and references therein. Specifically, in [41]
the Lie symmetries for multidimensional difference equations investigate, in
the case of a two-dimensional lattice. Moreover, in [42] the authors applied
the Lie symmetries for the study of multi-dimensional parabolic partial
differential equations with diffusion components.

In this work we are interested in the extension of the analysis performed in
[23]. Indeed we perform a complete classification of the Lie symmetries for
the generalized 1+1 $\left(gZE^{1}\right)$ and 2+1 $\left(gZE^{2}\right)$
Zoomeron equations of the form

$gZE^{1}:\left(\frac{u_{xt}}{f\left(u\right)}\right)_{tt}-\left(\frac{u_{xt}}{f\left(u\right)}\right)_{xx}+2\left(g\left(u\right)\right)_{xt}=0,$
(4)
$gZE^{2}:\left(\frac{u_{xy}}{f\left(u\right)}\right)_{tt}-\left(\frac{u_{xy}}{f\left(u\right)}\right)_{xx}+2\left(g\left(u\right)\right)_{xt}=0,$
(5)

where $f\left(u\right)$,$~{}g\left(u\right)\,\ $are nonconstant functions,
that is, $f_{,u}g_{,u}\neq 0$. In particular, inspired by the Ovsiannikov
classification scheme [43] we determine the unknown functions
$f\left(u\right)$,$~{}g\left(u\right)$ such that equations (4) and (5) to
admit Lie symmetries.

Such an analysis will provide us with important information about the relation
of the Lie symmetries with the existence of solitons. Furthermore, from the
Lie symmetries we can construct similarity transformations and determined
exact solutions for the latter equations. These exact solutions are mainly
related with the existence of travel-wave solutions or self-similar solutions.
Hence, the exact solutions provide the asymptotic behaviour of the general
solution near to the moving wall as the soliton “moves” in the space or time
variables. The plan of the paper is as follows.

The basic properties and definitions for the Lie symmetry analysis are
presented in Section 2. In Section 3 we present the Lie symmetry
classification for the generalized 1+1 and 2+1 Zoomeron equations. The
application of the Lie symmetries is presented in Section 4 in which we use
the similarity transformations in order to construct exact solutions. Finally,
in Section 5 we summarize our results and we draw our conclusions.

##  2 Preliminaries

In this Section we present the basic properties and definitions for the theory
of Lie symmetries of differential equations. Assume the function $\Phi$ which
describes the map of a one-parameter point transformation such as
$\Phi\left(\mathbf{u}\left(t,x^{i}\right)\right)=\mathbf{u}\left(t,x^{i}\right)\
$with infinitesimal transformation

$\displaystyle t^{\prime}$ $\displaystyle=$ $\displaystyle
t^{i}+\varepsilon\xi^{t}\left(t,x^{i},u\right),$ (6) $\displaystyle
x^{\prime}$ $\displaystyle=$ $\displaystyle
x^{i}+\varepsilon\xi^{x^{i}}\left(t,x^{i},u\right),$ (7) $\displaystyle
u^{\prime}$ $\displaystyle=$ $\displaystyle
u+\varepsilon\eta\left(t,x^{i},u\right),$ (8)

and generator $\mathbf{X}=\frac{\partial
t^{\prime}}{\partial\varepsilon}\partial_{t}+\frac{\partial
x^{\prime}}{\partial\varepsilon}\partial_{x}+\frac{\partial
u}{\partial\varepsilon}\partial_{u},~{}$where $\varepsilon$ is the parameter
of smallness; $u\left(t,x^{i}\right)\ $is the dependent function and
$\left(t,x^{i}\right)$ are the independent variables with
$x^{i}=\left(x,y\right)$.

Let $u\left(t,x^{i}\right)$ be a solution to the differential equation
$\mathcal{H}\left(t,x,u,u_{,t},u_{,x},...\right)=0.$ Then, under the one-
parameter map $\Phi$, function
$u^{\prime}\left(t^{\prime},x^{i\prime}\right)=\Phi\left(u\left(t,x^{i}\right)\right)$
is a solution for the differential equation, $\mathcal{H}$, if and only if the
differential equation is also invariant under the action of the map, $\Phi$,
that is, the following condition holds [25, 26, 27, 28]

$\Phi\left(\mathcal{H}\left(t,x,u,u_{t},u_{x}...\right)\right)=0.$ (9)

For every map $\Phi$ in which condition (9) holds, it means that the generator
$X$ is a Lie point symmetry for the differential equation, while the following
condition is true

$\mathbf{X}^{\left[n\right]}\left(\mathcal{H}\left(t,x,u,u_{t},u_{x}...\right)\right)=0.$
(10)

Vector field $\mathbf{X}^{\left[n\right]}$ describes the first extension of
the symmetry vector in the jet-space of variables,
$\left\\{t,x,u,u_{t},u_{x},....,u_{i_{1},...,i_{n}}\right\\}$.

The importance of the existence of a Lie symmetry for a given differential
equation is that from the associated Lagrange’s system,

$\frac{dt}{\xi^{t}}=\frac{dx^{i}}{\xi^{x^{i}}}=\frac{du}{\eta},$ (11)

the solution of this system provides the invariant functions which can be used
to reduce the number of the independent variables of the differential equation
and lead to the construction of similarity solutions In the case of partial
differential equations, the application of the Lie invariants reduces the
number of the independent variables.

The admitted symmetry vectors of a given set of differential equations
constitute a closed-group known as a Lie group. The main application of the
Lie symmetries is the determination of solutions known as similarity solutions
and follows from the application of the Lie invariants in the differential
equations. However, in order to classify all the possible similarity
transformations and solutions the one-dimensional optimal system should be
calculated [28].

Assume the $n$-dimensional Lie algebra $G_{n}$ with elements
$\left\\{X_{1},~{}X_{2},~{}...~{}X_{n}\right\\}~{}$and structure constants
$C_{jk}^{i}$. We define the two symmetry vectors

$Z=\sum_{i=1}^{n}a_{i}X_{i}~{},~{}W=\sum_{i=1}^{n}b_{i}X_{i}~{},~{}\text{\
}a_{i},~{}b_{i}\text{ are constants.}$ (12)

and we define the Adjoint operator

$Ad\left(\exp\left(\epsilon
X_{i}\right)\right)X_{j}=X_{j}-\epsilon\left[X_{i},X_{j}\right]+\frac{1}{2}\epsilon^{2}\left[X_{i},\left[X_{i},X_{j}\right]\right]+...~{},$
(13)

known as the adjoint representation, in which
$\left[X_{i},X_{j}\right]=C_{ij}^{k}X_{k}$ is the Lie Bracket.

We say that the vectors $Z$ and $W$ are equivalent if and only if [28]

$\mathbf{W}=\prod_{i=1}^{n}Ad\left(\exp\left(\epsilon_{i}X_{i}\right)\right)\mathbf{Z,}$
(14)

or

$W=cZ~{},~{}c=const\text{ that is }b_{i}=ca_{i}\text{.}$ (15)

As far as the constants $a_{i}$ are concerned, their independent constants are
invariant functions of the Adjoint operator. Indeed if
$\phi\left(a_{i}\right)$ is a function of $a_{i}$, the invariants are given by
the linear system of partial differential equations

$\Delta_{i}\left(\phi\right)=C_{ij}^{k}a^{j}\frac{\partial\phi}{\partial
a^{i}}\equiv 0.$ (16)

The one-dimensional subalgebras of $G_{n}$ which are not related through the
adjoint representation form the one-dimensional optimal system. The
determination of the one-dimensional system is essential in order to perform a
complete classification of all the possible similarity transformations and
solutions.

##  3 Lie symmetries for the Zoomeron equations

We apply the Lie symmetry condition (10) for the generalized 1+1
$\left(gZE^{1}\right)$ and 2+1 $\left(gZE^{2}\right)$ Zoomeron equations and
we determine all the functional forms of $f\left(u\right)$ and
$g\left(u\right)$ where the partial differential equations admit Lie point
symmetries. For simplicity on our presentation we omit the calculations. The
results are summarized in the following propositions.

For the $gZE^{1}$ equation it follows.

###### Proposition 1

The Lie point symmetry classification for the generalized 1+1 Zoomeron
equation (4) for arbitrary functions $f\left(u\right)$ and $g\left(u\right)$
are two, they are $X_{1}=\partial_{t}$ and $X_{2}=\partial_{x}$ with the
commutator, $\left[X_{1},X_{2}\right]=0$, that is, they form the $2A_{1}$ Lie
algebra. However for specific functional forms of $f\left(u\right)$ and
$g\left(u\right)$ additional Lie symmetries exist.

(A) For $f\left(u\right)=f_{0}u^{K}$ and
$g\left(u\right)=g_{0}+g_{1}u^{-\frac{2}{a_{1}}-\left(K-1\right)}$, where
$\alpha_{1}$ is a nonzero constant; in this case $gZE^{1}$ equation admits the
additional Lie symmetry
$X_{3}=t\partial_{t}+x\partial_{x}+\alpha_{1}u\partial_{u}$. The nonzero
commutators for the Lie algebra are $G_{3}=\left\\{X_{1},X_{2},X_{3}\right\\}$
are $\left[X_{1},X_{3}\right]=X_{1}$ and $\left[X_{2},X_{3}\right]=X_{2}$.
Thus, $G_{3}$ is the $A_{3,3}$ Lie algebra.

(B) For $f\left(u\right)=f_{0}u^{K}$ and $g\left(u\right)=g_{0}+g_{1}\ln u$,
$gZE^{1}$ equation admits the additional Lie
symmetry$~{}X_{4}=t\partial_{t}+x\partial_{x}+\frac{1}{1-K}u\partial_{u}$,
which, as above, the admitted Lie symmetry vectors
$\left\\{X_{1},X_{2},X_{4}\right\\}$ form the $A_{3,3}$ Lie algebra.

(C) For $f\left(u\right)=f_{0}\exp\left(Ku\right)$ and
$g\left(u\right)=g_{0}+g_{1}\exp\left(-\left(K+\frac{2}{\alpha_{2}}\right)u\right)$,
with $\alpha_{2}$ a nonzero constant; the additional Lie symmetry vector for
equation $gZE^{1}$ is the field
$X_{5}=t\partial_{t}+x\partial_{x}+a_{2}\partial_{u}$. The Lie symmetries
$\left\\{X_{1},X_{2},X_{5}\right\\}$ form the $A_{3,3}$ Lie algebra.

(D) For $f\left(u\right)=f_{0}\exp\left(Ku\right)$ and
$g\left(u\right)=g_{0}+g_{1}u$ the additional Lie symmetry vector is the
vector field $X_{6}=t\partial_{t}+x\partial_{x}-\frac{1}{K}\partial_{u}$. The
admitted Lie symmetries form the $A_{3,3}$ Lie algebra.

Similarly for the $gZE^{2}$ equation the Lie symmetry classification is as
follows.

###### Proposition 2

The generalized 2+1 Zoomeron equation (5) for arbitrary functional forms of
$f\left(u\right)$ and $g\left(u\right)$ is invariant under the four-
dimensional Lie algebra $G_{4}=\left\\{Y_{1},Y_{2},Y_{3},Y_{4}\right\\}$ which
consists of the vector fields $Y_{1}=\partial_{t}$, $Y_{2}=\partial_{x}$,
$Y_{3}=\partial_{y}$ and $Y_{4}=t\partial_{t}+x\partial_{x}-y\partial_{y}$.
The nonzero commutators for the elements of the $G_{4}$ are
$\left[Y_{1},Y_{4}\right]=Y_{1}$, $\left[Y_{2},Y_{4}\right]=Y_{2}$ and
$\left[Y_{3},Y_{4}\right]=-Y_{3}$. Hence, it follows that $G_{4}$ forms the
$A_{4,5}^{ab}$ Lie algebra. For specific forms of $f\left(u\right)$ and
$g\left(u\right)$ additional symmetries exist.
(I) When $f\left(u\right)=f_{0}u^{K}$ and
$g\left(u\right)=g_{0}+g_{1}u^{-\frac{1}{\alpha_{1}}-\left(K-1\right)}$, where
$\alpha_{1}$ is a nonzero constant; in this case $gZE^{2}$ equation admits the
additional Lie symmetry vector $Y_{5}=y\partial_{y}-\alpha_{1}u\partial_{u}$.
The additional nonzero commutator is the $\left[Y_{3},Y_{5}\right]=Y_{3}$.
Thus, the Lie algebra $G_{5}=\left\\{Y_{1},Y_{2},Y_{3},Y_{4},Y_{5}\right\\}$
is the $3A_{1}\otimes_{s}2A_{1}$.

(II) For $f\left(u\right)=f_{0}u^{K}$ and
$g\left(u\right)=g_{0}+g_{1}\ln\left(u\right)$, the generalized $gZE^{2}$
equation admits the additional Lie symmetry vector
$Y_{6}=y\partial_{y}+\frac{1}{1-K}u\partial_{u}$. The admitted Lie symmetries
form the $3A_{1}\otimes_{s}2A_{1}$ Lie algebra.

(III) Moreover, when $f\left(u\right)=f_{0}\exp\left(Ku\right)$ and
$g\left(u\right)=g_{0}+g_{1}\exp\left(-\left(K+\frac{2}{\alpha_{2}}\right)u\right)$
the additional Lie symmetry vector for equation $gZE^{2}$ is the vector field
$Y_{7}=y\partial_{y}+a_{2}\partial_{u}$, where again the admitted Lie algebra
is the $3A_{1}\otimes_{s}2A_{1}$.

(IV) Finally, for $f\left(u\right)=f_{0}\exp\left(Ku\right)$ and
$g\left(u\right)=g_{0}+g_{1}u$, equation $gZE^{2}$ is invariant under the
action of the one-parameter point transformation with generator the vector
field $Y_{8}=y\partial_{y}-\frac{1}{K}\partial_{u}$. Similarly, the admitted
Lie symmetries form the $3A_{1}\otimes_{s}2A_{1}$ Lie algebra.

As far as the admitted Lie algebras by the generalized Zoomeron equations the
following corollary holds.

###### Corollary 3

The generalized 1+1 Zoomeron equation (4) is invariant under the action of a
two-dimensional or three-dimensional Lie algebra, the $2A_{1}$ or the
$A_{3,3}$. Furthermore, the generalized 2+1 Zoomeron equation (5) admits four
or five Lie symmetries which form the $A_{4,5}^{ab}$ or the
$3A_{1}\otimes_{s}2A_{1}$ Lie algebras.

At this point it is important to mention that not all the comments $f_{0}$,
$g_{0}$ and $g_{1}$ are essential parameters. Without loss of generality in
the following we assume $g_{0}=0$ and $f_{0}=1$. We proceed with the
derivation of the one-dimensional optimal system for each of the admitted Lie
algebras.

###  3.1 One-dimensional optimal system

The main application of Lie symmetries is the definition of similarity
transformations which are used to simplify the given differential equation.
The derivation of the one-dimensional optimal system is essential for the
complete classification of all the independent similarity transformations,
consequently, of the independent similarity solutions.

In Tables 1, 2 and in Tables 3, 4 we present the commutators and the Adjoint
representation for the admitted Lie algebras of equations $gZE^{1}$ and
$gZE^{2}$ respectively.

Table 1: Commutators of the admitted Lie point symmetries for the generalized 1+1 Zoomeron equation, $A=3,4,5,6$ $\left[~{},~{}\right]$ | $\mathbf{X}_{1}$ | $\mathbf{X}_{2}$ | $\mathbf{X}_{A}$
---|---|---|---
$\mathbf{X}_{1}$ | $0$ | $0$ | $X_{1}$
$\mathbf{X}_{2}$ | $0$ | $0$ | $X_{2}$
$\mathbf{X}_{A}$ | $-X_{1}$ | $-X_{2}$ | $0$
Table 2: Adjoint representation for the admitted Lie point symmetries of the generalized 1+1 Zoomeron equation, $A=3,4,5,6$ $Ad\left(\exp\left(\varepsilon\mathbf{X}_{\mu}\right)\right)\mathbf{X}_{\nu}$ | $\mathbf{X}_{1}$ | $\mathbf{X}_{2}$ | $\mathbf{X}_{3}$
---|---|---|---
$\mathbf{X}_{1}$ | $X_{1}$ | $X_{2}$ | $X_{3}-\varepsilon X_{1}$
$\mathbf{X}_{2}$ | $X_{1}$ | $X_{2}$ | $X_{3}-\varepsilon X_{2}$
$\mathbf{X}_{3}$ | $e^{\varepsilon}X_{1}$ | $e^{\varepsilon}X_{2}$ | $X_{3}$

We assume now that the generic symmetry vector is

$X=a_{1}X_{1}+a_{2}X_{2}+a_{A}X_{A},$ (17)

for equation $gZE^{1}$, or

$Y=b_{1}Y_{1}+b_{2}Y_{2}+b_{3}Y_{3}+b_{4}Y_{4}+b_{A}Y_{A},$ (18)

for the generalized $gZE^{2}$ equation, where
$a_{1},~{}a_{2},~{}a_{3},~{}a_{A}$ and $b_{1},~{}b_{2}$, $b_{3},~{}b_{4}$ and
$b_{A}$ are constants.

For the admitted two-dimensional Lie algebra $2A_{1}\,\ $of equation
$gZE^{1},$ with $a_{A}=0$, we determine the invariants of the Adjoint
representation $a_{1}$ and $a_{2}$. Thus, the one-dimensional optimal system
consists of the one-dimensional Lie algebras

$OpS\left(2A_{1}\right):\left\\{X_{1}\right\\}~{},~{}\left\\{X_{2}\right\\}~{},~{}\left\\{X_{1}+\alpha
X_{2}\right\\}.$ (19)

In addition, for the $A_{3,3}$ Lie algebra we determine that $a_{A}$ is the
invariant for the Adjoint representation. Hence, the one-dimensional optimal
system consists of the vector fields

$OpS\left(A_{3,3}\right):OpS\left(2A_{1}\right)~{},~{}\left\\{X_{A}\right\\}.$
(20)

Similarly, for equation $gZE^{2}$ and the four-dimensional Lie algebra
$A_{4,5}^{ab}$ we derive the Lie invariants of the Adjoint representation to
be $a_{4}$. The one-dimensional optimal system is

$\displaystyle OpS\left(A_{4,5}^{ab}\right)$ $\displaystyle:$
$\displaystyle\left\\{Y_{1}\right\\}~{},~{}\left\\{Y_{2}\right\\}~{},~{}\left\\{Y_{3}\right\\}~{},~{}\left\\{Y_{4}\right\\}~{},~{}\left\\{Y_{1}+\alpha
Y_{2}\right\\}$ (21) $\displaystyle\left\\{Y_{1}+\alpha
Y_{3}\right\\}~{},~{}\left\\{Y_{2}+\alpha
Y_{3}\right\\}~{},~{}\left\\{Y_{1}+\alpha Y_{2}+\beta Y_{3}\right\\}.$

Finally, for the five-dimensional Lie algebra $3A_{1}\otimes_{s}2A_{1}$ the
invariants of the Adjoint operator are $a_{4}$ and $a_{5}$. Hence, with the
use of Table 4 we derive the one-dimensional system

$\displaystyle OpS\left(3A_{1}\otimes_{s}2A_{1}\right)$ $\displaystyle:$
$\displaystyle
OpS\left(A_{4,5}^{ab}\right)~{},~{}\left\\{Y_{A}\right\\}~{},~{}\left\\{Y_{4}+\alpha
Y_{A}\right\\}~{},~{}~{}$ (22) $\displaystyle\left\\{Y_{1}+\alpha
Y_{A}\right\\}~{},~{}\left\\{Y_{2}+\alpha
Y_{A}\right\\}~{},~{}\left\\{Y_{1}+\alpha Y_{2}+\beta Y_{A}\right\\}.$

Table 3: Commutators of the admitted Lie point symmetries for the generalized 2+1 Zoomeron equation, $A=5,6,7,8$ $\left[~{},~{}\right]$ |  $Y$1 |  $Y$2 |  $Y$3 |  $Y$4 |  $Y$A
---|---|---|---|---|---
$Y$1 | $0$ | $0$ | $0$ | $Y_{1}$ | $0$
$Y$2 | $0$ | $0$ | $0$ | $Y_{2}$ | $0$
$Y$3 | $0$ | $0$ | $0$ | $Y_{3}$ | $-Y_{3}$
$Y$4 | $-Y_{1}$ | $-Y_{2}$ | $-Y_{3}$ | $0$ | $0$
$Y$A | $0$ | $0$ | $Y_{3}$ | $0$ | $0$
Table 4: Adjoint representation for the admitted Lie point symmetries of the generalized 2+1 Zoomeron equation, $A=5,6,7,8$ $Ad\left(\exp\left(\varepsilon\mathbf{Y}_{\mu}\right)\right)\mathbf{Y}_{\nu}$ |  $Y$1 |  $Y$2 |  $Y$3 |  $Y$4 |  $Y$A
---|---|---|---|---|---
$Y$1 | $Y_{1}$ | $Y_{2}$ | $Y_{3}$ | $Y_{4}-\varepsilon Y_{1}$ | $Y_{5}$
$Y$2 | $Y_{1}$ | $Y_{2}$ | $Y_{3}$ | $Y_{4}-\varepsilon Y_{2}$ | $Y_{5}$
$Y$3 | $e^{\varepsilon}Y_{1}$ | $e^{\varepsilon}Y_{2}$ | $Y_{3}$ | $Y_{4}-\varepsilon Y_{3}$ | $Y_{5}+\varepsilon Y_{3}$
$Y$4 | $e^{\varepsilon}Y_{1}$ | $e^{\varepsilon}Y_{2}$ | $e^{\varepsilon}Y_{3}$ | $Y_{4}$ | $Y_{5}$
$Y_{A}$ | $Y_{1}$ | $Y_{2}$ | $e^{\varepsilon}Y_{3}$ | $Y_{4}$ | $Y_{5}$

##  4 Application of Lie invariants

We proceed with the application of the Lie symmetries to the derivation of
similarity solutions.

###  4.1 Equation $gZE^{1}$

For each element of the one-dimensional optimal system it follows.

####  4.1.1 $\left\\{X_{1}\right\\},~{}\left\\{X_{2}\right\\}$

The vector fields $\left\\{X_{1}\right\\}~{},~{}\left\\{X_{2}\right\\}$
provide static and stationary solutions of the form $u=u\left(x\right)$ and
$u=u\left(t\right)$ respectively. These are not solutions of special interest.

####  4.1.2  $\left\\{X_{1}+\alpha X_{2}\right\\}$

The similarity transformation provided by $\left\\{X_{1}+\alpha
X_{2}\right\\}$ is $\xi=x-\alpha t$, $\ u=u\left(\xi\right)$. Thus, by
replacing in (4) we find the reduced equation

$\left(a^{2}-1\right)\left(\frac{u_{\xi\xi}}{f\left(u\right)}\right)_{\xi\xi}+2\left(g\left(u\right)\right)_{\xi\xi}=0.$
(23)

For $\alpha^{2}=1$ it follows $g\left(u\right)=u_{1}\xi+u_{0}$. However, for
$\alpha^{2}\neq 1$ we find

$u_{\xi\xi}+\left(2\left(g\left(u\right)\right)+u_{1}\xi+u_{0}\right)f\left(u\right)=0.$
(24)

This is a second-order ordinary differential equation of the form
$u_{\xi\xi}+G\left(u\right)+f\left(u\right)\left(u_{1}\xi+u_{0}\right)=0$.
Such equations have been widely studied in the literature before.

When $u_{1}=0$, the analytic solution is expressed in terms of quadratures,
that is,

$\frac{1}{2}\left(u_{\xi}\right)^{2}-\int\left(G\left(u\right)+f\left(u\right)u_{0}\right)du=h~{},~{}h=const\text{.}$
(25)

In the case where $g\left(u\right)=u^{2}$, $f\left(u\right)=u$ and $u_{1}=0$,
$u_{0}=0$ we find the oscillator
$\frac{1}{2}\left(u_{\xi}\right)^{2}-\frac{1}{2}u^{2}=h$.

####  4.1.3 $\left\\{X_{3}\right\\}$

The Lie symmetry vector $X_{3}$ provides the similarity transformation
$u=U\left(\sigma\right)t^{a_{1}}$, with $\sigma=\frac{x}{t}$.

In the new variables we derive the closed-form solution of the latter system
is $U\left(\sigma\right)=U_{0}\sigma^{a_{1}}$, from which we conclude that

$u\left(t,x\right)=u_{0}x^{a_{1}}t^{2a_{1}}.$ (26)

where $u_{0}$ is an integration constant.

####  4.1.4 $\left\\{X_{4}\right\\}$

For case (B) of Proposition 1, from the vector field $X_{4}$ we find the
similarity transformation $u=U\left(\sigma\right)t^{\frac{1}{1-K}}$,
$\sigma=\frac{x}{t}$. We find that in the new variables the closed-form
solution is

$u\left(t,x\right)=u_{0}x^{\frac{1}{1-K}}t^{\frac{2}{1-K}}\text{.}$ (27)

We remark that this solution is of the form of that found from the application
of the Lie symmetry vector $X_{3}$, where $a_{1}=\frac{1}{1-K}$ and $u_{0}$ is
an integration constant.

####  4.1.5 $\left\\{X_{5}\right\\}$

For case (C) of Proposition 1 and the vector field $X_{5}$ we calculate the
similarity transformation $u=a_{2}\ln
t+U\left(\sigma\right),~{}\sigma=\frac{x}{t}$. Hence, by substituting into
equation (4) we find the trivial similarity solution

$u\left(t,x\right)=a_{2}\ln\left(U_{0}x\right),$

which is a static solution; and $U_{0}$ is an integration constant.

####  4.1.6 $\left\\{X_{6}\right\\}$

Finally, for case (D) of Proposition 1 and the vector field $X_{6}$ we derive
the similarity transformation $u=-\frac{1}{K}\ln
t+U\left(\sigma\right),~{}\sigma=\frac{x}{t}\,\ $and the closed-form solution

$u\left(t,x\right)=-\frac{1}{K}\ln\left(U_{0}x\right).$

The latter is a static solution, similar to that determined by the vector
field $X_{5}$, when $a_{2}=-\frac{1}{K}$ and $U_{0}$ is an integration
constant.

###  4.2 Equation $gZE^{2}$

We continue our analysis with the study of the similarity transformations for
the 2+1 partial differential equation $gZE^{2}$. At this point it is important
to mention that in order to reduce equation $gZE^{2}$ to an ordinary
differential equation we should make use of two elements of the one-
dimensional optimal system.

However, when we apply the similarity transformation of a given symmetry to
the differential equation, the reduced equation does not admit always the
remainder of the symmetries. Hence, if $Y_{i},~{}Y_{k}$ are two symmetry
vectors of $gZE^{2}$, with commutator $\left[Y_{i},Y_{k}\right]=cY_{k}$ where
$c$ may be zero, application of the invariants of $Y_{i}$ gives a reduced
equation for which $Y_{k}$ is not a point symmetry. Specifically $Y_{k}$
becomes a nonlocal symmetry [44]. On the other hand, reduction with respect to
the vector field $Y_{k}$ on the master equation provides a reduced equation
which possesses the symmetry vector $Y_{i}$ [45] .

Hence, in order to perform a double reduction we should determine all the two-
dimensional subalgebras which consist of the elements of the one-dimensional
optimal system of equation $gZE^{2}$. With the use of Table 3 we find the
following two-dimensional subalgebras

$\mathbf{A}^{1}=\left\\{\alpha_{1}Y_{1}+\alpha_{2}Y_{2}+\alpha_{3},\beta_{1}Y_{1}+\beta_{2}Y_{2}+\beta_{3}Y_{3}\right\\}~{},~{}\mathbf{A}^{2}=\left\\{\alpha_{1}Y_{1}+\alpha_{2}Y_{2},Y_{4}\right\\}~{},\mathbf{~{}A}^{3}=\left\\{Y_{3},Y_{4}\right\\}~{},$
$\mathbf{A}^{4}=\left\\{Y_{4},Y_{A}\right\\}~{},~{}\mathbf{A}^{5}=\left\\{\alpha_{1}Y_{1}+\alpha_{2}Y_{2}+\alpha_{A}Y_{A},\beta_{1}Y_{1}+\beta_{2}Y_{2}+\beta_{A}Y_{A}\right\\}~{}.$

####  4.2.1 Reduction with $\mathbf{A}^{1}$

Application of the similarity transformation provided by the two-dimensional
Lie algebra$\mathbf{~{}A}^{1}$ in equation (5) gives the travelling wave
similarity transformation

$u=u\left(\xi\right),~{}\xi=\frac{y\left(a_{1}b_{2}-b_{1}a_{2}\right)+x\left(b_{1}a_{3}-a_{1}b_{3}\right)+t\left(a_{2}b_{3}-a_{3}b_{2}\right)}{a_{1}b_{2}-a_{2}b_{1}}.$
(28)

Thus, the reduced equation is determined to be

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(\left(a_{2}b_{3}-a_{3}b_{2}\right)^{2}-\left(a_{3}b_{1}-a_{1}b_{3}\right)^{2}\right)\left(a_{1}b_{3}-b_{1}a_{3}\right)\left(\frac{u_{\xi\xi}}{f\left(u\right)}\right)_{\xi\xi}$
(29)
$\displaystyle+2\left(a_{2}b_{3}-a_{3}b_{2}\right)\left(a_{1}b_{3}-b_{1}a_{3}\right)\left(g\left(u\right)\right)_{\xi\xi}.$

Consequently, the reduced equation is similar to that for the generalized 1+1
Zoomeron equation found above, i.e. equation (24).

####  4.2.2 Reduction with $\mathbf{A}^{2}$

From the Lie algebra $\mathbf{A}^{2}$ we calculate the similarity
transformation

$u=u\left(\zeta\right)~{},~{}\zeta=\left(a_{2}t-a_{1}x\right)y.$ (30)

In the new variables the reduced equation is

$\left(a_{2}-a_{1}\right)\left(\frac{\left(\zeta
u_{\zeta}\right)_{\zeta}}{f\left(u\right)}\right)_{\zeta\zeta}+2\left(g\left(u\right)\right)_{\zeta\zeta}=0.$
(31)

Therefore for $\left(a_{2}-a_{1}\right)\neq 0$ it follows that

$\left(\zeta
u_{\zeta}\right)_{\zeta}+2\left(g\left(u\right)\right)f\left(u\right)+\left(u_{1}\zeta+u_{0}\right)f\left(u\right)=0.$
(32)

We assume now that $f\left(u\right)$ and $g\left(u\right)$ have the functional
forms of that of some specific cases of Proposition 2. For Case (I), we
determine the closed-form solution

$u\left(\zeta\right)=u_{0}\zeta^{a_{1}}.$ (33)

Similarly for Case (III) we derive the closed-form solution

$u\left(\zeta\right)=u_{0}\ln\zeta,u_{0}=-\frac{a_{2}}{a_{2}K+2}.$ (34)

####  4.2.3 Reduction with $\mathbf{A}^{3}$

From the Lie algebra $\mathbf{A}^{3}$ we find that
$u=u\left(\frac{x}{t}\right)$. Thus the analytic solution is

$g\left(u\right)=u_{1}\frac{x}{t}+u_{0}.$

####  4.2.4 Reduction with $\mathbf{A}^{4}$

We continue with the application of the elements of the two-dimensional Lie
algebra $\mathbf{A}^{4}$. For simplicity of our presentation we give the
similarity solution and the corresponding exact solution that we can derive.

##### Case (I)

The similarity transformation is
$u=U\left(\sigma\right)\left(ty\right)^{a_{1}},~{}\sigma=\frac{x}{t}$. A
closed-form exact solution is derived to be

$U\left(\sigma\right)=U_{0}\sigma^{a_{1}}~{}.$

##### Case (II)

The similarity transformation is
$u=U\left(\sigma\right)\left(ty\right)^{\frac{1}{1-K}},~{}\sigma=\frac{x}{t}$.
A closed-form exact solution is derived to be

$~{}U\left(\sigma\right)=U_{0}\sigma^{\frac{1}{1-K}}~{}.$

##### Case (III)

The similarity transformation is
$u=U\left(\sigma\right)+a_{2}\ln\left(ty\right),~{}\sigma=\frac{x}{t}$. A
closed-form exact solution is derived to be

$U\left(\sigma\right)=a_{2}\ln\sigma.$

##### Case (IV)

The similarity transformation is
$u=U\left(\sigma\right)+\frac{1}{1-K}\ln\left(ty\right),~{}\sigma=\frac{x}{t}$.
A closed-form exact solution is derived to be

$U\left(\sigma\right)=\frac{1}{1-K}\ln\sigma.$

We omit the presentation with reduction with respect to the two-dimensional
Lie algebra $\mathbf{A}^{5}$ because we were not able to find any closed-form
solution.

##  5 Conclusion

In this study we apply the theory of Lie symmetries in order to study the 1+1
and 2+1 generalized Zoomeron equations with two arbitrary functions. We found
that for nonconstant functions, the generalized 1+1 Zoomeron equation admits
two or three Lie point symmetries which form the $2A_{1}$ or the $A_{3,3}$ Lie
algebras respectively. Moreover, the generalized 2+1 Zoomeron equation admits
four or five Lie point symmetries for specific forms of the unknown functions.
The Lie symmetries form the $A_{4.5}^{ab}$ or $3A_{1}\otimes_{s}2A_{1}$ Lie
algebras respectively. It is important to mention that the maximum dimensional
Lie algebra corresponds, in both cases, to the non-generalized Zoomeron
equations.

For each admitted Lie algebra we determined the one-dimensional optimal
system. The elements of the latter system were used to define similarity
transformations in order to simplify the studied equations. Finally, new exact
solutions were found. The latter solutions are singular and can be seen as the
asymptotic solutions which describe the kink behaviour of the solitons.

In a future work we plan to investigate the existence of analytic solutions
for the reduced systems as also their integrability properties. Such an
analysis overpasses the purposes of this work.

Conflicts of interest/Competing interests: The authors declare no conflict of
interest.

Availability of data and material: Not applicable.

## References

  * [1] M. Braun, Differential Equations and Their Applications, Texts in Applied Mathematics, Springer, New York (1993)
  * [2] D.G. Crighton, Acta Applicandae Mathematica 39, 39 (1995)
  * [3] H. Leblond, Phys. Rev. A 78, 013807 (2008)
  * [4] D.A. Fogaça, F.S. Navarra and L.G. Ferreira Filho, Phys. Rev. D 84, 054011 (2011)
  * [5] L. Casetti, M. Cerruti-Sola, M. Pettini and E.G.D. Cohen, Phys. Rev. E 55, 6566 (1997)
  * [6] P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge University Press, Cambrdige (1992)
  * [7] N. Manton and P. Sutcliffe, Topological Solitons, Cambridge University Press, Cambrdige (2010)
  * [8] M. Gunaydin and H. Nicolai, Phys. Lett. B 351, 169 (1996)
  * [9] A. Giacomini, M. Lagos, J. Oliva and A. Vera, Phys. Lett. B 783, 193 (2018)
  * [10] F. Canfora, N. Dimakis and A. Paliathanasis, EPJC 79, 139 (2019)
  * [11] F. Calogero and A. Degasperis, Il Nuovo Cimento B 39, 1 (1997)
  * [12] W. Gao, H. Rezazadeh, Z. Pinar, H.M. Baskonus, S. Sarwar and G. Yel, Optical and Quantum Electronics 52, 52 (2020)
  * [13] A. Degasperis, C. Rogers and W.K. Schief, Stud. Appl. Math. 109, 39 (2002)
  * [14] R. Abazari, Appl. Math. Sci. 5, 2943 (2011)
  * [15] K. Khan and M.A. Akbar, Ain Shams Engineering Journal 5, 247 (2014)
  * [16] M. Alquran and K. Al-Khaled, Math. Sci. 6, 11 (2012)
  * [17] T. Motsepa, C.M. Khalique and M.L. Gandarias, Symmetry 9, 27 (2017)
  * [18] A. Degasperis, J. Phys. A.: Math. Theor. 43, 434001 (2010)
  * [19] F. Batool, G Akram, M. Sadaf and U. Mehmood, J. Nonl. Math. Phys. (2022) https://doi.org/10.1007/s44198-022-00097-y
  * [20] G. Verbist, D. Weaire and A.M. Kraynik, J. Phys. Condens. Matter 8, 3715 (1996)
  * [21] A. Irshad and S. T. Mohyud-Din, Walailak Journal of Science and Technology 10, 201 (2013)
  * [22] A. Qawasmeh, J. Math. Compt. Sci. 3, 1475 (2013)
  * [23] R.M. Morris and P.G.L. Leach, Phys. Scr. 90, 015202 (2015)
  * [24] J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, J. Math. Phys. 17, 986 (1976)
  * [25] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC, Florida (2000)
  * [26] G.W. Bluman and S. Kumei, Symmetries of Differential Equations, Springer-Verlag, New York (1989)
  * [27] H. Stephani, Differential Equations: Their Solutions Using Symmetry, Cambridge University Press, New York (1989)
  * [28] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1993)
  * [29] A.A. Chesnokov, Eur. J. Appl. Math. 20, 461 (2009)
  * [30] A. Paliathanasis, Symmetry 11, 1115 (2019)
  * [31] S. Jamal, A.H. Kara and A.H. Bokhari, Candian Journal of Physics 90, 667 (2012)
  * [32] U. Camci, S. Jamal and A.H. Kara, Int. J. Theor. Phys. 53, 1483 (2014)
  * [33] S. Jamal and A.G. Johnpillai, Math. Mod. Anal. 25, 198 (2020)
  * [34] D. Huang and N.M. Ivanova, J. Math. Phys., 48, 073507 (2007)
  * [35] J. Belmonte-Beitia, V.M. Pérez-García, V. Vekslerchik, and P.J. Torres, Phys. Rev. Lett. 98, 064102 (2007)
  * [36] S. Kumar, M. Niwas A.-M. Wazwaz, Phys. Scr. 95, 095204 (2020)
  * [37] S. Kumar, K.S. Nisar and A. Kumar, Results in Phys. 28, 104621 (2021)
  * [38] S. Kumar, A. Kumar and A.M. Wazamz, Eur. Phys. J. Plus 135, 870 (2020)
  * [39] S. Kumar, D. Kumar, A. Kumar, Chaos Solitons and Fractals 142, 110507 (2021)
  * [40] A. Paliathanasis, Quaestiones Mathematicae 45, 1099 (2021)
  * [41] D. Levi, S. Tremblay and P. Winternitz, J. Phys. A: Math. Gen. 44, 9507 (2001)
  * [42] M. Craddock and K. Lennox, J. Diff. Equations 252, 56 (2012)
  * [43] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, (1982)
  * [44] P.G.L. Leach, K.S. Govinder and K. Andriopoulos, Journal of Appl Maths 2012, 1 (2012)
  * [45] K.S. Govinder, J. Math. Anal. Appl. 258, 720 (2001)