# Lie symmetry classification and qualitative analysis for the fourth-order
Schrödinger equation

A. Paliathanasis1,2, G. Leon3,1 and P.G.L. Leach1,4 Email:
<EMAIL_ADDRESS>$~{}^{1}$Institute of Systems Science, Durban University
of Technology
PO Box 1334, Durban 4000, Republic of South Africa
2Instituto de Ciencias Físicas y Matemáticas,
Universidad Austral de Chile, Valdivia 5090000, Chile
3Departamento de Matemáticas, Universidad Católica del Norte,
Avda. Angamos 0610, Casilla 1280 Antofagasta, Chile
4School of Mathematical Sciences, University of KwaZulu-Natal,
Durban, Republic of South Africa

###### Abstract

The Lie symmetry analysis for the study of a $1+n~{}$fourth-order Schrödinger
equation inspired by the modification of the deformation algebra in the
presence of a minimum length is applied. Specifically, we perform a detailed
classification for the scalar field potential function where non-trivial Lie
symmetries exist and simplify the Schrödinger equation. Then, a qualitative
analysis allows for the reduced ordinary differential equation to be analyzed
to understand the asymptotic dynamics.

Keywords: Lie symmetries; invariants; fourth-order Schrödinger equation

##  1 Introduction

The Lie symmetry analysis is a systematic approach to the study of nonlinear
differential equations [1, 2]. The existence of a symmetry vector for a given
differential equation indicates the existence of invariant functions, then
used to simplify the differential equation and when it is possible to
determine exact or analytic solutions [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14]. Moreover, symmetries can be used for the determination of conservation
laws, as also to identify equivalent dynamical systems [15, 16, 17, 18].
Finally, the Lie symmetry analysis covers a wide range of applications in all
areas of applied mathematics. In this work, we are interested in the symmetry
classification of a higher-order differential equation.

Consider the fourth-order partial differential equations known as Schrödinger
equation

$i\frac{\partial\Psi}{\partial
t}+\alpha\Delta\Psi+\gamma\Delta^{2}\Psi+V\left(\Psi\right)=0,$ (1)

with $\gamma\neq 0$, $\Delta$ the Laplace operator
$\Delta=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{\left|g\right|}g^{\mu\nu}\right)\frac{\partial}{\partial
x^{\nu}}$, $g_{\mu\nu}$ is the metric tensor which describes the physical
space. The fourth-order Schrödinger equation was introduced in [19, 20] in
order to investigate the effects of the presence of small fourth-order
dispersion terms in the propagation of laser beams in a bulk medium with Kerr
nonlinearity. For $V\left(\Psi\right)=\left|\Psi\right|^{2p}\Psi$, the
stability of solitons investigated by Karpman in [19]. It was found that when
$g_{\mu\nu}$ is the Euclidian manifold, then for $p\dim\left(g\right)<4$, the
soliton solutions are stable. Moreover, as it has been shown in [20], equation
(1) can follow by a variational principle. Since then, the fourth-order
Schrödinger equation was the subject of study in various articles in the
literature, see for instance [21, 22, 23, 24, 25, 26, 27, 28].

However, equation (1) describes also the modified Schrödinger equation for a
particle in the context of the Generalised Uncertainty Principle (GUP). GUP
has its origin in the existence of a minimal length of the order of the Planck
length ($l_{PL}$). The latter is a standard prediction of different approaches
to quantum physics and gravity, that is, from string theory, noncommutative
geometry, and others [29, 30, 31, 32]. Specifically, the minimal length in
Heisenberg’s Uncertainty Principle [33] is introduced. For a review on GUP, we
refer the reader in [34].

In the simplest case of quadratic GUP, the modified Heisenberg’s Uncertainty
Principle reads

$\Delta X_{\mu}\Delta P_{\nu}\geqslant\frac{\hbar}{2}[\delta_{ij}(1+\beta
P^{2})+2\beta P_{\mu}P_{\nu}].$ (2)

Consequently, the deformed algebra follows [35, 36],

$[X_{\mu},P_{\nu}]=i\hbar[\delta_{ij}(1-\beta P^{2})-2\beta P_{\mu}P_{\nu}],$
(3)

where $\beta$ is the parameter of deformation defined by
$\beta={\beta_{0}}/{M_{Pl}^{2}c^{2}=\beta}_{0}\ell_{Pl}^{2}/2\hbar^{2}$, where
$M_{Pl}$ is the Planck mass, $\ell_{Pl}$ ($\approx 10^{-35}~{}m)$ is the
Planck length and $M_{Pl}c^{2}$ ($\approx 1.2~{}\times~{}10^{19}~{}GeV)$ the
Planck energy, such that $\beta^{2}\rightarrow 0$. Thus, we can consider the
coordinate representation of the modified momentum operator is
$P_{\mu}=p_{\mu}(1-\beta p^{2})$ [36], while keeping $X_{\mu}=x_{\mu}$
undeformed.

Thus the time-independent Schrödinger equation reads

$\left(g^{\mu\nu}P_{\mu}P_{\nu}-\left(mc\right)^{2}\right)\Psi=0.$ (4)

That is,

$-2\beta\hbar^{2}\Delta^{2}\Psi+\Delta\Psi+\left(\frac{mc}{\hbar}\right)^{2}\Psi=0,$
(5)

where we have assumed terms with $\beta^{2}\rightarrow 0$. The fourth-order
equation (5) is the static version of (1) for $V\left(\Psi\right)$ be a linear
function. For some recent applications of GUP in physical theories see [37,
38, 39, 40, 41] and references therein.

In the following, we perform a complete classification of function
$V\left(\Psi\right)$ according to the admitted Lie point symmetries of
equation (1). Such a classification scheme was proposed in the previous
century by Ovsiannikov, where the Lie point symmetries for the nonlinear
equation $u_{t}=\left(f\left(u\right)u_{x}\right)_{x}$ were classified [42],
leading to new interesting problems in applied mathematics and physics [43,
44, 45, 46, 47, 48, 49, 50]. Apart from the analysis of symmetries, the
concept of asymptotic solution and boundary layer is essential in our context
[51].

The plan of the paper follows. In Section 2, we present the basic properties
and definitions for the theory of Lie symmetries of differential equations,
and we introduce the concept of the boundary layer. In Section 3 we present
our classification scheme for the Lie point symmetries of the fourth-order
Schrödinger equation. We present some applications of the Lie point symmetries
for the construction of similarity solutions in Section 4. Finally, in Section
5, we summarise our results.

##  2 Preliminaries

A differential equation may be considered as a function
$H=H(x^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},...)$ in the space
$B=B(x^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},...)$, where $x^{i}$ are the
independent variables and $u^{A}$ are the dependent variables. In our
consideration for equation (1) $x^{i}=\left(t,x^{\mu}\right)$ and
$u^{A}\left(x^{i}\right)=\Phi\left(x^{i}\right).$

###  2.1 Lie symmetry vector

Consider now, the infinitesimal transformation

$\displaystyle\bar{x}^{i}$
$\displaystyle=x^{i}+\varepsilon\xi^{i}(x^{k},u^{B})~{},$ (6)
$\displaystyle\bar{u}^{A}$
$\displaystyle=\bar{u}^{A}+\varepsilon\eta^{A}(x^{k},u^{B})~{},$ (7)

with generator the vector field

$\mathbf{X}=\xi^{i}(x^{k},u^{B})\partial_{x^{i}}+\eta^{A}(x^{k},u^{B})\partial_{u^{A}}~{}.$
(8)

The generator $\mathbf{X}$ of the infinitesimal transformation (6)-(7) is a
Lie point symmetry for the function $H$ if there exists a function $\lambda$
such that the following condition holds [1, 2]

$\mathbf{X}^{[N]}(H)=\lambda H~{},~{}\mod H=0,$ (9)

where

$\mathbf{X}^{[N]}=\mathbf{X}+\eta_{\left[i\right]}^{A}\partial_{u_{i}^{A}}+\eta_{\left[ij\right]}^{A}\partial_{u_{ij}^{A}}+...+\eta_{\left[ij...j_{N}\right]}^{A}\partial_{u_{ij...j_{N}}^{A}}$
(10)

is the $n^{th}$ prolongation vector

$\eta_{\left[i\right]}^{A}=\eta_{,i}^{A}+u_{,i}^{B}\eta_{,B}^{A}-\xi_{,i}^{j}u_{,j}^{A}-u_{,i}^{A}u_{,j}^{B}\xi_{,B}^{j}~{},$
(11)

with

$\displaystyle\eta_{\left[ij\right]}^{A}$
$\displaystyle=\eta_{,ij}^{A}+2\eta_{,B(i}^{A}u_{,j)}^{B}-\xi_{,ij}^{k}u_{,k}^{A}+\eta_{,BC}^{A}u_{,i}^{B}u_{,j}^{C}-2\xi_{,(i|B|}^{k}u_{j)}^{B}u_{,k}^{A}$
$\displaystyle-\xi_{,BC}^{k}u_{,i}^{B}u_{,j}^{C}u_{,k}^{A}+\eta_{,B}^{A}u_{,ij}^{B}-2\xi_{,(j}^{k}u_{,i)k}^{A}-\xi_{,B}^{k}\left(u_{,k}^{A}u_{,ij}^{B}+2u_{(,j}^{B}u_{,i)k}^{A}\right),$
(12)

and in general

$\eta_{\left[ij...j_{N}\right]}^{A}=D_{j_{n}}\left(\eta_{ij...j_{n-1}}^{A}\right)-u_{ij...k}^{A}D_{j_{N}}\xi^{k}.$
(13)

The existence of a Lie point symmetry in a given differential equation is
essential in simplifying the differential equation through the similarity
transformations. Indeed from a specific Lie symmetry vector one may defines
the following Lagrange system

$\frac{dx^{i}}{\xi^{i}}=\frac{du^{A}}{\eta^{A}}=\frac{du_{i}^{A}}{\eta_{\left[i\right]}^{A}}=\frac{du_{ij}^{A}}{\eta_{\left[ij\right]}^{A}}=...$
(14)

whose solution provides the characteristic functions
$W^{\left[0\right]}\left(x^{k},u^{A}\right)$,
$W^{\left[1\right]}\left(x^{k},u^{A},u_{i}^{A}\right)$, etc. These functions
can be used to define the corresponding similarity transformation.

###  2.2 The Concept of a Boundary Layer

Consider the function $\psi_{\varepsilon}(\tau)$ defined on a domain
$D\subseteq\mathbb{R}^{n}$ (in our case $\tau\in D\subset\mathbb{R}$) and
$\varepsilon$ is a small parameter. Suppose there exists a connected subset
$S\subset D$ of dimension $\leq n$, with the property that
$\psi_{\varepsilon}(\tau)$ has no regular expansion in each subset of $D$
containing points of $S$. Then, a neighbourhood of $S$ in $D$ with a size to
be determined is a boundary layer of the function $\psi_{\varepsilon}(\tau)$.
A boundary layer, corresponding with the subset $S$, has been found near the
boundary point $\tau=0$ of the domain [51]. A fundamental technique to study
the behaviour of a function $\psi_{\varepsilon}(\tau)$ in a boundary layer is
to use a local analysis. Suppose that near a point $\tau_{0}\in S$ the
boundary layer is characterised in size by an order function
$\delta(\varepsilon)$. We “rescale” or “stretch” the variable $\tau$ by
introducing the local variable
$\xi=\frac{\tau-\tau_{0}}{\delta(\varepsilon)}$. If
$\delta(\varepsilon)=o(1)$, we call $\xi$ a local (stretched or boundary
layer) variable. The function $\psi_{\varepsilon}(\tau)$ transforms to
$\psi_{\varepsilon}(\tau)=\psi_{\varepsilon}(\tau)(\tau_{0}+\delta(\varepsilon)\xi)=\phi^{*}_{\varepsilon}(\xi).$
It is then natural to continue the local analysis by expanding the function
$\phi^{*}_{\varepsilon}(\xi)$ with respect to the local variable $\xi$; with
hope to find again a regular expansion. To be more precise, assume that
$\phi^{*}_{\varepsilon}=O_{s}(1)$ (that is, $O(1)$ and $\neq o(1)$ as
$\varepsilon\rightarrow 0$) near $\xi=0$. We wish to find local approximations
of $\phi^{*}_{\varepsilon}$ by a regular expansion of the form
$\phi^{*}_{\varepsilon}(\xi)=\sum_{n}\delta_{n}^{*}(\varepsilon)\psi_{n}(\xi)$
with $\delta_{n}^{*}(\varepsilon)$, $n=0,1,2,\ldots$ an asymptotic sequence.

In many problems, the function $\psi_{\varepsilon}(\tau)$ has been implicitly
defined as the solution of a system of differential equations with initial
and/or boundary conditions. For example, suppose that we have to study the
perturbation problem $L_{\varepsilon}\phi=f(\tau),\tau\in D+\text{other
conditions}$. $L_{\varepsilon}$ is an operator containing a small parameter
$\varepsilon$. For instance,
$L_{\varepsilon}=\varepsilon\frac{d^{2}}{d\tau^{2}}+\frac{d}{d\tau}$,
$D=[0,1]$, $f(\tau)=0$ and boundary conditions $\phi(0)=1$, $\phi(1)=0$. The
function $\tilde{\phi}(\tau)$ will be called a formal approximation or formal
expansion of $\psi_{\varepsilon}(\tau)$ if $\tilde{\phi}$ satisfies the
boundary conditions to a certain approximation, and if
$L_{\varepsilon}(\tilde{\phi}(\tau))=f(\tau)+o(1)$. In practise, the
requirement of $\tilde{\phi}$ to satisfy the boundary conditions in full is
sometimes relaxed to $\tilde{\phi}$ to satisfy the boundary conditions to a
certain approximation. To prove that if $\tilde{\phi}$ is a formal
approximation, it also is an asymptotic approximation of $\phi$ is, in
general, a complex problem. Moreover, one can give straightforward and
realistic examples that are not true.

##  3 Symmetry classification for the fourth-order Schrödinger equation

Before we proceed with the symmetry classification, we set without loss of
generality $\gamma=1$ and by a change of transformation on the variable $t$ we
can remove the coefficient $i$. Hence, equation (1) can be written in the
equivalent form

$\frac{\partial\Psi}{\partial
t}+\alpha\Delta\Psi+\Delta^{2}\Psi+V\left(\Psi\right)=0.$ (15)

Moreover, with the use of the new variable $\Phi=\Delta\Psi$, the fourth-order
differential equation (15) is written as the following Schrödinger-Poisson
system

$\displaystyle\frac{\partial\Psi}{\partial
t}+\Delta\Phi+\alpha\Phi+V\left(\Psi\right)$ $\displaystyle=$ $\displaystyle
0,$ (16) $\displaystyle\Phi-\Delta\Psi$ $\displaystyle=$ $\displaystyle 0.$
(17)

Assume now the generic vector field

$X=\xi^{t}\left(t,x^{\mu}.\Psi,\Phi\right)\partial_{t}+\xi^{\mu}\left(t,x^{\mu},\Psi,\Phi\right)\partial_{\mu}+\eta^{\Psi}\left(t,x^{\mu},\Psi,\Phi\right)\partial_{\Psi}+\eta^{\Phi}\left(t,x^{\mu},\Psi,\Phi\right)\partial_{\Phi},$
(18)

where in order to be the generator of an one-parameter point transformation in
the space of variables $\left\\{x^{\mu},\Psi\right\\},$ it should be
$\xi_{,\Phi}^{t}=0$, $\xi_{,\Phi}^{\mu}=0$ and $\eta_{,\Phi}^{\Psi}=0$.

The $2^{nd}$ prolongation vector reads

$X^{\left[2\right]}=X+\eta_{\left[t\right]}^{\Psi}\partial_{\Psi_{t}}+\eta_{\left[\mu\right]}^{\Psi}\partial_{\Psi_{\mu}}+\eta_{\left[t\right]}^{\Phi}\partial_{\Phi_{t}}+\eta_{\left[\mu\right]}^{\Phi}\partial_{\Phi_{\mu}}+\eta_{\left[\mu\nu\right]}^{\Psi}\partial_{\Psi_{\mu\nu}}+\eta_{\left[\mu\nu\right]}^{\Phi}\partial_{\Phi_{\mu\nu}}.$
(19)

Consequently, we apply the symmetry condition (9), and by using the geometric
approach described in [52], we summarise the classification scheme in the
following theorem.

Theorem 1: The generic Lie point symmetry vector for the Schrödinger-Poisson
system (16), (17) in an arbitrary background space $g_{\mu\nu}$, and for
arbitrary function $V\left(\Psi\right)$ is

$X_{G}=a_{1}\partial_{t}+a_{\sigma}\mathbf{K}\left(x^{\kappa}\right)\partial_{\mu},$
(20)

where $\mathbf{K}\left(x^{\mu}\right)$ is an isometry for the metric tensor
$g_{\mu\nu}$, that is
$\left[\mathbf{K}\left(x^{\kappa}\right),g_{\mu\nu}\left(x^{\kappa}\right)\right]=0$.

However for specific functional forms of the potential $V\left(\Psi\right)$
the classification scheme is described as follows.

Theorem 2: Let the metric tensor $g_{\mu\nu}\left(x^{\kappa}\right)$ and
$\mathbf{K}\left(x^{\kappa}\right)$ describe the isometries of
$g_{\mu\nu}\left(x^{\kappa}\right)$, and $\mathbf{H}\left(x^{\kappa}\right)$
is a proper Homothetic vector of $g_{\mu\nu}\left(x^{\kappa}\right)$, i.e.
$\left[\mathbf{H}\left(x^{\kappa}\right),g_{\mu\nu}\left(x^{\kappa}\right)\right]=2g_{\mu\nu}\left(x^{\kappa}\right)$.
Then for special functional forms of $V\left(\Psi\right)$ the generic symmetry
vector for the Schrödinger-Poisson system (16), (17) is:

For $\alpha\neq 0$,

I: For $V\left(\Psi\right)=V_{0}\Psi$, the symmetry vector is
$X_{G}^{I}=a_{1}\partial_{t}+a_{\sigma}\mathbf{K}\left(x^{\kappa}\right)\partial_{\mu}+a_{2}\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)+a_{3}\left(F\left(t,x^{\kappa}\right)\partial_{U}+F_{,\mu\nu}\left(t,x^{\kappa}\right)\partial_{\Phi}\right)$,
where $F\left(t,x^{\kappa}\right)$ is a solution of the original system. The
new coefficients in the vector field indicate the linearisation of the system.

For $\alpha=0$,

II: For $V\left(\Psi\right)=0$, the generic symmetry vector is
$X_{G}^{II}=a_{1}\partial_{t}+a_{\sigma}\mathbf{K}\left(x^{\kappa}\right)\partial_{\mu}+a_{2}\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)+a_{3}\left(F\left(t,x^{\kappa}\right)\partial_{U}+F_{,\mu\nu}\left(t,x^{\kappa}\right)\partial_{\Phi}\right)+a_{4}\left(4t\partial_{t}+\mathbf{H}\left(x^{\kappa}\right)\partial_{\mu}-2\Phi\partial_{\Phi}\right)$.

III: For $V\left(\Psi\right)=V_{0}\Psi$, the generic symmetry vector is
$X_{G}^{III}=a_{1}\partial_{t}+a_{\sigma}\mathbf{K}\left(x^{\kappa}\right)\partial_{\mu}+a_{2}\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)+a_{3}\left(F\left(t,x^{\kappa}\right)\partial_{U}+F_{,\mu\nu}\left(t,x^{\kappa}\right)\partial_{\Phi}\right)+a_{4}\left(4t\partial_{t}+\mathbf{H}\left(x^{\kappa}\right)\partial_{\mu}-2\Phi\partial_{\Phi}-4V_{0}t\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)\right)$.

IV: For$~{}V\left(\Psi\right)=V_{0}\Psi^{P+1},~{}P\neq-1,0$, the generic
symmetry vector is
$X_{G}^{IV}=a_{1}\partial_{t}+a_{\sigma}\mathbf{K}\left(x^{\kappa}\right)\partial_{\mu}+a_{4}\left(4t\partial_{t}+\mathbf{H}\left(x^{\kappa}\right)\partial_{\mu}-2\Phi\partial_{\Phi}-\frac{4}{P}\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)\right)$.

V: For $V\left(\Psi\right)=V_{0}\exp\left(P\Psi\right),~{}P\neq 0$, the
generic symmetry vector is
$X_{G}^{IV}=a_{1}\partial_{t}+a_{\sigma}\mathbf{K}\left(x^{\kappa}\right)\partial_{\mu}+a_{4}\left(4t\partial_{t}+\mathbf{H}\left(x^{\kappa}\right)\partial_{\mu}-2\Phi\partial_{\Phi}-\frac{4}{P}\left(\partial_{\Psi}\right)\right)$.

We proceed with our analysis by considering specific metric tensor
$g_{\mu\nu}$.

##  4 Application

Consider now that the metric tensor $g_{\mu\nu}$ is maximally symmetric and
admit a homothetic vector field. Hence, $g_{\mu\nu}$ is necessary for the flat
space. For simplicity of our calculations, assume further that $\dim
g_{\mu\nu}=1$. The one-dimensional flat space with line element
$ds^{2}=dx^{2}$ admits the isometry $\partial_{x}$ and the proper Homothetic
field $x\partial_{x}$.

Therefore the Schrödinger-Poisson system reads

$\displaystyle\frac{\partial\Psi}{\partial t}+\frac{\partial^{2}\Phi}{\partial
x^{2}}+\alpha\Phi+V\left(\Psi\right)$ $\displaystyle=$ $\displaystyle 0,$ (21)
$\displaystyle\Phi-\frac{\partial^{2}\Psi}{\partial x^{2}}$ $\displaystyle=$
$\displaystyle 0.$ (22)

In the case where $\alpha\neq 0$, the generic vector field is
$X^{I}=a_{1}\partial_{t}+a_{2}\partial_{x}$, for arbitrary potential function
$V\left(\Psi\right)$. From the elements of $X^{I}$ we can reduce the dynamical
system into the static and the stationary cases. However, from the vector
field $\partial_{t}+c\partial_{x}$ we reduce the dynamical system as follow

$\displaystyle-c\frac{\partial\Psi}{\partial\xi}+\frac{\partial^{2}\Phi}{\partial\xi^{2}}+\alpha\Phi+V\left(\Psi\right)$
$\displaystyle=$ $\displaystyle 0,$ (23)
$\displaystyle\Phi-\frac{\partial^{2}\Psi}{\partial\xi^{2}}$ $\displaystyle=$
$\displaystyle 0,$ (24)

where $\xi=x-ct$ is the new independent variable and $c$ describes the speed
of the travel wave. For a linear function $V\left(\Psi\right),$ the closed-
form solution of the system (23), (24) can be expressed in terms of
exponential functions.

However, for $V\left(\Psi\right)=V_{0}\Psi$, there exist the additional
possible reduction
$\partial_{t}+c\partial_{x}+\beta\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)$,
which provides the similarity transformation $\Psi=e^{\beta
t}\psi\left(\xi\right)$, $\Phi=e^{\beta t}\phi\left(\xi\right)$, $\xi=x-ct$
with reduced system

$\displaystyle-c\frac{\partial\psi}{\partial\xi}+\frac{\partial^{2}\phi}{\partial\xi^{2}}+\alpha\phi+\beta\psi+V_{0}\psi$
$\displaystyle=$ $\displaystyle 0,$ (25)
$\displaystyle\phi-\frac{\partial^{2}\psi}{\partial\xi^{2}}$ $\displaystyle=$
$\displaystyle 0.$ (26)

Let us focus now with the case where $\alpha=0$ and assume
$V\left(\Psi\right)=V_{0}\Psi^{P+1}$ and
$V\left(\Psi\right)=V_{0}\exp\left(P\Psi\right)$.

###  4.1 Powerlaw function $V\left(\Psi\right)=V_{0}\Psi^{P+1},\;P\neq 0$

For the power-law potential function, from the vector field
$\left(4t\partial_{t}+x\partial_{x}-2\Phi\partial_{\Phi}-\frac{4}{P}\left(\Psi\partial_{\Psi}+\Phi\partial_{\Phi}\right)\right)$
we define the similarity transformation

$\Psi\left(t,x\right)=\psi\left(\sigma\right)t^{-\frac{1}{P}}~{},~{}\Phi\left(t,x\right)=\phi\left(\sigma\right)t^{-\frac{2+P}{2P}}~{},~{}\sigma\left(t,x\right)=\frac{x}{t^{\frac{1}{4}}},$

and if $P\neq 0$, with reduced system

$\displaystyle\frac{\partial^{2}\phi}{\partial\sigma^{2}}+V_{0}\psi^{P+1}-\frac{1}{4}\sigma\frac{\partial\psi}{\partial\sigma}-\frac{1}{P}\psi$
$\displaystyle=$ $\displaystyle 0,$ (27)
$\displaystyle\phi-\frac{\partial^{2}\psi}{\partial\sigma^{2}}$
$\displaystyle=$ $\displaystyle 0.$ (28)

If $\phi=0$, we have

$\psi=\psi_{1}\sigma+\psi_{0}.$ (29)

Then, from compatibility conditions the only possible solution is the constant
solution $\psi=\psi_{0}$ such that

$V_{0}\psi_{0}^{P+1}-\frac{\psi_{0}}{P}=0\implies\psi_{0}=(PV_{0})^{-1/P}.$
(30)

Therefore, we assume the nontrivial case $\phi\neq 0$. Then, we have the
fourth-order equation

$\displaystyle\frac{\partial^{4}\psi}{\partial\sigma^{4}}+V_{0}\psi^{P+1}-\frac{1}{4}\sigma\frac{\partial\psi}{\partial\sigma}-\frac{1}{P}\psi$
$\displaystyle=$ $\displaystyle 0.$ (31)

We introduce the logarithmic independent variable

$\tau=\ln(\sigma),$ (32)

and redefine

$\displaystyle\psi(\sigma)=\bar{\psi}(\ln(\sigma)).$ (33)

That is, for any function $f(\sigma)$, define

$\bar{f}(\tau)=f(e^{\tau}).$ (34)

Then, using the chain rule and the relation $\sigma=e^{\tau}$ we obtain

$\displaystyle\frac{\partial
f}{\partial\sigma}=e^{-\tau}\bar{f}^{\prime}(\tau),$ (35)
$\displaystyle\frac{\partial^{2}f}{\partial\sigma^{2}}=e^{-2\tau}\left(\bar{f}^{\prime\prime}(\tau)-\bar{f}^{\prime}(\tau)\right),$
(36)
$\displaystyle\frac{\partial^{3}\psi}{\partial\sigma^{3}}=e^{-3\tau}\left({\bar{f}}^{(3)}(\tau)-3{\bar{f}}^{\prime\prime}(\tau)+2{\bar{f}}^{\prime}(\tau)\right),$
(37)
$\displaystyle\frac{\partial^{4}\psi}{\partial\sigma^{4}}=e^{-4\tau}\left(u^{(4)}(\tau)-6u^{(3)}(\tau)+11u^{\prime\prime}(\tau)-6u^{\prime}(\tau)\right).$
(38)

Then, (31) becomes

$\displaystyle\frac{\bar{\psi}(\tau)\left(PV_{0}\bar{\psi}(\tau)^{P}-1\right)}{P}+\left(-6e^{-4\tau}-\frac{1}{4}\right)\bar{\psi}^{\prime}+e^{-4\tau}\bar{\psi}^{\prime\prime}+e^{-4\tau}\bar{\psi}^{(3)}(\tau)+e^{-4\tau}\bar{\psi}^{(4)}(\tau)=0.$
(39)

Assuming that $\bar{\psi}$ is bounded with bounded derivatives as
$\tau\rightarrow+\infty$ we obtain the asymptotic equation

$\displaystyle\frac{\bar{\psi}_{+}(\tau)\left(PV_{0}\bar{\psi}_{+}(\tau)^{P}-1\right)}{P}-\frac{1}{4}\bar{\psi}_{+}^{\prime}(\tau)=0,$
(40)

which admits the first integral

$\displaystyle
c_{1}\frac{\bar{\psi}_{+}(\tau)^{P}}{\left(1-PV_{0}\bar{\psi}_{+}(\tau)^{P}\right)}=e^{-{4\tau}}\implies\bar{\psi}_{+}(\tau)=\left(PV_{0}+c_{1}e^{4\tau}\right){}^{-1/P}.$
(41)

Defining

$z_{+}(\tau):=\frac{\bar{\psi}_{+}(\tau)^{P}}{\left(1-PV_{0}\bar{\psi}_{+}(\tau)^{P}\right)},$
(42)

$z_{+}(\tau)$ is monotone decreasing as $\tau\rightarrow+\infty$ for $P>0$ and
monotone increasing as $\tau\rightarrow+\infty$ for $P<0$. In other words, the
asymptotic states of $\bar{\psi}_{+}(\tau)$ are

$\displaystyle\lim_{\tau\rightarrow+\infty}\bar{\psi}_{+}(\tau)=0\;\text{ if
}P>0,V_{0}>0,$ (43)
$\displaystyle\lim_{\tau\rightarrow-\infty}\bar{\psi}_{+}(\tau)=\psi_{0}:=(PV_{0})^{-1/P}\;\text{
if }P>0,V_{0}>0,$ (44)

and

$\displaystyle\lim_{\tau\rightarrow+\infty}\bar{\psi}_{+}(\tau)=\psi_{0}:=(PV_{0})^{-1/P}\;\text{
if }P<0,V_{0}<0,$ (45)
$\displaystyle\lim_{\tau\rightarrow-\infty}\bar{\psi}_{+}(\tau)=0\;\text{ if
}P>0,V_{0}>0.$ (46)

The cases of interest are as $\tau\rightarrow+\infty$. That is, the monotonic
function $z_{+}$ unveils the asymptotic behaviour as $\tau\rightarrow+\infty$.

Now, assuming that $\bar{\psi}$ is bounded with bounded derivatives as
$\tau\rightarrow-\infty$, we obtain the asymptotic equation

$\displaystyle-6\bar{\psi}_{-}^{\prime}(\tau)+11\bar{\psi}_{-}^{\prime\prime}(\tau)-6\bar{\psi}_{-}^{(3)}(\tau)+\bar{\psi}_{-}^{(4)}(\tau)=0,$
(47)

with solution

$\displaystyle\bar{\psi}_{-}(\tau)=c_{2}e^{\tau}+\frac{1}{2}c_{3}e^{2\tau}+\frac{1}{3}c_{4}e^{3\tau}+c_{5},$
(48)

such that

$\lim_{\tau\rightarrow-\infty}\bar{\psi}_{-}(\tau)=c_{5}.$ (49)

Substituting $\psi(\tau)=\bar{\psi}_{-}(\tau)$ in (39), and taking limit
$\tau\rightarrow-\infty$, we obtain the compatibility condition

$c_{5}\left(-1+PV_{0}c_{5}{}^{P}\right)=0.$ (50)

That is, $c_{5}\in\left\\{0,(PV_{0})^{-1/P}\right\\}.$ The choice
$c_{5}=(PV_{0})^{-1/P}$ gives the proper matching condition

$\lim_{\tau\rightarrow-\infty}\bar{\psi}_{+}(\tau)=\lim_{\tau\rightarrow-\infty}\bar{\psi}_{-}(\tau)=(PV_{0})^{-1/P}.$
(51)

Summarising, as $\tau\rightarrow-\infty$,
$\psi(\tau)\approx\bar{\psi}_{-}(\tau)$, whereas, as $\tau\rightarrow+\infty$,
$\psi(\tau)\approx\bar{\psi}_{+}(\tau)$.

Let be define the new time variable $s=(1+\tanh(\tau))/2$ that brings the
interval $(-\infty,\infty)$ to $(0,1)$. Then, the original layer problem
becomes a two-point problem, with endpoints $0$ and $1$. The asymptotic
solutions can be found as

$\Phi_{-}(s)=\bar{\psi}_{-}(-\text{arctanh}(1-2s)),$ (52)

that is,

$\displaystyle\Phi_{-}(s)=(PV_{0})^{-1/P}+\frac{c_{3}s}{2-2s}+\left(c_{2}\left(\frac{1}{s}-1\right)+\frac{c_{4}}{3}\right)e^{-3\text{arctanh}(1-2s)}.$
(53)

As $s\rightarrow 0^{+}$, we have the asymptotic behaviour
$\Phi_{-}\rightarrow(PV_{0})^{-1/P}$.

Moreover,

$\Phi_{+}(s)=\bar{\psi}_{+}(-\text{arctanh}(1-2s)),$ (54)

becomes

$\displaystyle\Phi_{+}(s)=\left(PV_{0}+\frac{c_{1}s^{2}}{(1-s)^{2}}\right)^{-1/P},$
(55)

such that

$\displaystyle\lim_{s\rightarrow 1^{-}}\Phi_{+}(s)=0\;\text{ if }P>0,V_{0}>0.$
(56)

And we have the matching condition

$\lim_{s\rightarrow 0^{+}}\Phi_{-}(s)=\lim_{s\rightarrow
0^{+}}\Phi_{+}(s)=(PV_{0})^{-1/P}.$ (57)

The next step is to introduce the stretched variables $\kappa=s/\varepsilon$
and $\lambda=(1-s)/\varepsilon$, and write a solution

$\Phi(s,\varepsilon)=\zeta(\kappa,\varepsilon)+\eta(\lambda,\varepsilon)$ (58)

where

$\zeta\rightarrow(PV_{0})^{-1/P}\;\text{as}\;\kappa=s/\varepsilon\rightarrow\infty$
(59)

and

$\eta\rightarrow 0\;\text{as}\;\lambda=(1-s)/\varepsilon\rightarrow\infty.$
(60)

Near $s=0$, $\eta$ and its derivatives will be asymptotically negligible, so
$d^{j}\Phi(s,\varepsilon)/ds^{j}\sim(1/\varepsilon^{j})\left[d^{j}\zeta(\kappa,\varepsilon)/d\kappa^{j}\right]$.
Take, for example,

$\zeta_{0}(\kappa,\varepsilon)=(PV_{0})^{-1/P}+\frac{c_{3}\kappa\varepsilon}{2-2\kappa\varepsilon}+\left(c_{2}\left(\frac{1}{\kappa\varepsilon}-1\right)+\frac{c_{4}}{3}\right)e^{-3\text{arctanh}(1-2\kappa\varepsilon)}.$
(61)

Using the notation

$\displaystyle\bar{\psi}(\tau,\varepsilon)=\Phi(\kappa,\varepsilon),\;\kappa=\frac{\tanh(\tau)+1}{2\varepsilon}$
(62)

the approximated equation (47), becomes

$\displaystyle
6\varepsilon(4\kappa\varepsilon(4\kappa\varepsilon-3)+1)\Phi^{\prime}(\kappa,\varepsilon)$
$\displaystyle+(\kappa\varepsilon-1)\Big{[}3(24\kappa\varepsilon(2\kappa\varepsilon-1)+1)\Phi^{\prime\prime}(\kappa,\varepsilon)$
$\displaystyle+4\kappa(\kappa\varepsilon-1)\left(\kappa\Phi^{(4)}(\kappa,\varepsilon)(\kappa\varepsilon-1)+3\Phi^{(3)}(\kappa,\varepsilon)(4\kappa\varepsilon-1)\right)\Big{]}=0,$
(63)

where primes means derivatives with respect to $\kappa$, that admits the exact
solution (61). Since we are taking $\varepsilon$ as an small parameter, we see
that the initial layer problem is of type

$\displaystyle\left(-3\Phi^{\prime\prime}(\kappa)-4\kappa\left(\kappa\Phi^{(4)}(\kappa)+3\Phi^{(3)}(\kappa)\right)\right)$
$\displaystyle+\varepsilon\left(6\Phi^{\prime}(\kappa)+3\kappa\left(25\Phi^{\prime\prime}(\kappa)+4\kappa\left(\kappa\Phi^{(4)}(\kappa)+6\Phi^{(3)}(\kappa)\right)\right)\right)+O\left(\varepsilon^{2}\right)=0.$
(64)

Taking the expansion

$\displaystyle\Phi(\kappa)=\Phi_{0}(\kappa)+\varepsilon\Phi_{1}(\kappa)+\ldots$
(65)

we obtain at first order

$\displaystyle-3\Phi_{0}^{\prime\prime}(\kappa)-4\kappa\left(\kappa\Phi_{0}^{(4)}(\kappa)+3\Phi_{0}^{(3)}(\kappa)\right)=0.$
(66)

Hence,

$\Phi_{0}(\kappa)=\frac{4}{3}\sqrt{\kappa}(d_{2}\kappa-3d_{1})+d_{4}\kappa+d_{3}.$
(67)

At second order we have

$60d_{2}\sqrt{\kappa}+6d_{4}-4\kappa^{2}\Phi_{1}^{(4)}(\kappa)-12\kappa\Phi_{1}^{(3)}(\kappa)-3\Phi_{1}^{\prime\prime}(\kappa)=0.$
(68)

Hence,

$\displaystyle\Phi_{1}(\kappa)=2d_{2}\kappa^{5/2}+\frac{4}{3}d_{6}\kappa^{3/2}+d_{4}\kappa^{2}+d_{8}\kappa-4d_{5}\sqrt{\kappa}+d_{7},$
(69)

and so on. Finally we replace the leading order and second order terms (67)
and (69), respectively, in (65) with the replacement (62).

Near $s=1$, $\zeta$ and its derivatives will be asymptotically negligible, so
$d^{j}\Phi(s,\varepsilon)/ds^{j}\sim(1/\varepsilon^{j})\left[d^{j}\eta(\lambda,\varepsilon)/d\lambda^{j}\right]$.
Take, for example,

$\eta_{0}(\lambda,\varepsilon)=\left(PV_{0}+c_{1}e^{4\text{arctanh}(1-2\lambda\varepsilon)}\right)^{-1/P}.$
(70)

Using the notation

$\displaystyle\bar{\psi}(\tau,\varepsilon)=\Phi\left(\lambda\right),\;\lambda=\frac{1-\tanh(\tau)}{2\varepsilon},$
(71)

the approximated equation (40), becomes

$\displaystyle
2V_{0}\Phi(\lambda)^{P+1}+\lambda(1-\lambda\varepsilon)\Phi^{\prime}(\lambda)=\frac{2\Phi(\lambda)}{P},$
(72)

which admits the solution (70).

###  4.2 Exponential function
$V\left(\Psi\right)=V_{0}\exp\left(P\Psi\right),\;P\neq 0$

On the other hand, for the exponential potential
$V\left(\Psi\right)=V_{0}\exp\left(P\Psi\right),\;P\neq 0$, the similarity
transformation which corresponds to the vector field
$\left(4t\partial_{t}+x\partial_{x}-2\Phi\partial_{\Phi}-\frac{4}{P}\left(\partial_{\Psi}\right)\right)$
is

$\Psi\left(t,x\right)=\frac{\ln
t}{P}+\psi\left(\sigma\right)~{},~{}\Phi=t^{-\frac{1}{2}}\phi\left(\sigma\right)~{},~{}\sigma\left(t,x\right)=\frac{x}{t^{\frac{1}{4}}},$

where the reduced system is

$\displaystyle\frac{\partial^{2}\phi}{\partial\sigma^{2}}+V_{0}e^{P\psi}-\frac{1}{P}-\frac{1}{4}\sigma\frac{\partial\psi}{\partial\sigma}$
$\displaystyle=$ $\displaystyle 0,$ (73)
$\displaystyle\phi-\frac{\partial^{2}\psi}{\partial\sigma^{2}}$
$\displaystyle=$ $\displaystyle 0.$ (74)

We introduce the logarithmic independent variable (32) and defining
$\psi(\sigma)$ by (33). Then, using the chain rule and the relation
$\sigma=e^{\tau}$ we obtain

$V_{0}e^{P\bar{\psi}(\tau)}+\left(-6e^{-4\tau}-\frac{1}{4}\right)\bar{\psi}^{\prime}+e^{-4\tau}\bar{\psi}^{\prime\prime}+e^{-4\tau}\bar{\psi}^{(3)}(\tau)+e^{-4\tau}\bar{\psi}^{(4)}(\tau)-\frac{1}{P}=0.$
(75)

Assuming that $\bar{\psi}$ is bounded with bounded derivatives as
$\tau\rightarrow+\infty$ we obtain the asymptotic equation

$\displaystyle
V_{0}e^{P\bar{\psi}_{+}(\tau)}-\frac{1}{P}-\frac{1}{4}\bar{\psi}_{+}^{\prime}(\tau)=0,$
(76)

with solution

$\bar{\psi}_{+}(\tau)=\ln\left(\left(PV_{0}+e^{4\tau+c_{1}P}\right){}^{-1/P}\right).$
(77)

Now, assuming that $\bar{\psi}$ is bounded with bounded derivatives as
$\tau\rightarrow-\infty$, we obtain, as in section 4.1, the asymptotic
equation (47), with solution (48). Substituting
$\psi(\tau)=\bar{\psi}_{-}(\tau)$ in (75), and taking limit
$\tau\rightarrow-\infty$, we obtain

$-\frac{1}{P}+V_{0}e^{c_{5}P}=0\implies
c_{5}=\ln\left[(PV_{0})^{-1/P}\right].$ (78)

That is, we have the matching condition

$\lim_{\tau\rightarrow-\infty}\bar{\psi}_{+}(\tau)=\lim_{\tau\rightarrow-\infty}\bar{\psi}_{-}(\tau)=\ln\left[(PV_{0})^{-1/P}\right].$
(79)

As in section 4.1, we have as $\tau\rightarrow-\infty$,
$\psi(\tau)\approx\bar{\psi}_{-}(\tau)$, whereas, as $\tau\rightarrow+\infty$,
$\psi(\tau)\approx\bar{\psi}_{+}(\tau)$. Using the same method, we define the
new time variable $s=(1+\tanh(\tau))/2$ that brings the interval
$(-\infty,\infty)$ to $(0,1)$. Then, the original layer problem becomes a two-
point problem, with endpoints $0$ and $1$. The asymptotic solutions can be
found as

$\displaystyle\Phi_{-}(s)=\ln\left[(PV_{0})^{-1/P}\right]+\frac{c_{3}s}{2-2s}+\left(c_{2}\left(\frac{1}{s}-1\right)+\frac{c_{4}}{3}\right)e^{-3\text{arctanh}(1-2s)}.$
(80)

As $s\rightarrow 0^{+}$, we have the asymptotic behaviour
$e^{\Phi_{-}}\rightarrow(PV_{0})^{-1/P}$.

Similarly, we have

$\displaystyle
e^{\Phi_{+}(s)}=\left(PV_{0}+\frac{c_{1}s^{2}}{(1-s)^{2}}\right)^{-1/P},$ (81)

such that

$\displaystyle\lim_{s\rightarrow 1^{-}}e^{\Phi_{+}(s)}=0\;\text{ if
}P>0,V_{0}>0.$ (82)

Finally, by introducing the stretched variables $\kappa=s/\varepsilon$ and
$\lambda=(1-s)/\varepsilon$, and write a solution

$\Phi(s,\varepsilon)=\zeta(\kappa,\varepsilon)+\eta(\lambda,\varepsilon),$
(83)

where

$\zeta\rightarrow\ln\left[(PV_{0})^{-1/P}\right]\;\text{as}\;\kappa=s/\varepsilon\rightarrow\infty,$
(84)

and

$\eta\rightarrow 0\;\text{as}\;\lambda=(1-s)/\varepsilon\rightarrow\infty.$
(85)

Then, the layer problem becomes a two-point problem, with endpoints $0$ and
$1$, and we obtain the asymptotic solutions following similar approaches as in
section 4.1.

##  5 Conclusions

Lie symmetry analysis is a powerful method for analysing nonlinear
differential equations. In this study, was applied the Lie symmetry analysis
to solve the group classification problem for a $1+n$-dimensional nonlinear
higher-order Schrödinger equation inspired by GUP.

The partial differential equation of our analysis admits an arbitrary
potential function which was a constraint according to the admitted Lie point
symmetries. For an arbitrary potential function, we found that the admitted
Lie symmetries are the Killing vectors of the $n$-dimensional space
additionally to the vector field $\partial_{t}$. However, a new symmetry
vector presented in Theorems 1 and 2 can exist for specific function forms of
the potential function.

To demonstrate the application of the Lie symmetry vectors, we use the
corresponding Lie invariants to define similarity transformations and reduce
the partial-differential equation into an ordinary differential equation.
Because of the non-linearity of the reduced equation, we studied the
asymptotic dynamics and evolution.

In relation to asymptotic analysis we have obtained asymptotic solutions

$\displaystyle\bar{\psi}_{-}(\tau)$
$\displaystyle=c_{2}e^{\tau}+\frac{1}{2}c_{3}e^{2\tau}+\frac{1}{3}c_{4}e^{3\tau}+\left\\{\begin{array}[]{cc}(PV_{0})^{-1/P}&\text{powerlaw
function}\\\ \ln\left[(PV_{0})^{-1/P}\right]&\text{exponential
function}\end{array}\right.,$ $\displaystyle\bar{\psi}_{+}(\tau)$
$\displaystyle=\left\\{\begin{array}[]{cc}\left(PV_{0}+c_{1}e^{4\tau}\right){}^{-1/P}&\text{powerlaw
function}\\\
\ln\left(\left(PV_{0}+e^{4\tau+c_{1}P}\right){}^{-1/P}\right)&\text{exponential
function}\end{array}\right.,$

with the proper matching condition

$\lim_{\tau\rightarrow-\infty}\bar{\psi}_{+}(\tau)=\lim_{\tau\rightarrow-\infty}\bar{\psi}_{-}(\tau)=\left\\{\begin{array}[]{cc}(PV_{0})^{-1/P}&\text{powerlaw
function}\\\ \ln\left[(PV_{0})^{-1/P}\right]&\text{exponential
function}\end{array}\right..$

Hence, as $\tau\rightarrow-\infty$, $\psi(\tau)\approx\bar{\psi}_{-}(\tau)$,
whereas, as $\tau\rightarrow+\infty$, $\psi(\tau)\approx\bar{\psi}_{+}(\tau)$.
Finally, the layer problem becomes a two-point problem, with endpoints $0$ and
$1$ by introducing the stretched variables $\kappa=s/\varepsilon$ and
$\lambda=(1-s)/\varepsilon$, and writing a formal solution

$\Phi(s,\varepsilon)=\zeta(\kappa,\varepsilon)+\eta(\lambda,\varepsilon),$
(86)

where

$\zeta\rightarrow\left\\{\begin{array}[]{cc}(PV_{0})^{-1/P}&\text{powerlaw
function}\\\ \ln\left[(PV_{0})^{-1/P}\right]&\text{exponential
function}\end{array}\right.,\;\text{as}\;\kappa=s/\varepsilon\rightarrow\infty,$
(87)

and

$\eta\rightarrow 0\;\text{as}\;\lambda=(1-s)/\varepsilon\rightarrow\infty.$
(88)

Then, it is interesting to analyse possible asymptotic solutions for different
initial/boundary conditions, but this numerical treatment is out of the scope
of the present research. In general, when solving the problem of approximating
a function $\psi_{\varepsilon}(\tau)$ depending on a small parameter
$\varepsilon$ in a domain $D$, the following program is implemented [51].

  1. 1.

Try to construct a regular expansion in the original variable $\tau$. That is
possible outside the boundary layers, and is usually called the outer
expansion.

  2. 2.

Construct in the boundary layer(s) a local expansion in an appropriate local
variable. A regular expansion is usually called the inner expansion or
boundary layer expansion.

  3. 3.

The inner and outer expansions should be matched to obtain a formal expansion
for the whole domain $D$.

In literature, several techniques combining the three stages were developed,
which makes the process more efficient. This formal expansion, which is valid
in the whole domain, is sometimes called a uniform expansion. Note, however,
that in the literature, expressions called “uniformly valid expansion” are
more often than not formal expansions. So we come to the next point.

  4. 4.

Prove that formal expansions, obtained in the stages 1–3, represent valid
asymptotic approximations of the function $\psi_{\varepsilon}(\tau)$ that we
set out to study.

This work contributes to the subject of the application of Lie point
symmetries on nonlinear differential equations. In this study, we considered
an Schrödinger equation constructed by the deformation algebra of the
quadratic GUP. However, that is not the unique proposed GUP, and other
deformations algebras exist. Therefore, in future work, we plan to perform a
detailed classification for the higher-order Schrödinger equation for
different models of GUP. Finally, we will present formal expansions,
representing valid asymptotic approximations of the function
$\psi_{\varepsilon}(\tau)$ for other initial conditions that we set out to
study by singular perturbations methods, boundary layers, and multiple time
scales.

## Acknowledgements

G.L. was funded by Vicerrectoría de Investigación y Desarrollo Tecnológico at
UCN.

## References

  * [1] H. Stephani, Differential Equations: Their Solutions Using Symmetry, Cambridge University Press, New York, (1989)
  * [2] G.W. Bluman and S. Kumei, Symmetries of Differential Equations, Springer-Verlag, New York, (1989)
  * [3] P.G.L. Leach and V.M. Gorringe, Phys. Lett. A, 133, 289 (1988)
  * [4] R. Gazinov and N.H. Ibragimov, Nonlinear Dynamics, 17, 387 (1998)
  * [5] N. H. Ibragimov, On the group classification of second order differential equations. (Russian) Dokl. Akad. Nauk SSSR, 183, 274, (1968)
  * [6] H. Azad and M.T. Mustafa, J. Math. Anal. Appl., 333, 1180, (2007)
  * [7] M. Tsamparlis and A. Paliathanasis, J. Phys. A: Math. Theor. 44, 175202 (2011)
  * [8] F.M. Mahomed. Math. Methods Appl. Sci. 30, 1995 (2007)
  * [9] S. Jamal, A.H. Kara, A.H. Bokhari, Canadian J. Phys. 90, 667 (2012)
  * [10] A.K. Halder, A. Paliathanasis, S. Rangasamy and P.G.L. Leach, Zeitschrift für Naturforschung A 74, 597 (2019)
  * [11] S. Jamal and A.H. Kara, Nonlinear Dynamics 67, 97 (2012)
  * [12] A.A. Chesnokov, J. Appl. Mech. Techn. Phys. 49, 737 (2008)
  * [13] S. Jamal, Quaestiones Mathematicae 41, 409 (2018)
  * [14] S. Jamal and N. Mnguni, Appl. Math. Comp. 335, 65 (2018)
  * [15] A.K. Halder, A. Paliathanasis and P.G.L Leach, Symmetry 10, 744 (2018)
  * [16] F. Schwarz, Computing 62, 1 (1999)
  * [17] G.J. Reid and A.D. Wittkopf, ISSAC ’00: Proceedings of the 2000 international symposium on Symbolic and algebraic computation, 272 (2000)
  * [18] S. Ali, M. Safdar and A. Qadir, J. Applied Mathematics 2014, 793247 (2014)
  * [19] V.I. Karpman, Phys. Lett. A 215, 254 (1996)
  * [20] V.I. Karpman and A.G. Shagalov, Physica D 144, 194 (2000)
  * [21] J. Segata, Math. Methods Appl. Sci. 26, 1785 (2006)
  * [22] B. Pausader, J. Functional Analysis 256, 2473 (2009)
  * [23] B. Pausader and S. Shao, J. Hyperbolic Differential Equations 7, 651 (2010)
  * [24] C. Baquet and E.J. Villamizar-Roa, Evolution Equations & Contrl Theory 9, 865 (2020)
  * [25] X. Liu and T. Zhang, J. Math. Phys. 62, 071501 (2021)
  * [26] B. Erdogan, W.R. Green and E. Torpak, J. Differential Equations 271, 152 (2021)
  * [27] G. Fibich, B. Ilan and G. Papanicolaou, SIAM J. Appl. Math. 62, 1437 (2002)
  * [28] G. Fibich, B. Ilan and S. Schochet, Nonlinearity 16, 1809 (2003)
  * [29] K. Konishi, G. Paffuti and P. Provero, Phys. Lett. B 234, 276 (1990)
  * [30] A. Camelia, Int. J. Mod. Phys. D 11, 35 (2002).
  * [31] P. Martinetti, F. Mercati and L. Tomassini, Rev. Math. Phys. 24, 1250010 (2012)
  * [32] Ashtekar, A. and Lewandowski, J., Class. Quantum Grav., 21, R53, (2004)
  * [33] M. Maggiore, Phys. Lett. B 304, 65 (1993)
  * [34] S. Hossenfelder, Living Reviews in Relativity 16, 2, (2013)
  * [35] S. Das and E.C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008)
  * [36] S.K. Moayedi, M.R. Setare and H Moayeri, Int. J. Theor. Phys. 49, 2080 (2010)
  * [37] B. Hamil, M. Merad and T. Birkandan, Eur Phys. J. Plus 134, 278 (2019)
  * [38] M.P. Dabrowski and F. Wagner, EPJC 80, 676 (2020)
  * [39] V. Nenmeli, S. Shankaranarayanan, V. Todorinov and S. Das, Phys. Lett. B 821, 136621 (2021)
  * [40] A. Das, S. Das and E.C. Vagenas, Phys. Lett. B 809, 135772 (2020)
  * [41] S. Aghababaei, H. Mordpour and E.C. Vagenas, Eur. Phys. J. Plus 136, 997 (2021)
  * [42] L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York, (1982)
  * [43] Z.-Y. Zhang and G.-F. Li, Physica A 540, 123134 (2020)
  * [44] S. Jamal, A. H. Kara and R. Narain, J. App. Math. 2012, 765361 (2012)
  * [45] V. Lahno, R. Zhdanov and O. Magda, Acta Appl. Math. 91, 253 (2006)
  * [46] V.A. Baikov, A.V. Gladkov and R.J. Wiltshire, J. Phys. A: Math. Gen. 31, 7483 (1998)
  * [47] D. Huang and N.M. Ivanova, J. Math. Phys., 48, 073507 (2007)
  * [48] R. Cherniha, M. Serov and Y. Prystavka, Comm. Nonl. Sci. Num. Sim. 92, 105466 (2021)
  * [49] S. Jamal, A. H. Kara and A. H. Bokhari, Can. J. Phys. 90, 667 (2012)
  * [50] M. Tsamparlis and A. Paliathanasis, J. Phys.A: Math. Theor. 44, 175202 (2011)
  * [51] Ferdinand Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Texts in Applied Mathematics, https://doi.org/10.1007/0-387-28313-7. Springer-Verlag, New York (2005)
  * [52] A. Paliathanasis and M. Tsamparlis, J. Geom. Phys. 107, 45 (2016)