\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9

\documentclass[twoside]{article}
\usepackage{amssymb, amsmath} 
\pagestyle{myheadings}

\markboth{\hfil Approximate equivalence transformations \hfil EJDE--2001/23}
{EJDE--2001/23\hfil R. N. Ibragimov \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 23, pp. 1--12. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Approximate equivalence transformations and invariant solutions of a
  perturbed nonlinear wave equation 
 %
\thanks{ {\em Mathematics Subject Classifications:} 58J90.
\hfil\break\indent
{\em Key words:} Perturbed nonlinear wave equation, 
approximate equivalence transformation, \hfil\break\indent
Lie algebra, invariant solutions. \hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted February 11, 2001. Published April 6, 2001.} }
\date{}
%
\author{ R. N. Ibragimov }
\maketitle

\begin{abstract} 
We discuss the properties of a perturbed nonlinear wave equation 
whose coefficients depend on the first-order spatial derivatives. 
In particular, we obtain a group of transformations which are stable 
with respect to the given perturbation, and derive the principal Lie 
algebra and its approximate equivalence transformation. The extension of 
the principal Lie algebra by one is obtained  by means of a well-known 
classification of low dimensional Lie algebras. We also obtain
some invariant solutions and classification of the perturbed equation.
\end{abstract}


\newtheorem{theorem}{Theorem}
\newtheorem{definition}[theorem]{Definition}


\section{Introduction}

We consider the nonlinear wave equation 
\begin{equation}
H(u)\equiv D\left\langle u\right\rangle -\varepsilon f( \frac{\partial u
}{\partial x}) =0,
\end{equation}
where $D=\partial _{t^{2}}^{2}-a\exp \left\{ \partial _xu\right\} \partial
_x^{2}$ is a nonlinear operator, $a$ is constant and $\varepsilon f$ is an
infinitesimal perturbation imposed on the principal part $D\left\langle
u\right\rangle =0$. For the sake of simplicity and  without loss of
generality, we put $a=1$.

The classification problem of a family of equations involves the
determination of the principal algebra $L_{\wp }$, the equivalence algebra $
E_{\wp }$ and extension of $L_{\wp }$ by subalgebras of $E_{\wp }$ to divide
the family into disjoint classes. In a recent paper [11], the nonlinear wave
equation $u_{tt}=f( x,u_x) u_{xx}+g( x,u_x) $ was
partially classified into thirty-three classes of equations and one of them
is given there as the family of the form of (1).

The main goal of this paper is to find invariant solutions to (1).
Since the perturbation destroys the group of transformations admitted by the
principal part, the analysis is rather difficult. However we shall construct
symmetries which are stable with respect to the perturbation. More
specifically, we shall find a mapping $\mathbf{F}:( x,t;u)
\to ( x,t;u') $ which transforms (1) into 
\begin{equation}
\frac{\partial ^{2}u'}{\partial t^{2}}-\frac{\partial
^{2}u'}{\partial x'^2 }\exp \left\{ \frac{\partial
u'}{\partial x'}\right\} -\varepsilon f'( \frac{\partial u'}{\partial x'})
+o( \varepsilon ) =0,
\end{equation}
i.e., the form of (1) is unchanged. Further we construct the principal
Lie algebra which enables us to classify (1). As the result of Lie group
classification, we determine the unknown function $f$ and, thus, find
invariant solutions admitted by (1).

The approximate method of group analysis, used in this paper, was developed
first by Ovsyannikov [7] while the problem of group classification of
partial differential equations according to their symmetries was first
considered by Sophus Lie [4]. The general approach to finding the symmetry
group of differential equations can be found, for example, in [2], [5], [8].

Recently, several papers, which are closely related to the present work,
were published. To name a few, Ames et al [1] investigated the group
properties of quasilinear hyperbolic equation of the form $u_{tt}=f(
u_x) u_{xx}$. The investigation was later generalized by Torrisi at
[9], [10] to equation of the form $u_{tt}=f( x,u_x) u_{xx}.$

\section{Group Classification}

We wish to find the approximate equivalence transformations for (1). In
this case, a natural modification of equivalence transformation that
involves approximate transformations (as in [3]) is used. Since an
equivalence transform is a nongenerate change of variables $x,t$ and $u$
which transform (1) to the same form as (2) (generally with
different function $f( u_x) $), we apply the Lie infinitesimal
method to calculate the group of equivalence transformations of the system 
\begin{eqnarray}
&D\left\langle u\right\rangle -\varepsilon f =o( \varepsilon ) ,&\\
&f_x =f_t=f_u=f_{u_t}=\varepsilon ^{-1}o( \varepsilon )&  \nonumber
\end{eqnarray}
and suppose that the operator for approximate transformation groups be given
in the form 
\begin{equation*}
\mathbf{X}=( \xi ^0+\varepsilon \xi ^1) \partial _x+(
\tau ^0+\varepsilon \tau ^1) \partial _t+( \eta
^0+\varepsilon \eta ^1) \partial _u+\varphi \partial _{f},
\end{equation*}
where $\xi ^\nu ,\tau ^\nu ,\eta ^\nu $ $( \nu =0,1) $ are
functions of $t,x$ and $u$ while $\varphi $ depends on variables 
$t,x,u,u_x,u_t$ and $f$. Thus we rewrite the generator of the group as 
$\mathbf{X}=\mathbf{X}^0+\varepsilon \mathbf{X}^1$, where $\mathbf{X}^0$
is a stable symmetry if it is admitted by unperturbed equation 
$D\left\langle u\right\rangle =0$. According to [3], we call the
corresponding symmetry generator $\mathbf{X}$ a deformation of the operator 
$\mathbf{X}^0$ which generates the elements of the principal Lie algebra 
$L_{\wp }$ for $D\left\langle u\right\rangle =0$.

In the extended space with variables $( x,t,u,u_x,u_t,\dots ) $,
the second prolongation of operator $\mathbf{X}$ is 
$$\displaylines{ 
\mathbf{X}^{\left[ 2\right] }=\mathbf{X+}( \zeta _0^{( x)
}+\zeta _1^{( x) }) \partial _{u_x}+( \zeta
_0^{( t) }+\zeta _1^{( t) }) \partial
_{u_t}+( \zeta _0^{( xx) }+\zeta _1^{( xx)
}) \partial _{u_{xx}} \cr
+( \zeta _0^{( tt) }+\zeta _1^{( tt) })
\partial _{u_{tt}}+\psi _x\partial _{f_x}+\psi _t\partial
_{f_t}+\psi _u\partial _{f_u}+\psi _{u_t}\partial _{f_{u_t}}+\dots 
}$$
Here the following notation is used $( \nu =0,1\text{ and }\theta \in
\left\{ x,t,u,u_t\right\} ) $: 
\begin{equation*}
\zeta _{\nu }^{( x) }=\eta _x^\nu +u_x( \eta _u^{\nu
}-\partial _x\xi ^\nu -u_x\partial _u\xi ^\nu ) -u_t(
\partial _x\tau ^\nu +u_x\partial _u\tau ^\nu ) ,
\end{equation*}
\begin{equation}
\zeta _{\nu }^{( t) }=\eta _t^\nu +u_t( \eta _u^{\nu
}-\partial _t\tau ^\nu -u_t\partial _u\tau ^\nu )
-u_x( \partial _t\xi ^\nu +u_t\partial _u\xi ^\nu ) ,
\quad
\end{equation}
\begin{eqnarray*}
\zeta _{\nu }^{( xx) } &=&\eta _{xx}^\nu +( 2\eta
_{xu}^\nu -\xi _{xx}^\nu ) u_x-\tau _{xx}^\nu u_t+(
\eta _u^\nu -2\xi _x^\nu ) u_{xx}\\
&&-2\tau _x^\nu u_{xt}+( \eta _{uu}^\nu -2\xi _{xu}^\nu )
u_x^{2}-2\tau _{xu}^\nu u_tu_x-\xi _{uu}^\nu u_x^{3}-\tau
_{uu}^\nu u_tu_x^{2}\\
&&-3\xi _u^\nu u_xu_{xx}-\tau _u^\nu u_tu_{xx}-2\tau _u^{\nu
}u_xu_{xt},
\end{eqnarray*}
\begin{eqnarray*}
\zeta _{\nu }^{( tt) } &=&\eta _{tt}^\nu +( 2\eta
_{tu}^\nu -\tau _{tt}^\nu ) u_t-\xi _{tt}^\nu u_x+(
\eta _u^\nu -2\tau \xi _t^\nu ) u_{tt}\\&&
-2\xi _t^\nu u_{xt}+( \eta _{uu}^\nu -2\tau _{tu}^\nu )
u_t^{2}-2\xi _{tu}^\nu u_tu_x-\tau _{uu}^\nu u_t^{3}-\xi
_{uu}^\nu u_t^{2}u_x\\&&
-3\tau _u^\nu u_tu_{tt}-\xi _u^\nu u_xu_{tt}-2\xi _u^{\nu}u_tu_{xt},
\end{eqnarray*}
\begin{eqnarray*}
\psi _{\theta } &=&d_{\theta }^{\ast }\left\langle \varphi \right\rangle
-f_td_{\theta }^{\ast }\left\langle \tau ^0+\varepsilon \tau
^1\right\rangle -f_xd_{\theta }^{\ast }\left\langle \xi ^0+\varepsilon
\xi ^1\right\rangle -f_ud_{\theta }^{\ast }\left\langle \eta
^0+\varepsilon \eta ^1\right\rangle \\&&
+ f_{u_t}d_{\theta }^{\ast }\left\langle \zeta _0^{( t)
}+\varepsilon \zeta _1^{( t) }\right\rangle +f_{u_x}d_{\theta
}^{\ast }\left\langle \zeta _0^{( x) }+\varepsilon \zeta
_1^{( x) }\right\rangle ,
\end{eqnarray*}
where 
\begin{equation*}
d_{\theta }^{\ast }=\partial _{\theta }+f_{\theta }\partial _{f}+f_{\theta
t}\partial _{f_t}+f_{\theta x}\partial _{f_x}+f_{\theta u}\partial
_{f_u}+f_{\theta u_t}\partial _{f_{u_t}}+\dots  .
\end{equation*}
Since $f_{\theta }=0$ $\forall \theta \in $ $\left\{ x,t,u,u_t\right\} ,$
i.e., $d_{\theta }^{\ast }=\partial _{\theta }$, the infinitesimal
invariance criterion for system (3) becomes 
\begin{equation}
\mathbf{X}^{\left[ 2\right] }\left[ H( u) \right] \mid
_{(M)}=o( \varepsilon ) ,
\end{equation}
\begin{equation}
\mathbf{X}^{\left[ 2\right] }( \varepsilon f_{\theta }) =o(
\varepsilon ) ,
\end{equation}
where the symbol $\mid _{(M)}$means evaluated on the manifold $M$, defined
by (1).

In zero-order approximation $( \varepsilon =0) $, system (5)--(6) yields the 
system of determining equations in the form 
\begin{equation}
\zeta _0^{( tt) }-\exp \left\{ u_x\right\} ( \zeta
_0^{( x) }u_{xx}+\zeta _0^{( xx) }) =0,\quad 
\varepsilon \psi _{\theta }=0
\end{equation}
which gives $\varphi _{\theta }=0$ and $( \zeta _0^{( x)
}) _{\theta }=0$ $( \forall \theta \in \left\{
x,t,u,u_t\right\} ) $ since $f$ is a differential variable which is
algebraically independent from $f_{u_x}$. Thus $\varphi $ is the function
of $u_x$ and $f$ only. Consequently, differentiation of (4) with
respect to $x$ and splitting it into independent parts, yields 
$$
\eta _{xx}^0=\tau _{xx}^0=\tau _{xu}^0=\eta _{xu}^0-\xi
_{xx}^0=\xi _{xu}^0=0.
$$ 
In a way similar to the above, we derive the following equations 
$$\displaylines{
\eta _{xt}^0=\tau _{xt}^0=\tau _{tu}^0=\eta _{tu}^0-\xi
_{xt}^0=\xi _{tu}^0=0, \cr
\eta _{xu}^0=\eta _{uu}^0-\xi _{xu}^0=\tau _{xu}^0=\tau
_{uu}^0=\xi _{uu}^0=0, \cr
\tau _x^0=\tau _u^0=0
}$$
from which we find 
$$\displaylines{
\xi ^0=a_1u+\alpha _1( t) x+\alpha _2( t) , \cr
\tau ^0=\tau ^0( t) , \cr
\eta ^0=\beta _1( t) u+a_2x+\beta _2( t) ,
}$$
where $a_1$and $a_2$ are constant coefficients. Similarly, splitting the
first of (7) we obtain 
$$\displaylines{
\xi ^0=( c_5+c_6) x+c_2, \cr
\tau ^0=c_5t+c_1, \cr
\eta ^0=2c_6x+( c_5+c_6) u+c_4t+c_3
}$$
with constants $c_1,\dots ,c_6$. Thus the unperturbed equation (1) is stable
with respect to the group $G^0$ of the transformations defined by the
following generators: 
$$\displaylines{
\mathbf{X}_1^0=\partial _t,\quad \mathbf{X}_2^0=\partial
_x,\quad \mathbf{X}_3^0=\partial _u,\quad \mathbf{X}
_4^0=t\partial _u, \cr
\mathbf{X}_5^0=t\partial _t+x\partial _x+u\partial _u,\quad 
\mathbf{X}_6^0=x\partial _x+( u+2x) \partial _u.
}$$
So equation $D\left\langle u\right\rangle =0$ is invariant with respect to a
group $G^0$, i.e., the unperturbed nonlinear wave equation admits $G^0$
whenever $u$ solves that equation. Note that if one rewrites (5) in the
form $( \mathbf{\xi \cdot \nabla }H) \mid _{(M)}=0$, it becomes
evident that (5) is the condition for the vector field $\mathbf{\xi }
=( \xi ,\tau )$ to be tangent to the manifold $M$.

In a way similar to the above, we write the invariance condition for (1)
up to the first order where $M=\left\{ u:\text{ }\partial _t^{2}u=\exp
\left\{ \partial _xu\right\} \partial _x^{2}u+\varepsilon f\right\} .$
After splitting of the determinant equation 
\begin{equation*}
Z( u) -\varepsilon \varphi =0,
\end{equation*}
where we denote 
\begin{equation*}
Z( u) =( \zeta _0^{( tt) }+\varepsilon \zeta
_1^{( tt) }) -( \zeta _0^{( x)
}+\varepsilon \zeta _1^{( x) }) u_{xx}\exp \left\{
u_x\right\} -( \zeta _0^{( xx) }+\varepsilon \zeta
_1^{( xx) }) \exp \left\{ u_x\right\} 
\end{equation*}
into independent parts and solving the resulting equations, we obtain 
$$\displaylines{
\tau ^1=a_1t+a_2,\quad \xi ^1=a_3x+a_4, \cr
\eta ^1=a_3u+2( a_1-a_3) x+\frac{a_5t^{2}}{2}+a_6t+a_{7}, \cr
\varphi =( c_1-2c_3) f+a_5
}$$
with constants $a_1,\dots ,a_{7}$. The last equations together with the group 
$G^0$ generate 13 dimensional approximate Lie algebra of approximate
equivalence transformations $G^1$spanned by generators 
$$\displaylines{
\mathbf{X}_1^1=\partial _t,\quad\mathbf{X}_2^1=\partial _x,
\quad\mathbf{X}_3^1=\partial _u,\quad\mathbf{X}
_4^1=t\partial _u, 
\cr
\mathbf{X}_5^1=t\partial _t+x\partial _x+u\partial _u-f\partial
_{f},\quad \mathbf{X}_6^1=x\partial _x+( u+2x)
\partial _u+f\partial _{f}, 
\cr
\mathbf{X}_{7}^1=\varepsilon \partial _t,\quad\mathbf{X}
_{8}^1=\varepsilon \partial _x,\quad\mathbf{X}_{9}^1=\varepsilon
\partial _u,\quad\mathbf{X}_{10}^1=\varepsilon t\partial _u,
\cr
\mathbf{X}_{11}^1=\varepsilon ( t\partial _t+2x\partial _x)
,\quad\mathbf{X}_{12}^1=\varepsilon ( x\partial _x+\left[ u-2x
\right] \partial _u) ,\quad\mathbf{X}_{13}^1=\varepsilon
( t^{2}\partial _u+2\partial _{f}) .
}$$
It is sufficient, for group classification, to consider the point
approximate equivalence transformations corresponding to nontrivial
generators $\mathbf{X}_5^1,\mathbf{X}_6^1$ and $\mathbf{X}_{13}^1$.
These transformations are given by 
$$\displaylines{
x'=a_1a_2x,\quad t'=a_1t,\quad f'=\varepsilon ( 2a_1a_2a_3+a_1a_2f) ,
\cr
u'=\varepsilon ( a_1a_2a_3t^{2}-2a_1(a_2-1) x+a_1a_2u) .
}$$

To find the principal Lie algebra $L_{\wp }$ for (1) and to find those
functions $f$ for which $L_{\wp }$ is extended, we seek the admitted
operator in the form $\mathbf{Y}=\mathbf{X}-\varphi \partial _{f}$. The
invariance condition for (1) 
\begin{equation*}
\mathbf{Y}^{( 2) }\left[ H( u) \right] \mid _{(M)}=0
\end{equation*}
yields the determining equation as 
\begin{equation*}
Z( u) -\varepsilon f'( \zeta _0^{( x)
}+\varepsilon \zeta _1^{( x) }) =0.
\end{equation*}
For the zero order approximation we obtain a similar result as above whereas
for the first order approximation the determining equation takes the form 
\begin{equation}
( \eta _u^0-2\tau _t^0) f+\zeta _1^{( tt)
}-\exp \left\{ u_x\right\} ( u_{xx_1}\zeta _1^{( xx)
}+\zeta _1^{( x) }) -f'\zeta _x^0=0.
\end{equation}
Substituting $u_{tt}=\exp \left\{ u_x\right\} u_{xx}$ and considering
arbitrary $f$ we split (8) into independent parts to obtain 
$$\displaylines{
\xi ^1=a_1x+a_2,\quad \tau ^1=a_3t+a_4,
\cr
\eta ^1=a_1u+2( a_1-a_2) x+a_5t+a_6.
}$$
Thus the principal 10 dimensional Lie algebra $L_{\wp }$ has the basis 
\begin{eqnarray}
&\mathbf{Y}_1=\partial _u,\quad \mathbf{Y}_2=\partial _x,\quad
\mathbf{Y}_3=\partial _t,\quad \mathbf{Y}_4=t\partial _u, &\nonumber
\\
&\mathbf{Y}_5=\varepsilon \mathbf{Y}_1,\quad \mathbf{Y}
_6=\varepsilon \mathbf{Y}_2,\quad \mathbf{Y}_{7}=\varepsilon 
\mathbf{Y}_3,\quad \mathbf{Y}_{8}=\varepsilon \mathbf{Y}_4, &
\\
&\mathbf{Y}_{9}=\varepsilon ( t\partial _t-2x\partial _u) ,
\quad \mathbf{Y}_{10}=\varepsilon ( x\partial _x+\left[ 2x+u
\right] \partial _u) .&\nonumber
\end{eqnarray}

We show that these symmetries are admitted by (1) in Appendix 1. If we
consider the function $f$ not arbitrary, then (8) reduces to 
\begin{equation*}
( \eta _u^0-2\tau _t^0) f+\eta _{tt}^1-f'\eta
_x^0=0
\end{equation*}
which is equivalent to relation 
\begin{equation}
\delta +( c_6-c_5) f-2c_6f'=0,
\end{equation}
where $\delta $ is a constant.

We further analyze the classifying relation (10) to obtain non-equivalent
forms of $f$. To this end we consider two different cases (see Appendix 2
for more details).

\paragraph{Case 1}. If $\delta =0$, then for $\gamma =\frac{c_6-c_5}{2c_6}
$ $( c_6\neq 0) $ we obtain the eleventh symmetry, namely 
\begin{equation*}
\mathbf{Y}_{11}=2( 1-\gamma ) x\partial _x+( 1-2\gamma
) t\partial _t+( 2x_1+2\left[ 1-\gamma \right] u)
\partial _u.
\end{equation*}
In other words, the equation 
\begin{equation}
\frac{\partial ^{2}u}{\partial t^{2}}-\exp \left\{ \frac{\partial u}{
\partial x}\right\} \frac{\partial ^{2}u}{\partial x^{2}}-\varepsilon A\exp
\left\{ \frac{\partial u}{\partial x}\right\} =0,\quad A>0,\text{ }
\gamma \in \mathbb{R}
\end{equation}
admits 11 dimensional Lie algebra.

In particular, for $\gamma =\frac{1}{2}$, we have\textbf{\ }$Y_{11}=\frac{1}{
\varepsilon }\mathbf{Y}_{10}.$

\paragraph{Case 2}.  If $\delta \neq 0$, the eleventh symmetry is given by 
\begin{equation*}
\mathbf{Y}_{11}=2( 1-\gamma ) x\partial _x+( 1-2\gamma
) t\partial _t+( 2x+2\left[ 1-\gamma \right] u-\frac{\delta }{
2c_6}t^{2}) \partial _u.
\end{equation*}

\section{The adjoint group and invariant solutions}

Now we are ready to construct the adjoint group of the algebra $L_{10}$ and
thus find some approximate invariant solutions. We start by giving the
definition of \emph{inner automorphism}. See [3] or [7] for more details.

\begin{definition} \rm
Let $X_1,\dots ,X_{r}$ be the selected basis of the vector space $L_{r}.$
Accordingly, the structure constants $c_{\mu \nu }^{\lambda }$ are known and
any $X$ $\in L$ is written as $X=e^{\mu }X_{\mu }$. Hence, the elements of $
L_{r}$ are represented by vectors $e=( e^1,\dots ,e^{r}) $. Let $
L_{r}^{A}$ be a Lie algebra spanned by the following operators 
\begin{equation}
E_{\mu }=c_{\mu \nu }^{\lambda }e^\nu \partial _{e^{\lambda }},\quad 
\mu =1,\dots ,r.
\end{equation}
with the Lie Bracket  defined by formula $\left[ X_1,X_2\right]
=X_1X_2-X_2X_1$. The algebra $L_{r}^{A}$ generates the group $G^{A}$
of linear transformations of $\left\{ e^{\mu }\right\} $. These
transformations determine the automorphism of the algebra $L_{r}$ known as
inner automorphism. The group $G^{A}$ is called group of automorphism of $
L_{r}$, or the adjoint group of $L_{r}.$
\end{definition}

We now consider the commutators of $L_{10}$ given in the follwiong table 
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\left[ X_{i}X_{j}\right] $ & $X_1$ & $X_2$ & $X_3$ & $X_4$ & $X_5$
\\ \hline
$X_1$ & $0$ & $-\varepsilon ( 2X_6-X_2) $ & $0$ & $0$ & $
\varepsilon X_5$ \\ \hline
$X_2$ & $\varepsilon ( 2X_6-X_2) $ & $0$ & $2\varepsilon
X_6$ & $0$ & $0$ \\ \hline
$X_3$ & $0$ & $-2\varepsilon X_6$ & $0$ & $\varepsilon X_4$ & $0$ \\ 
\hline
$X_4$ & $0$ & $0$ & $-\varepsilon X_4$ & $0$ & $-\varepsilon X_6$ \\ 
\hline
$X_5$ & $-\varepsilon X_5$ & $0$ & $0$ & $\varepsilon X_6$ & $0$ \\ 
\hline
$X_6$ & $-\varepsilon X_6$ & $0$ & $0$ & $0$ & $0$ \\ \hline
$X_{7}$ & $2X_6-X_2$ & $0$ & $2X_6$ & $0$ & $0$ \\ \hline
$X_{8}$ & $0$ & $0$ & $-X_4$ & $0$ & $-X_6$ \\ \hline
$X_{9}$ & $-X_5$ & $0$ & $X_5$ & $X_6$ & $0$ \\ \hline
$X_{10}$ & $-X_6$ & $0$ & $0$ & $0$ & $0$ \\ \hline
\end{tabular}
\\[2mm]
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\left[ X_{i}X_{j}\right] $ & $X_6$ & $X_{7}$ & $X_{8}$ & $X_{9}$ & $
X_{10} $ \\ \hline
$X_1$ & $\varepsilon X_6$ & $-( 2X_6-X_2) $ & $0$ & $
X_5 $ & $X_6$ \\ \hline
$X_2$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline
$X_3$ & $0$ & $-2X_6$ & $X_4$ & $-X_5$ & $0$ \\ \hline
$X_4$ & $0$ & $0$ & $0$ & $-X_6$ & $0$ \\ \hline
$X_5$ & $0$ & $0$ & $X_6$ & $0$ & $0$ \\ \hline
$X_6$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline
$X_{7}$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline
$X_{8}$ & $0$ & $0$ & $0$ & $-X_{10}$ & $0$ \\ \hline
$X_{9}$ & $0$ & $0$ & $X_{10}$ & $0$ & $0$ \\ \hline
$X_{10}$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ \hline
\end{tabular}
\end{center}

To find the transformations that give rise to the adjoint group of $L_{10},$
we seek the generators of the adjoint algebra $L_{10}^{A}$ in the form (12),
i.e., $E_{\mu }=c_{\mu \nu }^{\lambda }e^\nu \partial _{e^{\lambda }}$, \
\ $\mu =1,\dots ,10,$where the structure constants are given by $\left[ X_{\mu
},X_{\nu }\right] =c_{\mu \nu }^{\lambda }X_{\lambda }$. We find (see
Appendix 3) 
$$\displaylines{
\mathbf{E}_1=( \varepsilon e^{2}+e^{7}) \partial
_{e^{2}}+( \varepsilon e^{5}+e^{9}) \partial _{e^{5}}+(
\varepsilon e^{6}-2e^{7}-2\varepsilon e^{2}+e^{10}) \partial _{e^{6}},
\cr 
\mathbf{E}_2=2\varepsilon e^1\partial _{e^{6}}-\varepsilon e^1\partial
_{e^{2}}+2\varepsilon e^{3}\partial _{e^{6}},
\cr 
\mathbf{E}_3=-\varepsilon e^{2}\partial _{e^{6}}+\varepsilon e^{4}\partial
_{e^{4}}-2e^{7}\partial _{e^{6}}+e^{8}\partial _{e^{4}}-e^{9}\partial
_{e^{5}},
\cr
\mathbf{E}_4=\varepsilon e^{3}\partial _{e^{4}}-\varepsilon e^{5}\partial
_{e^{6}}-e^{9}\partial _{e^{6}},
\cr
\mathbf{E}_5=-\varepsilon e^1\partial _{e^{5}}+\varepsilon e^{4}\partial
_{e^{6}}+e^{8}\partial _{e^{6}},
\cr
\mathbf{E}_6=-\varepsilon e^1\partial _{e^{6}},
\cr
\mathbf{E}_{7}=2e^1\partial _{e^{6}}-e^1\partial _{e^{2}}+2e^{3}\partial
_{e^{6}},
\cr
\mathbf{E}_{8}=-e^{3}\partial _{e^{4}}-e^{5}\partial _{e^{6}}+e^{9}\partial
_{e^{10}},
\cr
\mathbf{E}_{9}=-e^1\partial _{e^{5}}+e^{3}\partial _{e^{5}}+e^{4}\partial
_{e^{6}}+e^{8}\partial _{e^{10}},
\cr
\mathbf{E}_{10}=-e^1\partial _{e^{6}}.
}$$

We further solve Lie equations for these operators to obtain the following
adjoint transformations which give rise to the adjoint group elements of the
algebra $L_{10}$: 
$$\displaylines{
e'^ 1=e^1, 
\cr
e`^2=2( a_2-a_1^{\varepsilon }a_3a_{7})
e^1-2\varepsilon a_1^{\varepsilon }a_3e^{2}+( \frac{
a_1^{\varepsilon }}{\varepsilon }-1) e^{7},
\cr
e'^3=e^{3},
\cr
e'^4=( \varepsilon +1) a_3e^{4}-( \left[
\varepsilon +1\right] a_3) ( a_{8}+a_4\varepsilon )
e^{3},
\cr
e'^5=-\varepsilon a_5e^1+a_1^{\varepsilon }e^{5}+( \frac{
a_1^{\varepsilon }}{\varepsilon }-1) e^{9},
\cr
e'^6=a_1^{\varepsilon }( 2\varepsilon \left[ e^1+e^{3}
\right] a_2-2a_3\left[ \varepsilon e^{2}+e^{7}\right] +a_4\left[
\varepsilon e^{5}+e^{10}\right] ) +a_5( \varepsilon
e^{4}+e^{8}) -
\cr
\varepsilon a_6e^1+2a_{7}( e^1+e^{3})
-a_{8}e^{5}+a_{9}e^{4}-a_{10}e^1-2e^{7}+e^{10}+e^{6},
\cr
e'^7=e^{7},
\cr
e'^8=e^{8},
\cr
e'^9=e^{9},
\cr
e'^10=a_{9}e^{8}-a_{8}e^{9}+e^{10}.
}$$

We now construct some regular invariant approximate solutions for  (11).
To this end we seek the approximate invariants for operator $\mathbf{X}$ in
the form 
\begin{equation*}
J( x,t,u,\varepsilon ) =J_0( x,t,u) +\varepsilon
J_1( x,t,u)
\end{equation*}
which are determined by equation 
\begin{equation}
\mathbf{X}J=o( \varepsilon ) .
\end{equation}
Thus (13) splits into 
$$ 
\mathbf{X}_0J_0\text{ }=0\quad \text{and}\quad \mathbf{X}_0J_1=-X_1J_0.
$$
Among other generators, (11) admits the generators 
\begin{eqnarray}
&\mathbf{Z}_1=\varepsilon ( t\partial _t-2x\partial _u) ,&\\
&\mathbf{Z}_2=( \partial _t+t\partial _u) +\mathbf{Z}_1,&\\
&\mathbf{Z}_3=( \partial _x+t\partial _u) +\varepsilon
( x\partial _x+( u+2x) \partial _u) .&
\end{eqnarray}
Operators (14)--(16) are linear combination of generators $\mathbf{Y}_2,
\mathbf{Y}_3,\mathbf{Y}_{7},\mathbf{Y}_{8}$ and $\mathbf{Y}_{9}$ given in
(9).

The operator (14) has the following functionally independent invariants: 
\begin{equation*}
h_1=x\quad\mbox{quad}\quad h_2=t\exp \left\{ \frac{u}{2x}\right\}
\end{equation*}
and the corresponding approximate invariant solution is given by 
\begin{equation*}
u=2x\ln ( \frac{y}{t}) ,
\end{equation*}
where $y$ satisfies the equation 
\begin{equation*}
y^{\prime \prime }+\frac{2}{x}y'-\frac{( y')
^{2}}{y}=\frac{\exp \left\{ -2x\frac{y'}{y}\right\} }{y}+\frac{
\varepsilon Ay}{2x}.
\end{equation*}

The functionally independent invariants 
\begin{equation*}
h_1=x+\varepsilon y_1( x,\frac{t^{2}}{2}-u) \text{ \ \ and \
\ }h_2=( \frac{t^{2}}{2}-u) +\varepsilon y_2( x,\frac{
t^{2}}{2}-u)
\end{equation*}
are determined by operator (15). Consequently, assuming that $y_1$ and $
y_2$ are equal to zero, the corresponding approximate invariant solution
is given by 
\begin{equation*}
u=\frac{t^{2}}{2}-y( x) ,
\end{equation*}
where $y$ satisfies the equation 
\begin{equation*}
y^{\prime \prime }( x) =\exp \left\{ y'( x)
\right\} -\varepsilon A.
\end{equation*}
Similarly, we find functionally independent invariants 
$$\displaylines{
h_1=t+\varepsilon g_1( t,tx-u) ,
\cr
h_2=( xt-u) +\varepsilon ( \left[ xt-u+2x\right] +g_2
\left[ t,xt-u\right] )
}$$
for the last operator (16) and thus, assuming the function $g_1$ and $
g_2 $ to be zero, find the corresponding approximate invariant solution 
\begin{equation*}
u=\frac{2\varepsilon }{1+\varepsilon }-xt+4\varepsilon A\exp \big\{ -\frac{t
}{2}\big\} +t+c,
\end{equation*}
where $A$ and $c$ are arbitrary constants.

\section{Conclusion}

In this paper a nonlinear wave equation with an infinitesimal perturbation
has been considered. The construction of the principal Lie algebra, the
equivalence transformation, the approximate principal Lie algebra, the
approximate equivalence transformation and the approximate invariant
solutions have been obtained. We have determined the function $f$ from which
the approximate principal Lie algebra extends by one and also we constructed
some approximate invariant solutions for (1).

The problem of finding the optimal system of one-dimensional subalgebras of $
L_{10}$ and the invariant solutions still remain open questions as well as
finding the Lagrangians and conservation laws for (1). We hope to return
to these questions in a forthcoming paper.

\paragraph{Acknowledgments} 
Motivation for this work was stimulated during discussion of Oron's paper
[6] with Gazizov in the light of his lectures notes [3] designed for the
seminars on Lie group analysis at the University of the Witwatersrand in
Johannesburg, South Africa.

The preparation of this paper was supported by Natural Sciences and Engineering
research Council of Canada (NSERC).

\subsection*{Appendix 1}
As an example, we show that approximate symmetries (9) leave (1)
invariant. Let us consider the last two generators 
\begin{equation*}
\mathbf{Y}_{9}=\varepsilon ( t\partial _t-2x\partial _u) 
\quad\mbox{quad}\quad\mathbf{Y}_{10}=\varepsilon ( x\partial _x+\left[
2x+u\right] \partial _u) \text{\ }
\end{equation*}
We have 
\begin{equation}
\mathbf{Y}_{9}^{\left[ 2\right] }( u_{tt}-\exp \left\{ u_x\right\}
u_{xx}-\varepsilon f( u_x) =0) \mid _{(u_{tt}=\exp
\left\{ u_x\right\} u_{xx})}  \notag
\end{equation}
\begin{equation}
=\zeta _1^{( tt) }-\exp \left\{ u_x\right\} ( \zeta
_1^{( x) }u_{xx}+\zeta _1^{( xx) }) , 
\tag{A 1}
\end{equation}
where we compute 
\begin{equation*}
\zeta _1^{( tt) }=u_{tt},\quad \zeta _1^{(
x) }=2,\quad \zeta _1^{( xx) }=-u_{xx}.
\end{equation*}
Hence the right hand side of (A1) becomes 
\begin{equation*}
u_{tt}-2\exp \left\{ u_x\right\} u_{xx}+\exp \left\{ u_x\right\}
u_{xx}\mid _{(u_{tt}=\exp \left\{ u_x\right\} u_{xx})}=0.
\end{equation*}
Similarly, 
\begin{equation}
\mathbf{Y}_{10}^{\left[ 2\right] }( u_{tt}-\exp \left\{ u_x\right\}
u_{xx}-\varepsilon f( u_x) =0) \mid _{(u_{tt}=\exp
\left\{ u_x\right\} u_{xx})}  \notag
\end{equation}
\begin{equation}
=\zeta _1^{( tt) }-\exp \left\{ u_x\right\} ( \zeta
_1^{( x) }u_{xx}+\zeta _1^{( xx) }) , 
\tag{A 2}
\end{equation}
where 
\begin{equation*}
\zeta _1^{( tt) }=-2u_{tt},\quad \zeta _1^{(
x) }=-2,\quad \zeta _1^{( xx) }=0.
\end{equation*}
Hence the right hand side of (A2) becomes 
\begin{equation*}
-2u_{tt}+2\exp \left\{ u_x\right\} u_{xx}\mid _{(u_{tt}=\exp \left\{
u_x\right\} u_{xx})}=0.
\end{equation*}
Thus the generators $\mathbf{Y}_{9}$ and $\mathbf{Y}_{10}$ leave (1)
invariant.

\subsection*{Appendix 2}
We use the relation (10) to determine non-equivalent forms of $f$.
\\
\textbf{Case 1}. If $\delta =0$, solution of (10) is given by 
\begin{equation*}
f=A\exp \left\{ \gamma \right\} u_x,
\end{equation*}
where $\gamma =\frac{c_6-c_5}{2c_6}$. Since $( 1-2\gamma )
c_6=c_5$, we obtain 
$$\displaylines{
\tau ^0=( 1-2\gamma ) c_6t+c_1,\quad \xi ^0=2(1-\gamma ) c_6x+c_2,
\cr
\eta ^0=2c_6x+2( 1-\gamma ) c_6u+c_4t+c_3,
\cr
\xi ^1=a_1x+a_2,\quad \tau ^1=a_3t+a_4,
\cr
\eta ^1=a_1u+2( a_1-a_3) +a_5t+a_6.
}$$
Thus for any $\gamma \in \mathbb{R}$ $( c_6\neq 0) $ the
extended symmetry is given by 
\begin{equation*}
\mathbf{Y}_{11}=2( 1-\gamma ) x\partial _x+( 1-2\gamma
) t\partial _t+( 2x+2\left[ 1-\gamma \right] u) \partial_u.
\end{equation*}
\\
\textbf{Case 2}. \ If $\delta \neq 0$, (10) yields 
\begin{equation*}
f=\frac{\delta }{2c_6\gamma }+2c_6\exp \left\{ \gamma \right\} u_x.
\end{equation*}
Then 
$$\displaylines{
\eta ^0=2c_6x+2c_6( 1-\gamma ) u-\gamma c_6t^{2}+c_4t+c_3,
\cr
\xi ^0=2( 1-\gamma ) c_6x+c_2,\quad \tau ^0=(
1-2\gamma ) c_6t+c_1.
}$$
Thus the eleventh symmetry is 
\begin{equation*}
\mathbf{Y}_{11}=2( 1-\gamma ) x\partial _x+( 1-2\gamma
) t\partial _t+( 2x+2\left[ 1-\gamma \right] u-\frac{\delta }{
2c_6}t^{2}) \partial _u.
\end{equation*}

\subsection*{Appendix 3}
As an example, we determine the generator $E_1$. The rest follow in the
similar manner. Let $\mu =1$ and $\lambda ,\nu =1,\dots ,10$. We write the Lie
brackets as 
\begin{equation*}
\left[ X_1,X_{\nu }\right] =c_{1\nu }^{\lambda }X_{\lambda }.
\end{equation*}
For $\nu =2$, we have 
\begin{equation*}
\left[ X_1,X_2\right] =c_{12}^{\lambda }X_{\lambda
}=c_{12}^1X_1+c_{12}^{2}X_2+\dots +c_{12}^{10}X_{10}
\end{equation*}
and so we obtain 
\begin{equation*}
c_{12}^{5}=\varepsilon \quad\mbox{quad}\quad c_{12}^{6}=-2\varepsilon .
\end{equation*}
Further we find 
$$\displaylines{
c_{15}^{5}=\varepsilon \quad ( \text{for }\nu =5) ,
\cr
c_{16}^{6}=\varepsilon \quad ( \text{for }\nu =6) ,
\cr
c_{17}^{6}=-2\quad\mbox{quad}\quad c_{17}^{2}=1\quad ( \text{for }
\nu =7) ,
\cr
c_{19}^{5}=1\quad ( \text{for }\nu =9) ,
}$$
and finally 
$c_{1,10}^{6}=1$ (for $\nu =10$) .
Thus generator (12) has the form 
\begin{equation*}
\mathbf{E}_1=( \varepsilon e^{2}+e^{7}) \partial
_{e^{2}}+( \varepsilon e^{5}+e^{9}) \partial _{e^{5}}+(
\varepsilon e^{6}-2e^{7}-2\varepsilon e^{2}+e^{10}) \partial _{e^{6}}.
\end{equation*}

\begin{thebibliography}{99}

\bibitem{a1}  Ames, W.F., Lohner, R.J., Adams, E., 
\emph{Group properties of $
u_{tt}=\left[ f( u) u_x\right] _x$}, Int. J. Nonlin.
Mech. 16, 439-447 (1981)

\bibitem{b1}  Bluman, G.W., Kumei, S., \emph{Symmetries and differential
equations}, Springer-Verlag, New York (1989)

\bibitem{g1}  Gazizov, R.K., \emph{Lecture notes on group classification of
differential equations with a small parameter}, University of the
Witwatersrand and University of North-West, South Africa (1996)

\bibitem{l1}  Lie, S., \emph{On integration of a class of linear partial
differential equations by means of definite integrals}, Arch. Math VI
(3), 1881, 328-368 (in German). Reprinted is S. Lie, Gesammelte
Abhandlundgen, Vol. 3, paper XXXV. (English translation published in CRC
handbook of Lie Group Analysis of differential equations, Vol. 2, 473-499,
Boca Raton, Florida: CRC Press. (1995)

\bibitem{o1}  Olver, P.J., \emph{Application of Lie Group Analysis to differential
equations}, Springer-Verlag, New York (1993)

\bibitem{o2}  Oron, A., Rosenau, P., \emph{Some symmetries of nonlinear heat and
wave equations}, Phys. Lett. A, 118 (4), 172-183. (1989)

\bibitem{o3}  Ovsyannikov, L.V., \emph{Group analysis of differential equations},
Academic Press, New York (1982)

\bibitem{s1}  Stephani, H., \emph{Differential equations}, Cambridge
University Press, Cambridge (1989)

\bibitem{t1}  Torrisi, M., Valenti, A., \emph{Group properties and invariant
solutions for infinitesimal transformations of a non-linear Wave Equation},
Int. J. Nonlin. Mech. 20, 135-144 (1985)

\bibitem{t2}  Torrisi, M., Tracina, R., Valenti, A., \emph{On equivalence
transformations applied to a non-linear Wave Equation}, in Modern
Group Analysis: Advanced Analytical and Computational Methods in
Mathematical Physics. Kluwer Academic Publishers, Dordrecht, 367-375 (1993) 

\bibitem{t3}  Torrisi, M., Valenti, A., \emph{Preliminary group classification 
of equation $v_{tt}=f( x,v_x) v_x+g( x,v_x) $}, 
J. Math. Phys. 32 (11) 2988 - 2995 (1991)

\end{thebibliography}

\noindent{\sc Ranis N. Ibragimov} \\
Department of Applied Mathematics\\
University of Waterloo\\
Waterloo, ON, N2L 3G1,  Canada\\
e-mail: ranis@nbnet.nb.ca

\end{document}