In general, any solution of (2) will be of the form w=w(t,x,\epsilon ) and the one-parameter Lie group of infinitesimal transformations in the (t,x,w)-space of equation (2) can be considered in the following form:

\stackrel{\u02c6}{t}=t+a{\xi}^{1}(t,x,w(t,x,\epsilon ),\epsilon )+\mathcal{O}\left({a}^{2}\right),

(4)

\stackrel{\u02c6}{x}=x+a{\xi}^{2}(t,x,w(t,x,\epsilon ),\epsilon )+\mathcal{O}\left({a}^{2}\right),

(5)

\stackrel{\u02c6}{w}=w+a\eta (t,x,w(t,x,\epsilon ),\epsilon )+\mathcal{O}\left({a}^{2}\right),

(6)

where *a* is the group parameter.

Let us suppose that w(t,x,\epsilon ) and \stackrel{\u02c6}{w}(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x},\epsilon ), analytic in *ε*, can be expanded in power series of *ε*, *i.e.*,

w(t,x,\epsilon )={w}_{0}(t,x)+\epsilon {w}_{1}(t,x)+\mathcal{O}\left({\epsilon}^{2}\right),

(7)

\stackrel{\u02c6}{w}(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x},\epsilon )={\stackrel{\u02c6}{w}}_{0}(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x})+\epsilon {\stackrel{\u02c6}{w}}_{1}(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x})+\mathcal{O}\left({\epsilon}^{2}\right),

(8)

where {w}_{0} and {w}_{1} are some smooth functions of *t* and *x*; {\stackrel{\u02c6}{w}}_{0} and {\stackrel{\u02c6}{w}}_{1} are some smooth functions of \stackrel{\u02c6}{t} and \stackrel{\u02c6}{x}.

Upon formal substitution of (7) in (2), equating to zero the coefficients of zero and first degree powers of *ε*, we arrive at the following system of PDEs:

{L}_{0}:={{w}_{0}}_{tt}-f({{w}_{0}}_{x}){{w}_{0}}_{xx}=0,

(9)

\begin{array}{c}{L}_{1}:={{w}_{1}}_{tt}-f({{w}_{0}}_{x}){{w}_{1}}_{xx}-{f}^{\prime}({{w}_{0}}_{x}){{w}_{0}}_{xx}{{w}_{1}}_{x}\hfill \\ \phantom{{L}_{1}:=}-{\lambda}^{\prime}({{w}_{0}}_{x}){{w}_{0}}_{xx}{{w}_{0}}_{tx}-\lambda ({{w}_{0}}_{x}){{w}_{0}}_{xxt}=0,\hfill \end{array}

(10)

where we have set

\begin{array}{c}f({{w}_{0}}_{x})=f({w}_{x}){|}_{\epsilon =0},\phantom{\rule{2em}{0ex}}{f}^{\prime}({{w}_{0}}_{x})={f}^{\prime}({w}_{x}){|}_{\epsilon =0},\hfill \\ \lambda ({{w}_{0}}_{x})=\lambda ({w}_{x}){|}_{\epsilon =0},\phantom{\rule{2em}{0ex}}{\lambda}^{\prime}({{w}_{0}}_{x})={\lambda}^{\prime}({w}_{x}){|}_{\epsilon =0}.\hfill \end{array}

Hence, {w}_{0} is a solution of nonlinear wave equation (9) which we call *unperturbed equation*, while {w}_{1} can be determined from the linear equation (10).

In order to have a one-parameter Lie group of infinitesimal transformations of the system (9)-(10), which is consistent with the expansions of the dependent variables (7) and (8), we introduce these expansions in the infinitesimal transformations (4)-(6). Upon formal substitution, equating to zero the coefficients of zero and first degree powers of *ε*, we get the following one-parameter Lie group of infinitesimal transformations in the (t,x,{w}_{0},{w}_{1})-space

\stackrel{\u02c6}{t}=t+a{\xi}_{0}^{1}(t,x,{w}_{0})+\mathcal{O}\left({a}^{2}\right),

(11)

\stackrel{\u02c6}{x}=x+a{\xi}_{0}^{2}(t,x,{w}_{0})+\mathcal{O}\left({a}^{2}\right),

(12)

{\stackrel{\u02c6}{w}}_{0}={w}_{0}+a{\eta}_{0}(t,x,{w}_{0})+\mathcal{O}\left({a}^{2}\right),

(13)

{\stackrel{\u02c6}{w}}_{1}={w}_{1}+a[{\eta}_{10}(t,x,{w}_{0})+{\eta}_{11}(t,x,{w}_{0}){w}_{1}]+\mathcal{O}\left({a}^{2}\right),

(14)

where we have set

{\xi}_{0}^{i}(t,x,{w}_{0})={\xi}^{i}(t,x,w(t,x,\epsilon ),\epsilon ){|}_{\epsilon =0},\phantom{\rule{1em}{0ex}}i=1,2,

(15)

{\eta}_{0}(t,x,{w}_{0})=\eta (t,x,w(t,x,\epsilon ),\epsilon ){|}_{\epsilon =0},

(16)

{\eta}_{10}(t,x,{w}_{0})+{\eta}_{11}(t,x,{w}_{0}){w}_{1}=\frac{d\eta}{d\epsilon}{|}_{\epsilon =0}.

(17)

We give the following definition: We call *approximate symmetries* of equation (2) the (exact) symmetries of the system (9)-(10) through the one-parameter Lie group of infinitesimal transformations (11)-(14). Consequently, the one-parameter Lie group of infinitesimal transformations (11)-(14), the associated Lie algebra and the corresponding infinitesimal operator

\begin{array}{rcl}X& =& {\xi}^{1}(t,x,{w}_{0})\frac{\partial}{\partial t}+{\xi}^{2}(t,x,{w}_{0})\frac{\partial}{\partial x}+\eta (t,x,{w}_{0})\frac{\partial}{\partial {w}_{0}}\\ +[{\eta}_{10}(t,x,{w}_{0})+{\eta}_{11}(t,x,{w}_{0}){w}_{1}]\frac{\partial}{\partial {w}_{1}}\end{array}

(18)

are called the approximate Lie group, the approximate Lie algebra and the approximate Lie operator of equation (2), respectively.

Moreover, after putting

{X}_{0}={\xi}_{0}^{1}(t,x,{w}_{0})\frac{\partial}{\partial t}+{\xi}_{0}^{2}(t,x,{w}_{0})\frac{\partial}{\partial x}+{\eta}_{0}(t,x,{w}_{0})\frac{\partial}{\partial {w}_{0}},

(19)

the approximate Lie operator (18) can be rewritten as

X={X}_{0}+[{\eta}_{10}(t,x,{w}_{0})+{\eta}_{11}(t,x,{w}_{0}){w}_{1}]\frac{\partial}{\partial {w}_{1}}

(20)

and {X}_{0} can be regarded as the infinitesimal operator of unperturbed nonlinear wave equation (9) (or (3)).

It is worthwhile noticing that, thanks to the functional dependencies of the coordinates of the approximate Lie operator (18) (or (20)), now we are able to work in hierarchy in finding the invariance conditions of the system (9)-(10): firstly, by classifying unperturbed nonlinear wave equation (9) through the operator (19) and after by determining {\eta}_{10} and {\eta}_{11} from the invariance condition that follows by applying the operator (20) to the linear equation (10). In fact the invariance condition of the system (9)-(10) reads

{X}_{0}^{(2)}({L}_{0}){|}_{{L}_{0}=0}=0,

(21)

{X}^{(3)}({L}_{1}){|}_{{L}_{0}=0,{L}_{1}=0}=0,

(22)

where {X}_{0}^{(2)} and {X}^{(3)} are the second and third extensions of the operators {X}_{0} and *X*, respectively.