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            {\centerline {#2}}  }
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\def\La{\Lambda}
\def\la{\lambda}
\def\D{\Delta}
\def\J{{\cal J}}
\def\Ga{\Gamma}
\def\B{{\cal B}}
\def\pa{\partial}
\def\eps{\varepsilon}
\def\phi{\varphi}


\def\section#1{\bigskip \bigskip {\bf #1} \bigskip}
\def\remark#1{{\it Remark #1.}}



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{\nopagenumbers

\titlea{Determining discrete symmetries of differential equations}{}

\centerline{Giuseppe Gaeta}
\bigskip
\centerline{\it Department of Mathematical Sciences}
\centerline{\it Loughborough University of Technology}
\centerline{\it Loughborough LE11 3TU (United Kingdom)}
\centerline{\tt G.Gaeta@lut.ac.uk}
\bigskip\bigskip
\centerline{Miguel A. Rodr\'{\i}guez}
\bigskip
\centerline{\it Departamento de F\'{\i}sica Te\'orica}
\centerline{\it Facultad de F\'{\i}sicas, Universidad Complutense}
\centerline{\it E-28040 Madrid (Spain)}
\centerline{\tt rodrigue@eucmos.sim.ucm.es}

\footnote{}{Work partially supported by C.N.R. (Italy) under grant
203-01-62 and DGICYT (Spain) under project PB92-0197}

\vskip 1 truecm
{\bf Summary} We propose a method to determine a special class of
discrete symmetries - which we call quantized - of differential
equations, based on solution of a linear functional equation. By the
same method, we can determine differential equations possessing a
given quantized symmetry by solving a linear (standard, i.e. non
functional) PDE. Geometrical motivation and interpretation are also
given, together with simple but physically significant examples.


\vfill \eject}
\pageno=1
\section{1. Introduction}

It is well known [1-6] that knowledge of
continuous symmetries of differential equations can be of great use in
finding particular or general solutions of the equations, or in
simplifying them.

In the determination of {\it continuous} symmetries of DEs, we proceed
by identifying the DE $\D$ with a manifold $S_\D$ in the appropriate
{\it jet space} $\J$; we then look for a vector field
$\eta$ such that its prolongation $\eta_*$ is tangent to $S_\D$. In
this way we are reduced to considerations on the {\it tangent space} to
$S_\D$, i.e. to a {\it linear} problem. Indeed, the condition $\eta_* :
S_\D \to T S_\D$ gives a system of linear PDEs, the {\it determining
equations} [1-6].

Knowledge of {\it discrete} symmetries would also be of great use in
the study of DEs; unfortunately, in the determination of general
discrete symmetries we cannot reduce to study the infinitesimal action
of vector fields, and we end up in general with a nonlinear problem.

In the present note, we point out that for some class of discrete
symmetries, which we will call {\it quantized} or {\it stroboscopic}
for reasons to be clear in the following, we can still reduce to a
linear problem, although considerably more difficult than the one to be
solved for continuous symmetries. The method we propose has a clear
geometric interpretation, and indeed we will present it geometrically,
starting from the identification of $\D$ with the solution manifold
$S_\D$.

The class of discrete transformations we will study is that of
transformations obtained by finite action of vector fields; i.e., if
$\eta$ is a vector field, we will consider transformations of the form
$T_\la = e^{\la \eta}$, where $\la \in R$ is a {\it finite} parameter;
we are interested in the case $\eta$ is {\it not} a symmetry of $\D$,
but there exist values $\la_0$ such that
$T_{\la_0} : S_\D \to S_\D$ (and $T_{\la_0} \not= I $). An example of
this would be an equation which is invariant under a lattice of discrete
translation, but not under continuous translations, see below.

It should be anticipated that the determining equation we obtain by the
method proposed here is a {\it functional} equation, so that in general
we cannot hope to find the most general solution, i.e. the complete set
of discrete symmetries to a given DE. It should also be remarked that
the problem is unavoidably underdetermined: indeed, two different
vector fields $\eta$ and
$\eta '$ such that $e^{\la_0 \eta} = e^{\la_0 \eta '}$ give raise to
the same discrete transformation; we will further comment on this in
the following.

It should also be mentioned that the problem of finding continuous
symmetries for discrete equations, which is in some sense dual to the
present one (see below), has been considered by several authors
(see e.g. [7-12] and references therein); the method proposed here is
however not related to methods used in such a problem.

In the following, we will proceed as follows. We will first consider
the geometric problem of quantized transformations leaving invariant a
manifold in $R^{n+1}$, and we will write explicitely the determining
equations for these.

We will then pause to shortly discuss the geometrical interpretation of
our method from a rather abstract point of view; this could be helpful in
establishing relations with differential geometric properties of the
manifold considered. Indeed, in this language we have to determine
connections for a certain fiber bundle.

Coming back to our main idea, we will then specialize to the case of
manifold in a jet space corresponding to a DE: in this case some extra
structure is present, and this results in some simplification of the
determining equation.  Even with this, we are not able to give the
complete solution to the determining equations, but we will show that
by restricting the form of the vector field to be quantized we can give
explicit solutions. We will then consider the inverse problem: that is,
given a quantized symmetry, determine the most general DE (for assigned
order and number of variables) invariant under that symmetry.

Finally, we will consider in detail some explicit examples, focusing in
particular on simple discrete symmetries of wide use, such as
translations, rotations and scale transformations.

We will suppose the reader is familiar with the problem and language of
determining continuous symmetries of differential equations, and in
general with the symmetry theory of differential equations [1-6].

In this note we will just present the main ideas underlying the
approach we propose, and give some simple examples; a more thorough
discussion will be presented elsewhere [13].



\section{2. Quantized symmetries for manifolds}

Let us consider $X = R^n$, $Y = R^1$. Let us consider a manifold
$\Ga$ in
$M = X \times Y = R^{n+1}$ defined by the equation
$$ y = f ( x ) \eqno(1) $$ with $f: X \to Y$ a smooth function.

Let us further consider a vector field in $M$,
$$ \eta = \phi^i (x , y) \pa_i + \psi (x , y) \pa_y \eqno(2) $$ Here and
in the following
$$ \pa_i \equiv {\pa \over \pa x^i } \eqno(3) $$ and summation over
repeated indices is understood.

The infinitesimal action $e^{\eps \eta}$ of $\eta$ in $M$ maps
$(x ,y)$ to a new point $(x ' , y')$, where
$$ x ' = x + \eps \phi (x ,y) \kern1cm ; \kern 1 cm y' = y + \eps
\psi (x ,y) \eqno(4) $$ so that the graph of $f$, $\Ga = \{ (x, f(x ) )
\}$ is tranformed into a new curve
$\Ga_\eps = \{ (x , f_\eps (x ) ) \}$ which is the graph of the function
$$ f_\eps (x ) = f(x ) + \eps [\psi (x ,y) - (\phi^i (x ,y)
\pa_i ) f(x ) ] \eqno(5) $$

Let us now introduce the function $F : R \times X \to R$ such that
$F( \la ,x ) = f_\la (x) $ is the transform of $f(x )$ under $e^{\la
\eta}$. Clearly, the infinitesimal transformation (5) yields that this
$F$ satisfies the {\it determining equation}
$$ {\pa F(\la ,x) \over \pa \la} + \phi^i (x,F(\la ,x)) {\pa F(\la ,x)
\over \pa x^i} = \psi (x, F(\la ,x) ) \eqno(6) $$ with initial condition
$$ F(0,x) = f(x) \eqno(7) $$

If $\eta$ is a continuous symmetry of $f$, then $\Ga$ must be invariant
under $\eta$, i.e. $f_\eps (x) = f(x)$, or equivalently
$( \pa F / \pa \la ) = 0$. Indeed in this case (6) gives just the usual
determining equations for symmetries of (1).

As mentioned in the introduction, we are interested in the case
$\eta$ is not a symmetry of (1), but there is a special value
$\la_0$ such that $f_{\la_0} = f$, i.e. such that
$$ F( \la_0 ,x ) = F (0,x) \eqno(8) $$

In other words, we are interested in determining $\eta$ such that the
solution to (6) with the initial condition (7) is {\it periodic} in
$\la$ (excluding the trivial case $\pa F / \pa \la = 0$). In this case,
$\La = e^{\la \eta} : M \to M$ maps $\Ga$ into itself and therefore
qualifies as a discrete symmetry of (1), or equivalently of $\Ga$.

It is now clear why we call such a symmetry, given by finite
action of a vector field which is not a continuous symmetry of (1)
itself, a quantized or stroboscopic symmetry, as anticipated in the
Introduction. It should be stressed that we should also require that
$\La \vert_{\Ga}$ is not the identity, or we would have a trivial
discrete symmetry.

Notice that, multiplying $\eta$ by a numerical constant, we can
always set $\la_0 = 1$, or $\la_0 = 2 \pi$. Thus, in the following we
will in general consider $\la_0$ as fixed. The ($2 \pi$) periodicity
requirement in $\la$ would suggest to expand $F(\la ,x)$ as a Fourier
series in $\la$. This is indeed possible, but is concretely useful only
if (6) is linear, as it will be discussed in detail in [13].

We stress that in (6) we have to determine not only
$F$, but $\phi$ and $\psi$ as well. Thus, we are not dealing with a
normal PDE, but with a functional PDE. The only data are the initial
condition (7) and the (arbitrary) $\la_0
\not= 0$ appearing in (8). We could - and will - consider the inverse
problem of determining the manifolds $\Ga$ which are invariant under
the action of a given prescribed symmetry. In this case $\phi , \psi$
are given, and (6) is a regular PDE, not a functional one. It is not
surprising that this inverse problem is much easier than the direct
one, as we will see in the following.

Although we have considered quite special $X,Y,M$ and
$\Ga$ for ease of notation, it is simple to generalize the above
discussion to the case $X$ and $Y$ are smooth submanifolds in real
spaces; for our purposes we will not need to consider the case
$\Ga$ is a generic smooth submanifold in $M = X \times Y$, and only need
to consider $\Ga$ as the graph of a function $f$ in an appropriate space.

It should be stressed that in introducing the function $F(\la ,x)$ we
are implicitely also passing to consider a vector field $\eta '$
associated to $\eta$ and acting in $M' = R \times M$, given explicitely
by
$$ \eta ' = \eta + \pa_\la \equiv \phi^i \pa_i + \psi \pa_y +
\pa_\la \eqno(9) $$

Then the graph $\Theta = {(x,y,\la ) : y = F(\la , x) } \subset M'$ of
$F(\la ,x )$ is by construction an invariant submanifold of
$M'$ under $\eta '$. We are therefore reconducted to the problem of
determining tangent vector field to a manifold, but now the manifold is
not given a priori, but depends itself on the vector field.

\section{3. A geometrical interpretation}

We would like to stress that $M'$ can be naturally seen as the total
space of a fiber bundle $B$ with base $R$ (corresponding to the $\la$),
fiber $M$ and projection $\pi :(\la, x,y) \to \la$;
$\eta '$ is then a connection on $B$. When we are looking for solutions
to (6), (7) which moreover satisfy the periodicity condition (8) - i.e.
we are looking for quantized symmetries - we can consider the analogous
bundle $\B$ with $S^1$ as base space,
$M$ as fiber and the same projection $\pi$.

Our problem can then be described in differential geometric language as
the search for a connection $\nabla$ on $\B$ such that there is a
section $\Theta$ invariant under this connection and such that the
restriction of $\Theta$ to $\pi^{-1} (0)$ is the prescribed $\Ga$.

In the above language, the determination of manifolds invariant under a
given quantized symmetry (the "inverse" problem, to be discussed below)
amounts to the determination of sections of $\B$ invariant under a given
connection
$\nabla$. Again, it is obvious that this is much easier than the
"direct" problem, although in general it is not trivial at all.

It should be stressed that although $\Theta$ is an invariant
manifold under the connection $\nabla$, this does not imply that
transporting a point $(x,f(x))$ around the base space
$S^1$ by $\nabla$ we get the same point. In general, we get a point $(x'
,f (x' ))$ with $x' \not= x$, and the discrete transformation $\La : \Ga
\to \Ga$ is related to the holonomy of the connection $\nabla$.

%\vfill\eject

\section{4. Quantized symmetries of differential equations}

As already recalled, a DE $\D$ is naturally identified with a manifold
$S_\D$ in an appropriate jet space $\J$ [1,5]. Therefore, if $\Ga$ is
$S_\D$ and $M$ is $\J$, the theory developed in the previous section can
be applied to differential equations as well.

However, the fact that we are dealing with jet spaces makes that an
extra structure (the contact structure) is now present. Due to this, we
have some extra constraint on the functions appearing in (6): e.g.,
$\eta$ should now be the prolongation to $\J$ of an underlying Lie-point
vector field $\eta_0$ acting in the space
$M_0$ of independent and dependent variables; due to this the
$\phi^i$ and
$\psi$ are not arbitrary smooth functions, but must satisfy some
relations, as we are now going to discuss.

Let us first consider the case of an autonomous ODE
$$ u_t = f(u) \eqno(10) $$ and time-independent Lie-point vector field
$$ \eta_0 = \phi (u) \pa_u \eqno(11) $$

Now $M_0 = R^2 = \{ (t,u)\}$, and (10) is identified with a manifold in
$\J = M = R^3 = \{ t , u , u_t \}$. The prolongation of $\eta_0 : M_0
\to T M_0$ to $M$ is given by
$$ \eta = \phi (u) \pa_u + \Phi (u,u_t ) \pa_{u_t} \eqno(12') $$ where
$\Phi = \phi_u u_t$ by the prolongation formula [1-6], so that on $\Ga$
we have
$$ \Phi = \phi_u f(u) \eqno(12'') $$

Now $u_t$ plays the role that was of $y$ in section 2, and $\Phi$
corresponds to $\Psi$; we get (6) in the form
$$ {\pa F(\la , u) \over \pa \la} + \phi (u) {\pa F(\la , u) \over
\pa u} = \Psi = \phi_u F(\la , u) \eqno(13) $$
(we have taken into account $t$-independence). Thus, $\Psi$ is now
determined by $\phi$, and we have only one arbitrary function in
$\eta$. Notice also that while the $\phi$ in the case of section 2 could
depend on $y$, now the request that $\eta_0$ be a Lie-point vector field
ensures that $\phi$ does not depend on $u_t$, and we end up with a
determining equation linear in $F$.

Thus, dealing with differential equations rather than algebraic
manifolds gives indeed a somewhat simpler problem !

It is worth considering also the case of nonautonomous ODEs and time
dependent vecor fields, i.e.
$$ u_t = f(t,u) \eqno(14) $$
$$ \eta_0 = \tau (t,u) \pa_t + \phi (t,u) \pa_u \eqno(15) $$
In this
case $\eta$ is still given by (12'), but the prolongation formula yields
$\Phi = \phi_u u_t - \tau_t u_t - \tau_u (u_t )^2$, and on $\Ga$ we have
$$ \Phi = \phi_t + [ \phi_u  - \tau_t ] f(t,u) - \tau_u [f(t,u)]^2
\eqno(16) $$
so that we end up with the determining equation
$$ {\pa F \over \pa \la} + \tau (t,u) {\pa F \over \pa t} + \phi (t,u)
{\pa F \over \pa u} = \Psi = \phi_t + [ \phi_u  - \tau_t ] F -
\tau_u F^2 \eqno(17) $$

It should be stressed that for higher order ODEs the right hand side
will not contain terms of order higher than one in $F$. This follows at
once from the recursive structure of prolongation coefficients, i.e. by
the general prolongation formula [1-6].

In the same way it is also easy to see, again by the prolongation
formula [1-6], that for  general DEs - ordinary or partial - of order
$n$ we will have on the left hand side a differential operator whose
coefficients do not depend on
$F$ (while in the case of algebraic manifolds, see section 2, they could
depend on $F$), applied to $F$, and on the right hand side an expression
which contains terms of order not higher than one in
$F$ if $n \not= 1$, and not higher than two in $F$ if $n=1$.

We would like to point out that in the study of continuous Lie-point
symmetries of first order ODEs such as (10), the determining equation
is just $\phi f_u - f \phi_u = 0$. In the case of discrete equations (see
13), we get an equation of this form, but with $\pa F / \pa \la$ on the
right hand side. Notice that indeed if $\pa F / \pa \la = 0$, then
$\eta_0$ is a continuous symmetry of (10).

The fact that first order differential equations lead to
more difficult determining equations than higher order ones, should not
be a surprise. Indeed, the same happens also in the case of determining
equations for continuous Lie-point symmetries [1,4,5].

The general procedure for writing the determining equations for
discrete symmetries of higher order ODEs or evolution PDEs should by now
be clear; it amounts to repeat the procedure illustrated in section 2
for manifolds and taking into account that $\eta : M \to T M$ is now
the prolongation of an $\eta_0 : M_0 \to T M_0$. We stress that this
only requires to apply the general prolongation formula [1-6].

%\vfill\eject
\section{5. Some special cases}

We will now consider shortly, but explicitely, the case of second order
ODEs and of evolution PDEs of order one or two in the spatial
derivatives.

{\it Autonomous second order ODEs}

For an autonomous second order ODE
$$ u_{tt} = f(u,u_t ) \eqno(18) $$
and time independent $\eta_0$ we have
$\eta_0$ as in (11), but now
$\eta$ is the second prolongation of $\eta_0$, i.e.
$$ \eta = \phi \pa_u + \phi_u u_t \pa_{u_t} + [ \phi_{uu} (u_t )^2 +
\phi_u u_{tt} ] \pa_{u_{tt}} \eqno(19) $$
and the determining equation is therefore
$$ { \pa F \over \pa \la } + \phi { \pa F \over \pa u} + {\pa \phi
\over \pa u} u_t { \pa F \over \pa u_t } = {\pa^2 \phi \over \pa u^2}
(u_t )^2 + {\pa \phi \over \pa u} F \eqno(20) $$
where $\phi = \phi (u)$ and $F = F(\la , u, u_t )$.

{\it First order PDEs}

For a first order PDE
$$ u_t = f(t,x,u,u_x ) \eqno(21) $$
with $x \in R^q$, we write
$$ \eta_0 = \tau (t,x,u) \pa_t + \xi^j (t,x,u) \pa_j + \phi (t,x,u)
\pa_u \eqno(22) $$
where $j$ runs from $1$ to $q$, sum over repeated
indices is understood, and $\pa_j = \pa / \pa x^j$. The first
prolongation of this yields
$$\eqalign{
\eta = \tau (t,x,u) \partial_t + \xi^j (t,x,u) \partial_j + \phi
(t,x,u) \partial_u & \cr + ( \phi_t + \phi_u u_t - \tau_t u_t - \tau_u
(u_t )^2 - \xi^j_t u_j - \xi^j_u u_j u_t ) \partial_{u_t} & \cr + (
\phi_j + \phi_u u_j - \tau_j u_t - \tau_u u_t u_j - \xi^i_j u_i -
\xi^i_u u_i u_j ) \partial_{u_j}} \eqno(23) $$
and therefore the determining equation is
$$\eqalign{ {\partial F \over \partial \lambda} + \tau {\partial F \over
\partial t} + \xi^j {\partial F \over \partial x^j} + \phi {\partial F
\over \partial u} + ( \phi_j + \phi_u u_j - (\tau_j + \tau_u u_j ) F -
\xi^i_j u_i - \xi^i_u u_i u_j ) {\partial F \over \partial u_j} & \cr =
\phi_t - \xi^j_t u_j + ( \phi_u - \tau_t - \xi^j_u u_j) F - \tau_u F^2 }
\eqno(24) $$

Notice that now $F$ appears also in the coefficients of
$F$ derivatives in the left hand side of the determining equation,
contrary to what happens in the case of ODEs.

{\it Autonomous first order PDEs}

For autonomous first order PDEs
$$ u_t = f (u,u_x) \eqno(21') $$
and $\eta_0$ independent of $x$ and $t$,
$$ \eta_0 = \phi (u) \pa_u \eqno(22') $$ we get
$$ \eta = \phi (u) \pa_u + (\phi_u u_x ) \pa_{u_x} + (\phi_u u_t )
\pa_{u_t} \eqno(23') $$ and the determining equation is
$$ {\pa F \over \pa \la} + \phi {\pa F \over \pa u} + (\phi_u u_x ) {\pa
F \over \pa u_x} = \phi_u F  \eqno(24') $$

We stress that in this case the coefficient of $F$ derivatives in the
l.h.s. of the determining equation do not depend on $F$, and the
determining equation is {\it linear}. By looking at the prolongation
formula, we see at once that this is a general fact; i.e., for
autonomous PDEs and $\eta_0$ of the form (22'), we always get linear
determining equations.

{\it Autonomous second order evolution PDEs}

In the case of autonomous second order evolution PDEs,
$$ u_t = f(u,u_x,u_{xx}) \eqno(25) $$
and $\eta_0$ independent of $x,t$,
i.e. as in (24'). In this case we get
$$ \eta = \phi \pa_u + \phi_u u_x \pa_{u_x} + ( \phi_{uu} (u_x )^2 +
\phi_u u_{xx} ) \pa_{u_{xx}} + \phi_u u_t \pa_{u_t}
\eqno(23'') $$ and the determining equation is therefore
$$ {\pa F \over \pa \la} + \phi {\pa F \over \pa u} + (\phi_u u_x ) {\pa
F \over \pa u_x} + ( \phi_{uu} (u_x )^2 + \phi_u u_{xx} ) {\pa F \over
\pa u_{xx} } = \phi_u F  \eqno(24'') $$

{\it A general remark}

We stress that if we have an autonomous evolution equation, the general
form of $\eta_0$ transforming it into autonomous evolution equations is
just $\eta_0 = \tau (t) \pa_t +  \xi^j (x) \pa_j +
\phi(t,x,u) \pa_u$ [5,14]. Notice that for autonomous equations, the
time and space translations correspond to continuous symmetries, so are
not interesting in the present setting. Notice also that the above
statement about linearity of the determining equations for autonomous
PDEs does also apply to $\eta_0 = \phi (t,x,u) \pa_u$.

%\vfill \eject
\section{6. Differential equations with prescribed discrete symmetries}

As already stressed, our main equation (13) (or (6) in the case of
general manifolds) yielding the condition to be satisfied to have a
quantized symmetry, is a {\it functional} equation, as it requires to
determine both the $F(\la , x)$ which is the "transported" of $f$
under the vector field, {\it and} the $\phi$ describing the vector
field (12) to be quantized to give a symmetry.

Our task is considerably simpler if we consider the ``inverse
problem'', i.e. if we assign a priori a vector field, namely the
$\phi$, and we ask to determine the differential equations which admit
this as a quantized symmetry.

In this case we have to determine the $f$ such that there exist a $F
(\la , u)$ periodic (in $\la$) solution of (13) with initial datum
$F(0,u) = f(u)$, namely the ``initial data'' in the space of
differential equation which lead to periodic evolution under the vector
field $\pa_\la + \phi (u) \pa_u$. Although this is not a completely
standard problem, it is nevertheless more tractable than the ``direct
problem'', as it will be concretely shown also by the examples in the
following section.

Indeed, if $\phi$ are given, then (13) - as an equation for $F$ - is a
quasilinear first order PDE, which can be solved in a standard way by
the method of characteristics. Notice that for the ``direct problem'',
eq.(13) can still be formally solved by characteristics, but now we
should have also to determine the characteristic vector field (hence,
the $\phi$) at the same time as $F$.

\section{7. Examples}

We will present here some simple examples, dealing with symmetries of
direct relevance in Physics; we will consider the case of
(discrete) translations, (discrete) rotations, (discrete) scale transformations,
and (discrete) conformal transformations. We will limit to consider first order
autonomous ODEs, $u_t = f(u) $, for the sake of simplicity, and we look for a
function $f(u)$ such that the equation admits a discrete symmetry, with
associated vector field $\eta_0 = \varphi (u) \pa_u $. The case of PDEs would
present the same kind of procedure; examples of this will be presented elsewhere
[13].


{\it A) Translations}

As the first -- and simplest -- example, we just consider a translation,
$ \eta_0 = \pa_u $. The determining equation (13) is now
$$ {\pa F \over \pa \la} +  {\pa F \over\pa u}  = 0 \eqno(26) $$

The general solution of this equation is
$$ F(\la , u ) = h (\la - u ) \eqno(27) $$
with $h$ a periodic function. The function $f(u)$ in our differential
equation is
equal to $F( \la , u)$ for $\la = 0$, i.e. we just obtain (with a sign change)
$$ u_t = h(u) \eqno(28) $$
with $h$ periodic, e.g. $h(u) = \sin (u)$.

The result in this case is pretty obvious, but we see how the construction of
this type of differential equations can be carried out in a straightforward way
(at least in the simplest cases).

\bigskip
{\it B) Rotations}

An example of nonlinear determining equations is given by discrete rotations in
the $(u,t)$ plane, i.e; by $\eta_0 = t \pa_u - u \pa_t$. In this case, the
determining equation is
$$ {\pa F \over \pa \la} - u {\pa F \over \pa t} + t {\pa F \over \pa u} \ = \ 1
+ F^2 \eqno(29) $$
The two invariants of this are
$$ z = \la - \arcsin \left( {u / \sqrt{t^2 + u^2} } \right) ~~;~~ w = t^2 +
u^2 \eqno(30) $$
and the general solution is then
$$ F(\la , t , u) = {u + h(z,w) \ t \over t - h(z,w) \ u } \eqno(31) $$
where again $h$ should be periodic in $\la$, i.e. in $z$. By taking e.g. $h(z,w)
= \sin (k z)$ (with $k \not= 0,1$) we have the differential equation
$$ u_t = {u + t \sin [k \ \arcsin (u / \sqrt{t^2 + u^2 } ) ] \over
t - u \sin [k \ \arcsin (u / \sqrt{t^2 + u^2 } ) ]} \eqno(32) $$
which is indeed invariant under the discrete rotation of angle $ \alpha = (2
\pi / k ) $,
$$ u \to u \cos (\alpha ) - t \sin (\alpha ) ~~;~~ t \to u \sin (\alpha ) +
t \cos
(\alpha)  \eqno(33) $$


\bigskip
{\it C) Dilations}

The next example we consider is that of a  dilation, i.e. $\eta_0=u\pa_u$; then
the determining equation is
$$ {\pa F \over \pa \la} +  u{\pa F \over
\pa u}  = F(\la,u) \eqno(34) $$
with general solution
$$F(\la,u)=h(ue^{-\la})u \eqno(35) $$
in which again $h$ should be periodic (but not constant) in $\la$, and $f(u)$
will be given by $f(u) = u h(u)$. We can take e.g.
$$ h (z) = \sin (2\pi-\log z) \eqno(36)$$ which yields
$$f(u) = - u \sin (\log u) \eqno(37)$$

\bigskip
{\it D) Conformal Transformations}

In the last example we consider a conformal transformation, i.e. we use a
nonlinear vector field, $\eta_0=u^2\pa_u$. The deermining equation
is in this case
$${\pa F \over \pa \la} +  u^2{\pa F \over \pa u}  = 2uF(\la,u)
\eqno(38)$$
with solution
$$F(\la,u)=h\left(\la+{1\over u}\right)u^2 \eqno(39) $$
where again $h$ should be periodic (but not constant) in $\la$.
Choosing, e.g.,
$$h(z) = \sin (z) \eqno (40)$$
we have for $f(u)$, i.e. for the differential equation,
$$f(u)=u^2\sin \left({1\over u}\right)\eqno(41)$$

%\vfill \eject
\section{8. Some final remarks}

We would like at this point to present some short remarks about the applications
and possible extensions of the method proposed here. Some of these will be
developed in a future work [13].

While continuous symmetries allow for a reduction in the order of an ODE,
this is
not the case for a discrete symmetry (just think of a system invariant under
left-right inversion), so that in this sense the information about discrete
symmetries is less valuable than the one on continuous symmetries. On the other
side, some kind of symmetry reduction can still be applied even for discrete
symmetries.

First of all, in the case of ODEs a discrete symmetry implies in general the
invariance (under the flow of the ODE) of certain subspaces, pointlike invariant
under the symmetry [15]; one can then consider the restriction of the ODE
to such
subspaces, and on this apply the usual symmetry reduction.

Moreover, in the case of PDEs a continuous symmetry does not allow to reduce the
equation, but only to consider specific classes of solutions, invariant
under the
given symmetry. From this point of view, discrete symmetries are equivalent to
continuous ones, in the sense we can still look for solutions which are
invariant
under the given (discrete) symmetry of the equation.

As it is well known (see e.g. [6]), it is possible that a given solution (or a
given set of solutions) is also invariant under a transformation which is not a
symmetry of the equation; one says then that this transformation is a {\it
conditional symmetry} [6,7] of the equation. Here again, discrete
symmetries would
play the same role as continuous ones, in that one can consider discrete
conditional symmetries and determine solutions which are invariant under them.

It is clear that in the present work we have considered only discrete symmetries
corresponding to diffeomorphisms (in the base space $M$) which are
homotopic to the
identity, i.e. which can be obtained as a continuous deformation of the
identity,
so that we cannot deal with general discrete tranformations. On the other side,
if we consider the different homotopy classes of diffeomorphisms of $M$ and
fix a
representative $g_0$ -- simple enough, e.g. an inversion -- for each of
these, we
could then apply the ideas of the present work to discrete transformations which
are obtained as continuous deformations of this $g_0$.

Finally, we would like to mention that in recent years some attention has been
devoted to a problem which is in some sense dual to the one considered
here, i.e.
to the study of continuous symmetries of discrete equations [8-12]; the
problem of
determining the differential equations which admit a given discrete symmetry can
simply be mapped into this one by reversing the role of the discrete map (to be
considered as the time evolution rather than the symmetry) and of the continuous
equation (to be considered as the symmetry rather than the time evolution law).
Notice, however, that not all the differential equations can be considered as
(Lie-point) vector fields, and not all the discrete maps describing time
evolution are of the class considered here.


\vfill\eject
\section{9. Conclusions}

We have discussed "quantized" symmetries of manifolds in $R^n$, and
found the equations these have to satisfy. On the basis of this
discussion we have considered the case of the manifolds identifying a
differential equation in the appropriate jet space, and found the
corresponding equations to be satisfied; these are the determining equations
for discrete (quantized) symmetries of a differential equation.

Such equations are functional ones; they can also be used to determine
the differential equations possessing a given discrete (quantized)
symmetry, in which case they are standard, i.e. non functional, linear
PDEs.

We have considered some simple cases, discussing quantized
translations, dilations, and conformal transformations; these examples
show that our method is concretely viable.

The computation of discrete symmetries of differential equations is a
very interesting problem from the point of view of explicit solutions
and properties of these equations. The method we present here allows a
systematic approach to this problem using well known techniques from
the computation of continuous symmetries. Although the determining
equations are very undetermined and not very easy to solve, they can be
used to find explicit discrete symmetries or particular equations having
these symmetries, and provide insights into the structure of
differential equations with these properties.

In a future work [13] we will present a more complete analysis
together with some further results and applications to integrable
models.

\vfill

\section{Acknowledgements} The work of M.A.R. was supported in part by
DGICYT (Spain) under project PB92-0197. The research described in this
work was performed during a six months visit of G.G. at the Departamento
de Fisica Teorica of the Universidad Complutense in Madrid, made
possible by the italian C.N.R. through grant 203-01-62. G.G. would like
to thank the researchers and the personnel of the Departamento de Fisica
Teorica for the warm hospitality extended to him in this occasion.

\vfill\eject

\section{References}

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[5] G. Gaeta, {\it Nonlinear symmetries and nonlinear equations};
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[6] P. Winternitz, ``Lie groups and solutions of nonlinear partial
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[15] M. Golubitsky, D. Schaeffer and I. Stewart, {\it Singularities and
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\bye