\documentclass{article}
\def\giorno{12 August 2004}
\def\nome{CGM2.tex}
\def\pn{\par\noindent}
\def\.#1{\dot #1}
\def\D{{\cal D}}
\def\G{{\cal G}}
\def\H{{\cal H}}
\def\I{{\cal I}}
\def\J{{\cal J}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\R{{\bf R}} %% reals
\def\S{{\cal S}}
\def\T{{\rm T}}
\def\V{{\cal V}}
\def\PT{\~\Phi}
\def\XT{\~X}
%%%%%%% notaz matem
\def\sse{\subseteq}
\def\ss{\subset}
\def\pa{\partial}
\def\pd{\partial}
\def\=#1{{\widetilde #1}}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
\def \wt#1{{\widetilde #1}}
\def\sse{\subseteq}
\def\eb{{\bf e}}
\def\d{{\rm d}} %% derivative
\def\xd{{\dot x}}
\def\yd{{\dot y}}
\def\grad{\nabla} %% gradient
\def\lapl{\triangle} %% laplacian
\def\cd{\cdot}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
%%%%%%%% greco
\def\a{\alpha}
\def\al{\alpha}
\def\b{\beta}
\def\be{\beta}
\def\g{\gamma}
\def\ga{\gamma}
\def\de{\delta} %% NON ridefinire come \d !!!!
\def\eps{\varepsilon}
\def\phi{\varphi}
\def\la{\lambda}
\def\La{\Lambda}
\def\s{\sigma}
\def\om{\omega}
\def\vth{\vartheta}
\def \ep{\varepsilon}
\def \eps{\ep}
\def\Om{\Omega}
\def\phi{\varphi}
\def\Ga{\Gamma}
\def\De{\Delta}
\def\th{\theta}
\def\z{\zeta}
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}} }}
\def\mapleft#1{\smash{\mathop{\longleftarrow}\limits^{#1}}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\w{\wedge}
\def\interno{\hskip 2pt \vbox{\hbox{\vbox to .18 truecm{\vfill\hbox to
.25 truecm {\hfill\hfill}\vfill}\vrule}\hrule}\hskip 2 pt}
\def\EOP{~\hfill $\diamondsuit$} % end of proof
\def\EOD{~\hfill $\clubsuit$} % end of definition
\def\EOR{~\hfill $\odot$} % end of remark
\begin{document}
\title{\bf On the relation between standard and $\mu$-symmetries for
PDEs}
\author{Giampaolo Cicogna\footnote{Dipartimento di Fisica ``E.
Fermi'', Universit\`a di Pisa, and INFN Sez. di Pisa, Largo B.
Pontecorvo 2, I--56127 Pisa (Italy); e-mail: cicogna@df.unipi.it},
Giuseppe Gaeta\footnote{Dipartimento di Matematica, Universit\`a di
Milano, via Saldini 50, I--20133 Milano (Italy); e-mail:
gaeta@mat.unimi.it}, and Paola Morando\footnote{Dipartimento di
Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24,
I--10129 Torino (Italy); e-mail: paola.morando@polito.it} }
\date{ \giorno }
\maketitle
\noindent {\bf Summary.}
We give a geometrical interpretation of the notion of
$\mu$-prolongations of vector fields and of the related concept of
$\mu$-symmetry for partial differential equations (extending to PDEs
the notion of $\lambda$-symmetry for ODEs). We give in particular a
result concerning the relationship between $\mu$-symmetries and
standard exact symmetries. The notion is also extended to the case of
conditional and partial symmetries, and we analyze the relation between
local $\mu$-symmetries and nonlocal standard symmetries.
\par\noindent
{\tt PACS: 02.20.-a ; 02.30.Jr \ . \ MSC: 58J70 ; 35A30 \ . }
\section*{Introduction}
Symmetry analysis and reduction methods are well established tools to
study nonlinear differential equations, see e.g.
\cite{Gae,Kra,Olv1,Olv2,Ste,Win}. It was recently pointed out that the
class of transformations which are useful for the symmetry reduction --
and for finding solutions -- of ODEs is not limited to (Lie-point,
generalized, non-local...) proper symmetries, but extends to a wider
class of transformations, which were christened ${\cal
C}^\infty$-symmetries or $\la$-symmetries, as they depend on a ${\cal
C}^\infty$ function $\la$ \cite{MuR1}. (See \cite{GMM,MuR2} for some
applications).
It is remarkable that for such transformations one can perform a
``symmetry reduction'' in exactly the same way as for standard
symmetries; transformations enjoying this property were then studied in
full generality -- for scalar ODEs~-- and it was shown that standard
and $\la$-symmetries are essentially the only possibility, up to
considering contact version of both \cite{Sac}.
The concept of $\lambda$-symmetries was extended to the PDE frame (both
scalar equations and systems) with $p$ independent variables $x^i$ and
$q$ dependent ones $u^a$ in \cite{GaMo}; in order to do this, the
central object is a horizontal one-form $\mu = \La_i \d x^i$, where
$\La_i (x,u^{(n)}) \in g \ell (q)$, and thus one speaks of
$\mu$-symmetries. We stress that $\mu$ is not entirely arbitrary:
indeed its coefficients must satisfy a compatibility condition, which
reads $ D_i \La_j - D_j \La_i + [\La_i , \La_j] = 0 $ for all $i,j =
1,...,p$ (note this is automatically satisfied for $p=1$, and takes a
simpler form for $q=1$).
This means that $\mu$ satisfies the horizontal Maurer-Cartan equation
$D \mu + (1/2) [\mu , \mu] = 0$, with $D$ the total horizontal external
derivative operator (defined on functions as $D f = (D_i f) \d x^i$,
with $D_i$ the total derivative with respect to $x^i$, and accordingly
on forms; see e.g. \cite{Olv1,Olv2} for details). Equivalently, the
(matrix) operators $\nabla_i := D_i + \La_i$ commute, $[\nabla_i ,
\nabla_j ] = 0 $.
It should be stressed that $\mu$-symmetries are {\it not} symmetries in
proper sense \cite{Gae,Olv1,Olv2,Ste}, i.e. they do not, in general,
map solutions into solutions; nevertheless, it was shown in \cite{GaMo}
that they can be used to perform ``symmetry reduction'' of PDEs and
systems of PDEs -- that is, to obtain invariant solutions~-- via
exactly the same method used for standard symmetries.
%\medskip
The main goal of this note is to give a geometrical interpretation of
the concept of $\mu$-prolongations, and hence of $\mu$-symmetries.
These concepts will be recalled, together with some related results
which are relevant in the following, in section 1. The geometrical
interpretation mentioned above will be provided in section 2 in the
form of a result describing the relationship between
$\mu$-prolongations and standard prolongations, and hence of
$\mu$-symmetries and standard symmetries. With this interpretation we
will also be able to extend to the framework of $\mu$-symmetries the
concepts of conditional \cite{CGspri,Gae,LeWi,Win} and partial
\cite{CGpar} symmetries, in section 3. Starting again from the
interpretation obtained in section 2 we also study, in section 4, the
interrelation between local $\mu$-symmetries and nonlocal standard
symmetries (of exponential type). After a short discussion on our
approach and results in section 5, several examples are considered in
section 6.
Finally, we would like to remark that the geometrical characterization
of $\mu$-prolongations given below only applies in full when there are
at least two independent variables, i.e. the ODE case is in this
respect a strongly degenerate one (see remark 7). This also explains
why such an interpretation -- and generalization to conditional and
partial symmetries -- could not be obtained in the seemingly simpler
ODE case.
\bigskip
\noindent {\bf Acknowledgements.} This work was partially supported by
GNFM-INdAM through the project {\it ``Simmetrie e tecniche di riduzione
per equazioni differenziali di interesse fisico-matematico''}.
\section{Prolongations, $\mu$-prolongations, and \\ $\mu$-symmetries}
In this section we fix notation and collect some background material,
either standard \cite{Kra,Olv1,Ste} or specific to $\mu$-symmetries
\cite{GaMo}, of use in the following.
We will denote by $M$ the space of independent and dependent variables;
these will be denoted respectively as $x \in B$ and $u \in F$, with $x
= (x^1 , ... , x^p )$ and $u = (u^1,...,u^q)$ in local coordinates.
Thus $M$ is the total space of a fiber bundle $(M,\pi,B)$ over $B$,
with fiber $\pi^{-1} (x) = F$. In the standard case, $B \sse \R^p $, $F
= \R^q$, and $M = B \times \R^q$; in this note we will mainly stick to
this setting, for ease of discussion.
\subsection{Prolongations and $\mu$-prolongations}
The bundle $M$ can be prolonged to the $n$-th jet bundle $(J^{(n)} M ,
\pi_n , B)$; the local coordinates $(x,u)$ in $M$ naturally induce
local coordinates $(x,u^{(n)} )$ in $J^{(n)} M$, given by $u^a_J$,
where $J = (j_1 , ... , j_p)$ is a multiindex of order $|J| := j_1 +
... + j_p$ and $u^a_J = \pa^{|J|} u / [(\pa x^1)^{j_1} ... (\pa
x^p)^{j_p} ]$. We also use the notation $u_{J,i} := \pa u^a_J / \pa
x^i$.
The jet space $J^{(n)} M$ is naturally endowed with a contact
structure, spanned by the set of $F$-valued contact one-forms $\vth_J$
with $|J| < n$; the reader is referred to \cite{Mal,Str} for
vector-valued forms. In the local coordinates introduced above the
contact forms are given by $\vth_J = (\vth^1_J , ... , \vth^q_J )$, with
$$ \vth^a_J \ := \ \d u^a_J \ - \ u^a_{J,m} \, \d x^m \ . \eqno(1.1) $$
The contact structure $\Theta$ is the $C^\infty [J^{(n)} M]$ module
spanned by the $\vth_J$, $|J| < n$.
A Lie-point vector field $X$ in $M$ is naturally prolonged to a vector
field $X^{(n)}$ in $J^{(n)} M$. This is the unique vector field which
projects to $X$ when restricted to $M$, and which preserves the contact
structure. The latter condition means that for any $\vth \in \Theta$,
there is a (possibly zero) $\^\vth \in \Theta $ such that $ \L_X (\vth
) = \^\vth$.
If $X$ and its prolongation $X^{(n)}$ are written in local coordinates
as
$$ \begin{array}{ll}
X &= \ \ \xi^i \, (\pa / \pa x^i) \ + \ \phi^a \, (\pa / \pa u^a) \
, \\
X^{(n)} &= \ \ X \ + \ \Phi^a_J \, (\pa / \pa u^a_J) \ , \end{array}
\eqno(1.2) $$
then the coefficients $\Phi^a_J$ satisfy the {\it prolongation formula}
(with $\Phi^a_0 \equiv \phi^a$)
$$ \Phi^a_{J,i} \ = \ D_i \Phi^a_J \ - \ u^a_{J,m} D_i \xi^m \ .
\eqno(1.3) $$
In the case of $\mu$-prolongations, we equip $M$ with a horizontal
one-form $\mu = \La_i \d x^i$ with values in the algebra $\G := g \ell
(q)$; this is the Lie algebra of the group $G := GL (q)$ of
$q$-dimensional nonsingular matrices (we refer again to \cite{Mal,Str}
for matrix-valued forms). That is, the $\La_i$ are matrix functions
defined on a suitable jet space $J^{(k)} M$ (if $k \le 1$ we will have
proper vector fields in each jet space, otherwise we deal with
generalized vector fields).
Given the (possibly, generalized) vector field $X$, its
$\mu$-prolongation is the unique vector field which projects to $X$
when restricted to $M$, and which ``$\mu$-preserves'' (see \cite{GaMo})
the contact structure $\Theta$. By this we mean that for any $\vth \in
\Theta$ there is a (possibly zero) $\^\vth \in \Theta$ such that $ \L_Y
(\vth^a ) + ( Y \interno [ (\La_i)^a_b \vth^b ] ) \d x^i = \^\vth^a$.
If $X$ is written in local coordinates as in (1.2), and its
$\mu$-prolongation as
$$ Y \ := \ X \ + \ \Psi^a_J \, (\pa / \pa u^a_J) \ , \eqno(1.4) $$
then the coefficients $\Psi^a_J$ satisfy the {\it $\mu$-prolongation
formula}
$$ \Psi^a_{J,i} \ = \ (\nabla_i)^a_b \, \Psi^b_J \ - \ u^b_{J,m} \,
(\nabla_i )^a_b \, \xi^m \ . \eqno(1.5) $$
Here $\nabla_i$ is the matrix operator defined by $\nabla_i = D_i +
\La_i$, i.e.
$$ (\nabla_i)^a_b \ := \ \de^a_b \, D_i \ + \ (\La_i )^a_b \ .
\eqno(1.6) $$
Note that, as discussed in \cite{GaMo}, $Y$ is well defined by (1.5) if
and only if the $\La_i$ satisfy the compatibility condition
$$ D_i \La_j \ - \ D_j \La_i \ + \ [ \La_i , \La_j ] \ = \ 0 \ ,
\eqno(1.7') $$
which is equivalently written as
$$ \[ \, \nabla_i \, , \, \nabla_j \, \] \ = \ 0 \ . \eqno(1.7'') $$
If (1.7) is satisfied only on a submanifold $S \ss J^{(n)} M$, then $Y$
is properly defined by (1.5) only on $S$; in case $Y : S \to \T S$
(i.e. $S$ is $Y$-invariant), it still makes sense to study the
$Y$-action on $S$, see remark 2 below.
\medskip\noindent
{\bf Remark 1.} The coefficients of the $\mu$-prolongation $Y$ can also
be described in terms of the coefficients of the ordinary prolongation
$X^{(n)}$ of the same vector field $X$. If we write $\Psi^a_J =
\Phi^a_J + F^a_J$, the difference terms $F^a_J$ satisfy (with of course
$F^a_0 = 0$) the recursion relations $F^a_{J,i} = [ \de^a_b D_i +
(\La_i)^a_b ] F^b_J + (\La_i)^a_b D_J Q^b$, where $Q$ is the
characteristic vector for $X$, i.e. $Q^a := \phi^a - u^a_i \xi^i$.
It follows from this that $\Psi^a_J = \Phi^a_J$ on the space $\I_X \ss
J^{(n)} M$ of $X$-invariant functions, identified by the vanishing of
$D_J Q^a$ for all the multiindices $J$, $0 \le |J| \le (n-1)$. This
also implies that the space $\I_X$ is invariant under $Y$, and
$\mu$-symmetries (see below) can be used for symmetry reduction in the
same way as standard symmetries. See \cite{GaMo} for details. \EOR
\medskip
Similarly to what happens for standard prolongations (and symmetries),
discussing $\mu$-prolongations is simpler when we consider vector
fields in evolutionary form; we recall that for $X$ given by (1.3), the
evolutionary representative is $X_Q = Q^a (\pa / \pa u^a)$, with $Q$ as
given above. Its $\mu$-prolongation $Y_Q$ is given by (1.4) with
$\Psi^a_J = (\nabla_J )^a_b Q^b$; here we have written $\nabla_J :=
(\nabla_1)^{j_1} \cdots (\nabla_p)^{j_p}$, which is legitimate in view
of (1.7$''$).
In the following, we will only deal with vector fields in evolutionary
form. Note that in geometrical terms an evolutionary vector field is a
(generalized) vector field on $(M,\pi,B)$ which is vertical for the
fibration $\pi$.
The (standard or $\mu$) $k$-th prolongation of such a vector field will
be a vector field on $(J^{(k)} M , \pi_k , B)$ which is vertical for
the fibration $\pi_k$, and this for any $k$. We will denote by $\V_k$
the set of vertical vector fields on $(J^{(k)} M , \pi_k , B)$.
\subsection{Symmetries and $\mu$-symmetries}
The vector field $X$ is a {\bf $\mu$-symmetry} (with a given $\mu$) of
the system of $r \ge 1$ differential equations $\De := (\De_1 , ... ,
\De_r )$ of order $n$ if and only if its $\mu$-prolongation $Y$ in the
jet space $J^{(n)} M$ is tangent to the solution manifold $S_\De \ss
J^{(n)} M$, $Y : S_\De \to \T S_\De$. Equivalently (under standard
nondegeneracy assumptions), $X$ is a $\mu$-symmetry of $\De$ if there
is a smooth matrix function $R : J^{(n)} M \to {\rm Mat}(r) $ such that
$Y(\De_\a ) = R_\a^\b \, \De_\b$.
If $Y$ is tangent to all level manifolds for $\De$, i.e. if in the
previous notation the function $R$ is identically zero, we say that $Y$
is a {\it strong} $\mu$-symmetry for $\De$.
\medskip\noindent
{\bf Remark 2.} Note that in order to consider $\mu$-symmetries of
$\De$, it suffices that the compatibility condition (1.7) be verified
on $S_\De$, not necessarily in the whole jet space; we would then have
``internal $\mu$-symmetries'', in analogy with standard internal
symmetries \cite{Kam}. This point will be relevant in section 4 below.
\EOR
\medskip\noindent
{\bf Remark 3.} Note also that, as mentioned above, if $\La_i = \La_i
(x,u^{(1)})$, then the $\mu$-prolongation of a Lie-point vector field
in $M$ is a proper (rather than generalized) vector field in each jet
space $J^{(k)} M$. \EOR
\section{The relation between $\mu$-prolongations and \\ standard
prolongations}
In this section we show that $\mu$-prolonged vector fields are
locally related to vector fields which are standard prolongations. This
will entail an equivalence between $\mu$-symmetries and (local or
nonlocal, see sect.4) standard symmetries for a given equation.
We will denote by $G$ the group $GL(q,\R)$ of $q$-dimensional
non-singular real matrices, and by $\G$ its Lie algebra. We denote by
$\Ga_n$ the space of smooth maps $\ga: J^{(n)} M \to G$. The space
$\Ga_n$ has a natural structure of Lie group.\footnote{In physical
language, it is a {\it gauge group} modelled on $G$ \cite{Ish,Nak}. To
avoid any confusion, note that usually -- that is, in Yang-Mills
theories -- one considers gauge transformations depending on the base
space variables only, i.e. $\ga : B \to G$; our setting is thus
considerably more general.}
When $F$ is $q$-dimensional there is a natural action of the group
$G=GL(q)$ on $T_f F$ for any point $f \in F$. As $(M,\pi,B)$ is a
vector bundle, the jet bundle $(J^{(n)} M , \pi_n , B)$ is also a
vector bundle.
The group $G$ acts naturally on the fiber $F = \R^q$; it also acts on
the tangent space $\T_p F^{(n)}$ to the fibers $F^{(n)} = \pi_n^{-1} x$
of $J^{(n)} M$ at any point $p = (x,u^{(n)} ) \in \pi_n^{-1} x$. This
is a vector space whose dimension $d=s q$ is an integer multiple of
$q$. We will consider the $G$ action on $\T_p F^{(n)}$ via the
$d$-dimensional representation which is the direct sum of $s$ copies of
the defining one, the invariant subspaces spanned by $u^a_K$ for
$a=1,..,q$ and a given multiindex $K$. We refer to this action as the
{\it jet representation} of $G$.
If the jet representation of $G$ is given by the $d$-dimensional
matrices $T : G \to {\rm Mat} (d)$, $T : g \mapsto T_g$, the function
$\ga \in \Ga_n$ is represented by the matrix function $T \circ \ga : z
\mapsto T_{\ga (z)} $ (with $z \in J^{(n)} M$). In the following we
will use, with a standard abuse of notation, the same symbol $\gamma$
for the map $\ga : M \to G$ and for its $q\times q$ matrix
representation. Thus $T_\ga = \ga \oplus ... \oplus \ga$.
A map $\ga \in \Ga_n$ acts naturally in $\V_n$ by the jet action. In
fact, rewriting now a generic element $Y \in \V_n$ as
$$ Y \ = \ \Psi^a_J \, (\pa / \pa u^a_J) \ , $$
$\ga$ acts on $Y$ to give $$ \ga \cdot Y \ := \ \[ \ga^a_b \ \Psi^b_J
\] \ (\pa / \pa u^a_J) \ . \eqno(2.1) $$
This action obviously maps $\V_n$ into itself.
As mentioned above, we will prove that $\mu$-prolongations and ordinary
ones are related by an action of the (gauge) group $\Ga_n$ on vertical
vector fields. In order to do this, we need the following notions,
which are obviously equivalence relations, due to the group property of
$\Ga_n$.
\medskip\noindent
{\bf Definition 1.} Given two vector fields $Y$ and $W$ in $\V_n$, we
say that they are {\bf $G$-equivalent} if there exists a function $\ga
\in \Ga_n$ such that $Y = \ga \cdot W$, globally in $J^{(n)} M$. We say
that $Y$ and $W$ are {\bf locally $G$-equivalent} if for any $z \in
J^{(n)}(M)$ there exists a neighborhood $A_z$ of $z$ and a local
function $\ga_z : A_z \to G$ such that $W = \ga_z \cdot Y$ in $A_z$.
\EOD
\medskip
We will consider, together with the function $\ga \in \Ga_n$, the
function $\ga^{-1} \in \Ga_n$. We stress that $\ga^{-1}$ is not the
inverse of the function $\ga$, but a function which at each point
$(x,u^{(n)})$ provides the element of $G$ inverse to the element $\ga
(x,u^{(n)})$.
\subsection{Compatibility condition}
We start by identifying the geometrical meaning of the compatibility
condition (1.7) in lemma 1. We will then recall proposition 1 (which we
quote from \cite{Sha}, see chapter 3, theorem 6.1 there). Proposition 2
is given in \cite{Mar}.
\medskip\noindent
{\bf Lemma 1.} {\it The compatibility condition (1.7) is the expression
in coordinates of the horizontal Maurer-Cartan equation
$$ D \mu \ + \ (1/2) \, [ \mu , \mu ] \ = \ 0 \ ; \eqno(2.2') $$
equivalently, it states that the Maurer-Cartan equation is satisfied up
to a form $\rho = - (\pa \La_j / \pa u^a_K ) \d x^j \vth^a_K$ in the
Cartan ideal ${\cal J} (\Theta )$ generated by $\Theta$,
$$ \d \mu \ + \ (1/2) \, [ \mu , \mu ] \ = \ \rho \, \in \, {\cal J}
(\Theta ) \ . \eqno(2.2'') $$}
\medskip\noindent
{\bf Proof.} The ``proof" amounts to recalling some definitions for
$\G$-valued forms \cite{Sha,Str}. Let $\{ e_i \}$ be a basis in $\G$,
so that any $\G$-valued form $\om$ (of any degree $k$) is written
uniquely as $\om = e_i \otimes \om^i$, with $\om^i$ a $k$-form; one
usually writes simply $\om = e_i \om^i$, omitting the tensor product
symbol, for short. Given two $\G$-valued forms $\a = e_i \otimes \a^i$
and $\b = \b^j \otimes e_j$, we define as usual \cite{Sha,Str}
$$ [ \a , \b ] \ := \ [ e_i , e_j ] \, \otimes \, (\a^i \w \b^j ) \ . $$
Denote by $D$ the total exterior derivative operator; for a horizontal
form $\a = A_\s \d x^\s$ (this is a shorthand for $\a = A_{\s_1 ...
\s_k} \d x^{\s_1} \w ... \w \d x^{\s_k}$) its action is given by $ D \a
= (D_i A_\s) \d x^i \w \d x^\s$. Using these definitions, we have indeed
$$ \begin{array}{rl}
D \mu \, + \, {1 \over 2} [ \mu , \mu ] \ =& \( D_i \La_j + {1 \over 2}
\, [\La_i , \La_j ] \) \, \otimes \, (\d x^i \w \d x^j ) \ = \\
=& \ {1 \over 2} \, \( D_i \La_j - D_j \La_i + [\La_i , \La_j ] \) \,
\otimes \, ( \d x^i \w \d x^j ) \ , \end{array} $$ as claimed.
Using (1.1) we have $D \mu = \d \mu - (\pa \La_j / \pa u^a_K ) \vth^a_K
\w \d x^j$, hence the equivalent form of the statement. \EOP
\medskip\noindent
{\bf Proposition 1.} {\it Let $E$ be a smooth manifold, $G$ a Lie group
with Lie algebra $\G$, and $\mu$ a $\G$-valued one form on $E$
satisfying the structural (Maurer-Cartan) equation $$ \d \mu \ + \ {1
\over 2} \, [\mu, \mu] \ = \ 0 \ . $$ Then for each point $z \in E$
there are a neighborhood $A_z \sse E$ of $z$ and a local function
$\ga_z : A_z \to G$ such that $\mu = \ga_z^{-1} \d \ga_z$ locally in
$A_z$.}
\medskip\noindent
{\bf Proposition 2.} {\it Let $(E,\pi,A)$ be a fiber bundle, $G$ a Lie
group with Lie algebra $\G$, and $\mu$ a horizontal $\G$-valued one
form on $E$ satisfying the horizontal Maurer-Cartan equation (2.2$'$).
Then for each point $z \in E$ there are a neighborhood $A_z \sse E$ of
$z$ and a local function $\ga_z : A_z \to G$ such that $\mu =
\ga_z^{-1} D \ga_z$ locally in $A_z$.}
\medskip\noindent
{\bf Definition 2.} Let $f : E \to G$ be a smooth map, $\om$
(respectively $\om_h$) the Maurer-Cartan (respectively, the horizontal
Maurer-Cartan) form on $G$. Then the {\bf Darboux derivative} of $f$ is
the $\G$-valued form $f^* (\om) = f^{-1} \d f \in \La^1 (E,\G )$
\cite{Mal,Sha}. The {\bf horizontal Darboux derivative} of $f$ is the
horizontal $\G$-valued form $f^* (\om_h) = f^{-1} D f \in \La^1 (E,\G
)$. \EOD
\medskip\noindent
{\bf Remark 4.} If we denote by $\om$ the Maurer-Cartan form on $G$,
proposition 1 guarantees that any $\G$-valued one-form $\mu$ satisfying
the structural equation is (locally) the Darboux derivative of some
(local) function $\ga$. \EOR
\subsection{Prolongations}
We proceed now, equipped with the previous results, to prove lemma 2
and theorem 1 (and 2) which establish the (local) $G$-equivalence of
standard and $\mu$-prolongations.
\medskip\noindent
{\bf Lemma 2.} {\it Let $\G$ be the Lie algebra of the group $G$, and
let $\mu$ be a $\G$-valued horizontal one form on $(J^{(n)} M,\pi_n
,B)$, written in local coordinates as $\mu = \La_i \d x^i $. Then $\mu$
satisfies the compatibility condition (1.7) if and only if for any $z
\in J^{(n)} M$ there are a neighborhood $A_z \sse J^{(n)} M$ and a
function $\ga_z : A_z \to G$ such that
$$ \La_i \ := \ \ga_z^{-1} (D_i \ga_z) \ . \eqno(2.3) $$}
\medskip\noindent
{\bf Proof.} With the choice (2.3), we have $D_i \La_j = (D_i
\ga_z^{-1} ) (D_j \ga_z ) + \ga_z^{-1} (D_i D_j \ga_z )$; moreover,
$\La_i \La_j = \ga_z^{-1} (D_i \ga_z ) \ga_z^{-1} (D_j \ga_z) = - (D_i
\ga_z^{-1} ) (D_j \ga_z )$. These show at once that (1.7) is satisfied
when $\La_i$ are given by (2.3).
As for the converse, i.e. that if (1.7) is satisfied then locally $\mu$
is written as $\mu = \La_i \d x^i$ with $\La_i$ as in (2.3), this
follows from lemma 1 and proposition 2. \EOP
\medskip\noindent
{\bf Remark 5.} If the function $\ga$ is given, the horizontal form
$\mu = \La_i \d x^i$ is uniquely determined by (2.3). On the other
hand, if a horizontal $\mu$ -- satisfying (1.7) -- is given, the
function $\ga$ is not uniquely determined by (2.3), even locally in
$A_z$. Indeed, if $\ga_1$ satisfies (2.3), we can always consider
$\ga_2 = h \ga_1$, with $h$ any constant element of $G$, which still
satisfies (2.3). \EOR
\medskip\noindent
{\bf Theorem 1.} {\it Let $Y=\Psi^a_J (\pa / \pa u^a_J) \in \V_n$ be
the $\mu$-prolongation of the (possibly generalized) vector field $X =
\Psi^a_0 (\pa / \pa u^a ) \in \V_0$. Suppose that $\mu= \ga^{-1} (D
\ga) = \ga^{-1} (D_i \ga) \d x^i$ for a smooth function $\ga \in
\Ga_n$. Then the vector field $W = \ga \cdot Y \in \V_n$ is the
standard prolongation of the (possibly generalized) vector field $\XT =
\ga \cdot X \in \V_0$.
Conversely, let $W := \XT^{(n)} = \PT^a_J (\pa / \pa u^a_J)$ be the
standard prolongation of the evolutionary vector field $\XT = \PT^a_0
(\pa / \pa u^a) \in \V_0$; let $\ga \in \Ga_n$ be a smooth function.
Then the vector field $Y \in \V_n$ defined by $Y = \ga^{-1} \cdot W$ is
the $\mu$-prolongation of the (possibly generalized) vector field $X =
\ga^{-1} \cdot \XT$, with $\mu = \ga^{-1} (D \ga)$.}
\medskip \noindent
{\bf Proof.} Saying that $W = \XT^{(n)}$ is a standard prolongation
means that $$ \PT^a_{J,i} \ = \ D_i ( \PT^a_J ) \ . $$
On the other hand, $W = \ga \cdot Y$, hence $\PT^a_K = \ga^a_b
\Psi^b_K$ for any multiindex $K$, see (2.1). In vector notation,
$$ \PT^a_{J,i} := \ \ga^a_b \Psi^b_{J,i} \ = \
D_i ( \ga^a_b \Psi^b_J ) \ = \ \ga^a_b (D_i \Psi^b_J) \ + \ \ga^a_b
[(\ga^{-1})^b_c (D_i \ga^c_m)] \, \Psi^m_J \ . $$
As the matrices $\ga (x,u^{(n)} )$ are invertible, this shows that $$
\Psi^a_{J,i} \ = \ \[ \de^a_b D_i \, + \, (\ga^{-1})^a_k (D_i \ga^k_b)
\] \Psi^b_J \ , $$
which is just the $\mu$-prolongation formula with the identification $
\La_i := \ga^{-1} (D_i \ga) $, as claimed in the statement.
In order to prove the converse, it suffices to perform the computation
the other way round, i.e. starting with $\Psi^a_{J,i} = \nabla_i (
\Psi^a_J )$. \EOP
\medskip
In the theorem above, $\mu$ is known to be in the form $\mu = \ga^{-1}
(D \ga )$; we will now consider the general case.
\medskip\noindent
{\bf Theorem 2.} {\it Let $Y \in \V_n$ be the $\mu$-prolongation of the
vector field $X \in \V_0$, with $\mu$ satisfying (2.2). Then for any $z
= (x , u^{(n)} ) \in J^{(n)} M$ there are a neighborhood $A_z \sse
J^{(n)} M$ and a local map $\ga_z : A_z \to G$ such that, locally in
$A_z$:
\par\noindent {\tt (i)} the form $\mu \in \La^1 (J^{(n)} M, \G )$ is
given by $\mu = \ga_z^{-1} (D \ga_z)$;
\par\noindent {\tt (ii)} $Y$ is $G$-equivalent to the vector field $W
:= \ga_z \cdot Y$, which is the standard prolongation of the vector
field $\XT := \ga_z \cdot X \in \V_0$.}
\medskip\noindent
{\bf Proof.} This merely states that theorem 1, which deals with the
case where $\mu = \ga^{-1} (D \ga)$ for some map $\ga \in \Ga_n$,
actually holds {\it locally} for all $\mu$ satisfying (2.2), or
equivalently (1.7). In fact, lemma 2 guarantees that any such $\mu$ can
be written, locally in $A_z$, in the form $\mu = \ga_z^{-1} (D \ga_z)$.
\EOP
\medskip
We stress that proposition 1, and hence lemma 2 and theorem 2, make
only local claims. Let us now consider the global setting.
\medskip\noindent {\bf Theorem 3.} {\it Let $\mu = \La_i \d x^i$ be a
$\G$-valued horizontal one form on $J^{(n)} M$ satisfying (1.7),
equivalently (2.2). Let $Y=\Psi^a_J (\pa / \pa u^a_J) \in \V_n$ be the
$\mu$-prolongation of the (possibly generalized) vector field $X =
\Psi^a_0 (\pa / \pa u^a ) \in \V_0$. Then the following are equivalent:
\par\noindent {\tt (i)} There exists a vector field $W \in \V_n$ which
is $G$-equivalent to $Y$ and which is the standard prolongation of a
vector field $\XT \in \V_0$. The vector fields $W$ and $\XT$ are given
by $W = \ga \cdot Y$ and $\XT = \ga \cdot X$, with $\La_i = \ga^{-1}
(D_i \ga)$.
\par\noindent {\tt (ii)} $\mu$ is the horizontal Darboux derivative of
a $\ga \in \Ga_n$, $\mu = \ga^{-1} D (\ga)$.}
\medskip\noindent {\bf Proof.} This follows at once from theorem 1 and
lemma 2. \EOP
\medskip
We denote by $\M$ the set of $\G$-valued horizontal one-forms
satisfying the horizontal Maurer-Cartan equation and by $\D$ the set of
one-forms which are the Darboux derivative of maps $\ga \in \Ga_n$; by
lemma 2, $\D \sse \M$. We define $\H := \M / \D$; this amounts to
factorization with respect to gauge equivalence. Note that $\H$ depends
on the topology (in particular, the first homotopy group) of $J^{(n)}
M$; when the fibers of $M$ and hence of $J^{(n)} M$ are contractible --
as in our case, where $M$ is a vector bundle -- this is just the
topology of the base space $B$ (see examples 2 and 3 below in this
respect).
\medskip\noindent {\bf Corollary 1.} {\it If $\H = 0$, then any
$\mu$-prolonged vector field, for any $\mu \in \M$, is globally
$G$-equivalent to a standard prolongation. In particular, this holds if
$J^{(n)} M$ is contractible.}
\medskip\noindent {\bf Proof.} Obvious. \EOP
\medskip
The previous results, in particular theorem 3 and its corollary, could
also be obtained from the global version of proposition 1, see again
\cite{Sha} (in particular chapter 3, theorem 7.14). This states that
$\mu$ is the Darboux derivative of a map $\ga : J^{(n)} M \to G$ if and
only if the monodromy representation $\Phi_\mu : \pi_1 (J^{(n)} M, z)
\to G$ is trivial (for any point $z \in J^{(n)} M$) and $\mu$ satisfies
the Maurer-Cartan equation.
\medskip\noindent
{\bf Remark 6.} When we consider the scalar case $q=1$, the group $G$
reduces to the multiplicative group $\R_+$, hence the $\La_i$ reduce to
nonzero smooth functions $\la_i : J^{(n)} M \to \R$; two vertical
vector fields $Y$ and $W$ in $\V_n$ are $G$-equivalent if and only if
they are collinear, i.e. if there is a (scalar) nowhere zero function
$\ga : J^{(n)} M \to \R$ such that $W = \ga Y$. Note that in this case
$G$ is an abelian group, and (1.7) reduces to $D_i \la_j = D_j \la_i$,
where the $\la_i$ are real functions; this means that locally there is
a ``potential'' $V$ such that $\la_i = D_i V = D_i(\ga)/\ga$. \EOR
\medskip\noindent
{\bf Remark 7.} If we consider ODEs rather than PDEs, we have $B=\R$
and $p = 1$. In this case condition (1.7) is automatically satisfied,
and does not impose any restriction on the $\La_i$. \EOR
\medskip\noindent
{\bf Remark 8.} The work \cite{Sac} considered scalar ODEs; it was
shown there that vector fields $Y$ on $J^{(n)} M$ which are the
$\la$-prolongation of a vector field $X$ in $M$ (with standard
prolongation $W = X^{(n)}$) can be identified by the property that the
characteristics of $W$ and of $Y$ coincide. In the frame of remark 6
above, having a one-dimensional fiber ($q=1$) makes that $GL(q)$
reduces to the group of nowhere vanishing real-valued functions $\la
(x,u^{(n)} )$, and vertical vector fields are just scaled by $\la$,
hence preserving the characteristics. In the general case, our result
says that the characteristics are acted upon by the group $G$, in a
covariant way: that is, acting on the vectors $u_K = (u^1_K , ... ,
u^q_K)$ (for any multiindex $K$, $|K| \ge 0$) in the same way. This is
precisely the requirement that $G$ acts by the jet action, see above.
\EOR
\medskip\noindent
{\bf Remark 9.} In the language of gauge theories, $\ga = \ga (x)$ and
the $\La_i$ satisfying (1.7) identify a flat connection, i.e. a zero
strength Yang-Mills field. Thus proposition 1 is a generalization (see
footnote 1) of the statement that any such field reduces locally to a
pure gauge field; see e.g. theorem 6.4 in \cite{Ish}. Similarly, lemma
2 above identifies conditions to have a pure gauge field (in our
extended sense) at the global level as well. \EOR
\subsection{Symmetries}
We now pass to discuss symmetries. As pointed out in remark 2, it would
actually suffice that $\mu$ satisfy (1.7) on $S_\De \ss J^{(n)} M$. On
the other hand, the distinction between local and global (in $S_\De$)
$G$-equivalence still applies. The result obtained earlier on and these
observations lead to the following result.
\medskip\noindent
{\bf Theorem 4.} {\it Let $\De$ be an equation or system of equations
in $J^{(n)} M$. Let $\mu = \La_i \d x^i$ be a horizontal $\G$-valued
one-form on $J^{(n)} M$, such that (1.7) are verified on a manifold
$C_\mu \sse J^{(n)} M$, with $S_\De \sse C_\mu$. Let the (possibly
generalized) vector field $X = \Psi^a_0 (\pa / \pa u^a ) \in \V_0$ be a
$\mu$-symmetry for $\De$. Then the following are equivalent:
\par\noindent {\tt (i)} There exists a (possibly, generalized and/or
nonlocal) vector field $\XT \in \V_0$ which is $G$-equivalent to $X$
and which is a standard symmetry of $\De$; this is given by $\XT = \ga
\cdot X$, where $\mu = \ga^{-1} (D \ga)$ on $C_\mu$ and hence in
particular on $S_\De$.
\par\noindent {\tt (ii)} The form $\mu$ is the horizontal Darboux
derivative of $\ga : C_\mu \to G$.}
\medskip\noindent
{\bf Proof.} For $C_\mu = J^{(n)} M$, this follows at once from theorem
3 and the definitions of standard and $\mu$-symmetry of $\De$. In the
general case, we just have to restrict all forms, operators (and
considerations) to the manifold $S_\De \sse C_\mu \sse J^{(n)} M$ of
interest when discussing symmetries of $\De$. \EOP
\medskip
Note that the possibility of having a nonlocal vector field, mentioned
in this statement, is present only in the case where $C_\mu \not=
J^{(n)} M$, as discussed in detail in section 4.
\section{Conditional and partial $\mu$-symmetries}
In this section we define conditional and partial $\mu$-symmetries, and
discuss the relation of these with ``standard'' conditional and partial
symmetries; the relevant definitions for the latter will also be
recalled. We will work at the local level only; discussion of global
problems would go along the same lines of section 2.2 (in particular,
our results hold also globally if $\mu$ is the Darboux derivative of a
function $\ga \in \Ga_n$, see theorems 3 and 4 above) and is left to
the interested reader.
\subsection{Conditional symmetries}
Let us consider first of all conditional symmetries: they are most
frequently considered in the scalar case $q=1$, but we can deal with
the general case $q>1$.
\medskip\noindent
{\bf Definition 3.} The vector field $X$ is a (standard) {\bf
conditional symmetry} of the differential system $\De$ if there are
solutions $u^a = f^a(x)$ to $\De$ which are invariant under $X$, i.e.,
such that $X : \sigma_f \to \T_{\sigma_f}$, where $\sigma_f$ is the
associated section in $M$. If $X$ is not a symmetry of $\De$, we say it
is a {\it proper} conditional symmetry. \EOD
\medskip
A particularly convenient way to look at conditional symmetries of a
differential system of $r$ equations $\De := \De_\alpha (x,u^{(n)} ) =
0$ is the following (cf. \cite{LeWi,Win}).
We consider the system made of $\De$ and of the equations expressing
the $X$-invariance of the solution $u^a = f^a(x)$. The latter are
simply $Q^a = 0$, with $Q^a$ the characteristic of the vector field
$X$, see above. That is, we are considering the system of $r +q$
equations
$$ \cases{ \De_\alpha (x,u^{(n)}) \ = \ 0 \ , & \cr
\phi^a - u^a_i \xi^i \ = \ 0 \ . & } \eqno(3.1) $$
Any solution to (3.1) is $X$-invariant and a solution to $\De$, and
conversely all the $X$-invariant solutions to $\De$ are solutions to
the system (3.1). Then, the conditional symmetries for $\De$ are simply
those vector fields $X$ for which (3.1) admits solutions. Note that $X$
turns out to be a symmetry of the system (3.1) (see however
\cite{OR,PS,Sac} and also \cite{CiK} for a careful discussion on the
notion of conditional symmetries).
\medskip
If $X$ is a symmetry of $\De$, it can happen -- as well known, also in
the case $q=1$ (see \cite{OR}) -- that $\De$ does not admit any
$X$-invariant solution (i.e., that the system (3.1) does not admit
solutions, for a given $X$), even when $X$ is an exact symmetry of
$\De$. In this sense, standard (exact) symmetries are not necessarily
conditional symmetries as well.
In the computation of conditional symmetries one often, to obtain
simpler formulas, normalizes to one a nonzero coefficient in the vector
field $X$ (see, e.g., \cite{Win}). This procedure can be seen as a
special case (see remark 6) of the following simple lemma.
\medskip\noindent
{\bf Lemma 3.} {\it Let $X_0$ be a conditional symmetry of $\De$, in
evolutionary form; let $\ga \in \Ga_n$. Then $X := \ga \cdot X_0$ is
also a conditional symmetry of $\De$.}
\medskip\noindent
{\bf Proof.} The prolongations of $X$ and of $X_0$ differ by terms
which vanish on the set where $D_J Q^a = 0$ for all $|J|=0,...,n-1$.
\EOP
\subsection{Conditional $\mu$-symmetries}
A natural definition of conditional $\mu$-symmetries would be the one
below.
\medskip\noindent
{\bf Definition 4.} The vector field $X$ is a {\bf conditional
$\mu$-symmetry} of the differential system $\De$ if it is a
$\mu$-symmetry of the system (3.1). If $X$ is not a $\mu$-symmetry of
$\De$, we say that it is a {\it proper} conditional $\mu$-symmetry. \EOD
\medskip\noindent
{\bf Remark 10.} As in the case of standard conditional symmetries, a
$\mu$-symmetry $X$ of the system $\De$ could fail to be also a
conditional $\mu$-symmetry (as $\De$ may do not admit solutions
invariant under the $\mu$-prolongation of $X$); an example with $q=1$
is $\De := e^x u_x + u_y- 1=0$, with $X = \pa_x + e^{-x} \pa_y $ and
$\mu = \d x$. \EOR
\medskip
One could prove the analogue of theorem 4, i.e. that for any
conditional $\mu$-symmetry $X$ there is a $G$-equivalent vector field
$\XT$ which is a standard conditional symmetry. However, as stated by
lemma 3, standard conditional symmetries already come in
$G$-equivalent families, so that considering conditional
$\mu$-symmetries does not give anything new with respect to standard
conditional symmetries:
\medskip\noindent
{\bf Corollary 2.} {\it Let $\De$ be a differential system; the vector
field $X$ is a conditional $\mu$-symmetry for $\De$ if and only if it
is a standard conditional symmetry for $\De$.}
\medskip\noindent
{\bf Proof.} By definition, a vector field $X$ is a conditional
$\mu$-symmetry (resp. a standard conditional symmetry) if it is a
$\mu$-symmetry (resp. a standard symmetry) of the system (3.1). Now,
the $\mu$-prolongations and the standard prolongations coincide on the
invariant set $\I_X\sse J^{(n)}M$ identified by the conditions $Q^a=0$
and their differential consequences, but in looking for conditional
symmetries one restricts precisely to this set. \EOP
\medskip
We can also investigate the relation between standard conditional
symmetries and ``full'' (rather than conditional) $\mu$-symmetries. It
turns out that there is a correspondence between $\mu$-symmetries and
a special subset of (standard) conditional symmetries.
\medskip\noindent
{\bf Corollary 3.} {\it Let $X$ be a full $\mu$-symmetry for a system
of PDEs $\De$, and let $\De$ admit a $X$-invariant solution. Then $X$
is a (standard) conditional symmetry for $\De$.
More precisely, $X$ is a full $\mu$-symmetry of $\De$, admitting
$X$-invariant solutions, if and only if: {\rm (i)} $X$ is a (standard)
conditional symmetry of $\De$, and {\rm (ii)} there is $\ga \in \Ga_n$
such that $\ga \cdot X$ is an exact symmetry of $\De$.}
\medskip\noindent
{\bf Proof.} According to theorem 4, if $X$ is a (full) $\mu$-symmetry
of $\De$, there is a $G$-equivalent exact symmetry $\XT = \ga \cdot X$.
Conversely, if $\XT$ is an exact symmetry in evolutionary form, then,
for any $\be \in \Ga_n$, one has that $X = \be \cdot \XT$ is a
$\mu$-symmetry, with $\mu= \be (D \be^{-1})$. The conclusion follows
observing that if $\XT$ is an exact symmetry (admitting some invariant
solution) then $\be\cdot\XT$ is in general no longer an exact symmetry
but only a conditional symmetry. \EOP
\medskip\noindent
{\bf Remark 11.} It is well known that conditional symmetries of a
given equation do not transform solutions of the given equation into
other solutions; the same is true, as already remarked, for
$\mu$-symmetries. It should be stressed that the latter have a more
convenient property: indeed, under the conditions of theorem 4, the
presence of a $\mu$-symmetry $X$ guarantees that there exists a true
$G$-equivalent symmetry $\XT$ which maps solutions into solutions. \EOR
\subsection{Partial symmetries}
Partial symmetries were introduced in \cite{CGpar}, and represent a
generalization of conditional symmetries. The vector field $X$ is a
(proper) {\bf partial symmetry} of the differential equation $\De$ if
there is a (proper) subset ${\cal S}$ of solutions which is mapped into
itself under $X$. In a sense, these interpolate between standard and
conditional symmetries: if ${\cal S}$ is the set of all solutions to
$\De$, we have a standard symmetry (in this case $X$ is a trivial
partial symmetry), and if there is some solution which is invariant
under $X$ (and hence in ${\cal S}$) we have a conditional symmetry.
Note that we are {\it not} requiring there is any $X$-invariant
solution. See \cite{CGpar} for details.
For the sake of simplicity, we restrict here our discussion to the case
$q=1$; the extension to the general case is once again completely
straightforward (example 7 below will consider the case $q=2$).
We can look at partial symmetries in a way similar to the Levi and
Winternitz approach to conditional symmetries \cite{LeWi}. We apply the
vector field $X^{(n)}$, where $n$ is the order of $\De$, to $\De$ and
restrict the obtained expression to the solution manifold $\S_\De$; we
call this $\De^{(1)}$. Note that $\De^{(1)}$ cannot vanish unless $X$
is a symmetry of $\De$. We repeat then the procedure, i.e. define $$
\De^{(k+1)} \ := \[ X^{(n)} (\De^{(k)} ) \]_{\S_k} \eqno(3.2) $$ where
$\S_k$ is the intersection of the solution manifolds for $\De\equiv
\De^{(0)}, \De^{(1)} ,..., \De^{(k)}$. We are interested in the case
there is some finite $\ell$ such that $\De^{(\ell+1)} \equiv 0$; in
this case we say that $X$ is a partial symmetry of order $\ell$. Note
that the set ${\cal S}={\cal S}_\ell$ corresponds to solutions to the
system $\{ \De^{(0)} , \De^{(1)} , ... , \De^{(\ell)} \}$, and that $X$
is a standard symmetry for this system.
\medskip\noindent
{\bf Lemma 4.} {\it Let $\De$ be a differential equation of order $n$,
and $X_0$ any vector field, on $M$; denote by $W:= X_0^{(n)}$ the
standard $n$-th order prolongation of $X_0$. Consider a vector field $Y
= \be \cdot W$ in $J^{(n)} M$, with $\be \in \Ga_n$, and write
$\^\De^{(k)}$ for the restriction to $\^S_{k-1}$ of $(Y)^k [ \De ]$,
with $\^S_k$ the intersection of the solution manifolds for $\De^{(0)},
\^\De^{(1)} ,..., \^\De^{(k)}$. Then $S_k = \^S_k$, and $\^\De^{(k+1)}
\equiv \De^{(k+1)}$ on $\S_k$, for all $k \ge 0$.}
\medskip\noindent
{\bf Proof.} The action of $Y$ on $\De$ will produce $\^\De^{(1)} := Y
[\De] = \be \De^{(1)}$. Further applying $Y$ we obtain $\^\De^{(2)} :=
Y[\^\De^{(1)}]=\be W[ \be \De^{(1)}] = \be^2 \De^{(2)} + [\be W(\be)]
\De^{(1)}$. In this way it is easy to see that we obtain, for all $k$,
$$ \^\De^{(k)} \ := \ \[ (Y)^k [\De] \]_{\^S_{k-1}} \ = \ \[ \be^k
\De^{(k)} + \sum_{m=0}^{k-1} h_m \De^{(m)} \]_{\^S_{k-1}} \eqno(3.3) $$
where $h_m$ are smooth functions.
It is obvious from (3.3) that $\^\De^{(k+1)} = \De^{(k+1)}$ on $\S_k$,
and actually that $\S_k = \^S_k$ for all $k$. \EOP
\subsection{Partial $\mu$-symmetries}
As for conditional symmetries, we can parallel the definition of
(standard) partial symmetries and define partial $\mu$-symmetries.
Roughly speaking, theorem 4 extends also to this case.
\medskip\noindent
{\bf Definition 5.} The vector field $X$ is a (proper) {\bf partial
$\mu$-symmetry} of the differential equation $\De$ if there is a
(proper) subset ${\cal S}$ of solutions $u = f(x)$ to $\De$, invariant
under the $\mu$-prolongations $Y$ of $X$. \EOD
\medskip
This definition can be recast in a constructive way in this form [cf.
(3.2)]:
\medskip\noindent
{\bf Definition 5$'$}. Let $\De=0$ be a PDE, $X$ a vector field and $Y$
a $\mu$-prolongation of $X$. Assume that $X$ is not a $\mu$-symmetry
for $\De$, i.e. $\De^{(1)}:=[Y(\De)]|_{\De=0}\not= 0$, but, defining
$$\De^{(k+1)}:=[Y(\De^{(k)})|_{\S_k} \quad {\rm for}\quad
k=0,1,\ldots,\ell \eqno(3.4)$$ with the same notations as above, assume
that $\De^{(\ell+1)}\equiv 0$. We then say that $X$ is a (proper) {\bf
partial $\mu$-symmetry of order $\ell>1$}. \EOD
\medskip\noindent
{\bf Corollary 4.} {\it Let $\De$ be a differential equation of order
$n$, and $X$ a (proper) partial $\mu$-symmetry for $\De$; denote by $Y$
the $\mu$-prolongation of order $n$ of $X$. Then there is $\ga \in
\Ga_n$ such that $\XT := \ga \cdot X$ is a (proper) standard partial
symmetry of $\De$.}
\medskip\noindent {\bf Proof}. This is a consequence of theorem 4 and
of lemma 4, with $X_0=\XT$ and $\be=\ga^{-1}$. \EOP
\bigskip
Partial $\mu$-symmetries differ from full $\mu$-symmetries also for
what concerns the reduction of PDEs, i.e. the problem of finding
$X$-invariant solutions. In the case of $\mu$-symmetries the reduction
-- which, as already remarked, is performed via the same method as
for standard symmetries \cite{GaMo} -- leads to a new equation, say
$\De^{[0]}(z,w)=0$, which involves the $X$-invariant variables $z$ and
$w$ determined by the invariance condition $Q=0$. In the case of
partial $\mu$-symmetries, instead, the PDE is transformed into an
equation of the form (cf. \cite{OR,PS,Sac}, see also \cite{CiK})
$$ R_1(s,z) \, \De^{[0]}_1 (z,w) \, + \, R_2 (s,z) \, \De_2^{[0]} (z,w)
\, + \, \ldots \ = \ 0 \eqno(3.5) $$ with coefficients $R_a$ depending
also on some non-invariant coordinate (denoted by $s$); now the
invariant solutions of the PDE can be obtained solving the {\it
system} of equations $\De^{[0]}_a=0$. In the case of a PDE involving a
single function $u$ depending on two independent variables only, this
becomes a system of ODEs (resp. a single ODE when $X$ is a full
$\mu$-symmetry).
\medskip\noindent {\bf Remark 12.} It follows immediately from this
argument and from lemma 4 that if $X$ is a partial $\mu$-symmetry of
order $\ell$, and $\S=\S_\ell$ is the corresponding subset of solutions
to $\De$, then there exists a standard partial symmetry $\XT$ of the
same order $\ell$, and with the same set $\S$ of solutions being
globally invariant under $\XT$ as well. \EOR
\section{The relation between local $\mu$-symmetries and \\
nonlocal standard symmetries}
In this section we want to point out relations between $\mu$-symmetries
and ordinary symmetries by using results of section 2 above. We will
show that there is (locally) a one to one correspondence between local
$\mu$-symmetries and nonlocal standard symmetries of exponential form.
We recall that in the definition of a $\mu$-symmetry for a system of
differential equations $\Delta=(\Delta_1, \ldots \Delta_r)$, in order
to apply the prolonged vector field on the solution manifold of the
system, we only need that compatibility conditions (1.7) hold on the
solution manifold $S_{\Delta}$ and not necessarily for all points in
$J^{(n)}M$. In this case we cannot guarantee that for each point $z \in
J^{(n)}(M)$ there are a neighborhood $A_z \subset J^{(n)} (M)$ of $z$
and a function $\ga : A_z \to G$ such that $\mu = \ga^{ -1 } (D_i \ga)
\d x^i$ on $A_z$. In other words, if (1.7) holds only on the solution
manifold $S_\De$, or however on a subset $C_\mu \not= J^{(n)} M$ of the
jet space (with $S_\De \sse C_\mu$), we cannot have $G$-equivalence of
$\mu$-symmetries with ordinary symmetries in the usual sense.
What we can prove is that there is a $G$-equivalence between
$\mu$-symmetries and nonlocal symmetries of exponential form. We
discuss here this equivalence only in the scalar case (i.e for the case
$q=1$ of a single PDE), for ease of notation; the case of systems of
PDEs is analogous.
Let $(M, \pi, B)$ a vector bundle with $1$-dimensional fiber
$F=\pi^{-1}(x)=\R$ and $B=\R^p$. We call {\bf non-local exponential
vector field} $X$ the formal vector field on $M$ given by
$$ \XT \ = \ e^{\int P_i(x, u^{(n)}) \d x^i } \ X \ , \eqno (4.1) $$
where $X = \xi^i (\pa / \pa x^i) + \varphi (\pa / \pa u)$ is a
(possibly generalized) vector field on $M$, and $P : J^{(n)} M \to
\R^p$ is a vector function defined on $J^{(n)}M$, satisfying
$$ D_k \, \[\int P_i (x, u^{(n)}) \, dx^i \] \ = \ P_k (x, u^{(n)}) \
. \eqno(4.2) $$
The integral $\int P_i(x, u^{(n)}) \, dx^i$ is in general a formal
expression. This generalizes the standard definition of nonlocal
exponential vector field for ODEs \cite{Olv1}.
\medskip\noindent
{\bf Definition 6.}
Given a scalar PDE $\De$ we say that the exponential vector field (4.1)
is a {\bf nonlocal exponential symmetry} of $\De$ if $D_i P_j = D_j
P_i$ on the solution manifold $ S_{\De}$, and $[X^{(n)}(\De)]_{S_\De} =
0 $. \EOD
\medskip
Now we can state our result relating $\mu$-symmetries and nonlocal
exponential symmetry of $\Delta$. This extends theorem 5.1 in
\cite{MuR1}.
\medskip\noindent
{\bf Theorem 5.} {\it Let $X$ be a vector field on $M$, and $\mu = P_i
(x,u^{(n)}) \d x^i$ a horizontal form, such that $D_i P_j = D_j P_i$ in
$C_\mu \sse J^{(n)} M$. Let $\De$ be a PDE of order $n$ on $M$, such
that $S_\De \sse C_\mu$. Let $X$ be a $\mu$-symmetry of $\De$. Then the
nonlocal exponential vector field $\XT = \exp [\int P_i (x,u^{(n)} ) \d
x^i ] \, X$ is a nonlocal exponential symmetry of $\De$. Conversely,
if $\XT$ as above is a nonlocal exponential symmetry of $\De$ with $X$
a vector field in $M$, then $X$ is a $\mu$-symmetry for $\De$.}
\medskip\noindent {\bf Proof.}
By using the results of section 2, it suffices to show that the
ordinary prolongation $W=\XT^{(n)}$ of $\XT$ and the $\mu$-prolongation
$Y$ of $X$ (with $\mu= P_i \d x^i$) are $G$-equivalent through the
function $$ \ga \ := \ \exp \[ \int P_i (x, u^{(n)}) \, \d x^i \] \ .
$$ We know from Theorem 1 that $\mu$-prolongations (with $\mu = P_i \d
x^i$) are $G$-equi\-va\-lent to ordinary prolongations by a function
$\ga \in \Ga_n$ satisfying $\ga^{-1} D_i \ga = P_i$; the general
(formal) solution of the previous equation for the unknown function
$\ga$ is given by $\ga = \exp[\int P_i (x, u^{(n)}) \d x^i ]$. \EOP
\medskip\noindent
{\bf Remark 13.} Determining the standard nonlocal symmetries of a
differential equation is in general a very difficult problem. On the
other hand, determination of local $\mu$-symmetries goes essentially
through the same procedure as determination of standard (Lie-point or
generalized) symmetries, i.e. we have a set of determining equations;
see \cite{GaMo}. \EOR
\medskip\noindent {\bf Remark 14.} If $\mu$ is written globally (on
$S_\De$) as $\mu = \ga^{-1} D \ga$ with $\ga : S_\De \to G$, see
section 2, we have $\XT = \ga \cdot X$ on $S_\De$. \EOR
\section{Discussion}
In this short section we present some further general remarks on our
approach.
\medskip\noindent
(1) First of all we note that although our motivation resided in
getting a sound understanding of $\lambda$ and $\mu$-symmetries, and
their relation with standard ones, which we believe is reached here,
our result can also be seen the other way round. That is, we have
actually obtained a description of how {\it gauge symmetries}, so
important and pervading in modern Physics based on variational
principles (e.g., Yang-Mills theories), enter in the theory of
symmetries of -- in general, non-variational -- general differential
equations.
\medskip\noindent
(2) We recall that it was shown in \cite{GaMo} that the determination
of $\mu$-symmetries goes through the solution of determining equations
pretty much as for standard symmetries. Note that even if one is
interested only in standard symmetries, obtaining $\mu$-symmetries
allows then to determine standard symmetries which are gauge-equivalent
to these.
\medskip\noindent
(3) The key role in our discussion -- and so in gauge symmetries of
(non variational) differential equations -- is played by horizontal
forms satisfying the horizontal Maurer-Cartan condition. These are also
known as zero-curvature representations, and considerable work has been
devoted to their study also from quite differents points of view. Here
we just refer the reader to \cite{Mar} and to the recent volume
\cite{AAM}, with the references in the works included therein, for
this.
\medskip\noindent
(4) The examples we provide (see next section) are mainly intended to
show that $\mu$-symmetries are -- in a suitable neighbourhood --
reducible to ordinary symmetries, maybe nonlocal. This corresponds to
our main motivation in enterprising this work, see above, but could
seem a bit disappointing to the reader seeking ``new'' symmetries. We
would thus like to stress that one could as well manifacture examples
of $\mu$-symmetries which are {\it not} reducible to ordinary ones,
e.g. by making sure that a nontrivial topology (in particular,
horizontal cohomology) is present, as in examples 2 and 3.
\medskip\noindent
(5) In the analysis of Yang-Mills theories, one is naturally led to
study gauge fields in terms of the topology of a certain principal
fiber bundle. The same applies here; more precisely, $\mu$-symmetries
are locally equivalent to standard ones, and the way in which different
local standard symmetries -- hence neighbourhoods -- are patched
together to make a $\mu$-symmetry depends on the topology of the
associated principal bundle $(P,\pi_G,J^{(n)} M)$ of fiber $G$ over the
relevant jet space; see section 2.
\medskip\noindent
(6) For what concerns the application to nonlocal symmetries, not only
the approach based on $\mu$-symmetries appears to be somewhat simpler
(see also remark 13 above), but it may even happen that generalized
vector fields with trivial characteristic vector provide nontrivial
$\mu$-symmetries, and hence nontrivial nonlocal symmetries. This is
shown in example 9, where we recover in this way some of the nonlocal
symmetries of the Calogero-Degasperis-Ibragimov-Shabat equation studied
by Sergyeyev and Sanders \cite{SeS}.
\medskip\noindent
(7) Finally, we would like to point out two possible directions of
further developement. One of these concerns potential symmetries
\cite{BK}, as suggested in example 10 below. A different direction (we
thank one of the referees for suggesting this) concerns the relation of
$\mu$-symmetries with pseudosymmetries \cite{Sok}; the latter are
related to factorization with respect to symmetries rather than to
invariant solutions.
\section{Examples}
In this section we consider examples illustrating our discussion of
previous sections, both for the scalar and the vector case, considering
full $\mu$-symmetries and partial $\mu$-symmetries as well.
Example 1 shows a global $G$-equivalence between a $\mu$-symmetry and a
standard one, while in examples 2 and 3 the $G$-equivalence is only
local. In examples 4 and 5 we deal with partial $\mu$-symmetries for
scalar equations (KdV and Boussinesq, respectively), while in examples
6 and 7 we deal with full and partial $\mu$-symmetries of a system of
PDEs. Finally, examples 8 and 9 illustrate the correspondence between
local $\mu$-symmetries and nonlocal standard symmetries of exponential
type, and example 10 recovers a well known case of potential symmetry
in terms of $\mu$-symmetries.
\subsubsection*{Example 1.}
In example 1 (sect.6.2) of \cite{GaMo}, one considered the vector field
$X = x \pa_x + 2 t \pa_t + u \pa_u$ and the form $\mu = \la \d x$,
where $\la$ is a real constant.
In this case, $\ga = e^{\la x}$; hence $$ \XT \ = \ e^{\la x} \ \( x
\pa_x + 2 t \pa_t + u \pa_u \) \ . $$
It is easy to check that the functions on $J^{(2)} M$ which have been
shown in \cite{GaMo} to be invariant under the $\mu$-prolongation of
$X$, are in fact also invariant under the standard prolongation of
$\XT$.
\subsubsection*{Example 2.}
We consider an example in the punctured plane, with $X$ given by the
standard rotation generator $ X = y \pa_x - x \pa_y $ and (writing for
ease of notation $r^2 = x^2 + y^2$) with $\mu$ given by
$$ \mu = (-y/r^2) \d x + (x/r^2) \d y \ . $$
This is singular for $r=0$, and now the domain on which $\mu$ is a
proper form is $B_0 = \R^2 \backslash \{ O \}$, which has a nontrivial
cohomology.
Writing $\theta = \arctan(y/x)$, this corresponds to $$ \ga \ =\
\exp\[\arctan(y/x)\] \, = \, \exp[\theta] \ , $$ and then to the vector
field $\XT = [\exp(\theta)] \pa_\theta$. Note that here $\ga$ is well
defined only locally, as it corresponds to a multivalued function. One
can check that $$ \zeta_1 \ := \ e^\theta u_\theta \quad {\rm and}
\quad \zeta_2 \ := \ e^{2 \theta} (u_{\theta \theta} + u_\theta ) $$
are invariant under the $\mu$-prolongation $Y$ of the vector field $X$.
In the $x,y$ coordinates (but retaining the notation $\theta := \arctan
(y/x)$ for ease of writing) these read
$$ \begin{array}{l}
\zeta_1 \ = \ (\exp \theta)\ (x u_y - y u_x) \ , \\
\zeta_2 \ = \ (\exp(2 \theta)) \ \( y^2 u_{xx} + x^2 u_{yy} - 2 x y
u_{xy} - x u_x - yu_y + x u_y -y u_x \) \ . \end{array} $$
Let $\zeta_3$ be any smooth nontrivial function $\zeta_3 = \zeta_3 (r,
u_r , u_{rr})$. Then any PDE of the form $$ \De \ :=\ F ( \zeta_1 ,
\zeta_2 , \zeta_3 ) \ = \ 0 $$ is a second-order equation invariant
under the $\mu$-prolongation $Y$. Its symmetry reduction gives an ODE
of the form $$\^\De \ = \ \^F(r,w_r,w_{rr}) \ := \ F (0,0,\zeta_3) \, =
\, 0 $$ for the function $u=w(r)$.
\subsubsection*{Example 3}
A variant of the previous example is obtained considering an equation
of the form $\Delta := F( \zeta_1 , \zeta_2 , \zeta_3) = 0$ where,
using again polar coordinates $r,\theta$ for notational convenience,
$$ \zeta_1 := r u_r \ , \ \zeta_2 := r u_r \log r - 2 i u_\theta \ , \
\zeta_3 := r^2 u_{rr} \ . $$
which contains also imaginary terms. This admits the scaling vector
field $X = x \pa_x + y \pa_y = r \pa_r$ as a $\mu$-symmetry, with
$$ \mu \ = \ {i\over 2} \( (-y/r^2) \d x \, + \, (x/r^2) \d y \) \ . $$
As in Example 2, we have a singularity in the origin, but here the
$\mu$-symmetry corresponds to a standard symmetry $\~X$ via a {\it
double-valued} function $\gamma$; we have indeed $\~X = \exp (i \theta
/ 2) X$.
\subsubsection*{Example 4}
Consider the KdV equation for $u=u(x,t)$
$$ \De \, := \, u_t + u_{xxx} + u \, u_x \ = \ 0 \ , $$
and the vector field
$$ X \ = \ {1\over t} \, {\pa\over{\pa x}} \, + \,
{1\over x} \, {\pa \over{\pa t}} \ . $$ We consider $\mu = (1/x) \d x +
(1/t) \d t$; this corresponds to $\ga = x t$.
Applying the $\mu$-prolongation $Y$ of $X$, we get
$$ \begin{array}{rl}
\De^{(1)} := & Y(\De)|_{\De=0} \, = \, (-2/ \ga) \, u_{xxx} \ , \\
\De^{(2)} := & Y(\De^{(1)})|_{\De^{(1)}=0} \, = \, \rho \
u_{xxx}|_{\De^{(1)}=0}\, =\, 0 \ , \end{array} $$
where $\rho$ is a function of $x$ and $t$. Therefore, $X$ is a partial
$\mu$-symmetry of order $\ell=2$.
The set ${\cal S}$ of solutions of the system $\De=\De^{(1)}=0$ is
given by $u=(x+c_1)/(t+c_2)$, where $c_1$ and $c_2$ are arbitrary
constants; this includes the solution $u=x/t$ which is invariant under
the $\mu$-prolongation of $X$.
As stated by remark 12, the set of solutions ${\cal S}$ turns out to be
a globally invariant symmetric set of solutions under $\XT=\ga \cdot
X$, which is indeed a standard partial symmetry for the KdV equation
\cite{CGpar}.
Finally, if one looks for the reduction of the KdV equation under the
above partial $\mu$-symmetry $X$, it is easy to see
that the invariance condition $Q=0$ gives the invariant variable
$z=x/t$, and substituting $u=w(z)$ into the equation one obtains
$$ w_{zzz} \ + t^2 ( \ w \, w_z\ -\ z\, w_z\ ) \ = \ 0 $$
which, as expected, is not a ``pure'' ODE for the unknown $w(z)$, but
has the form (3.5) (the role of the variable $s$ is played here by
$t$). Solutions invariant under the $\mu$-prolongation of $X$ are
provided by the system $ w_{zzz} = w_z (w-z) = 0$, and from this one
obviously recovers the solution $w=z=x/t$ found above.
\subsubsection*{Example 5.}
As another example, consider the Boussinesq equation
$$\De\, :=\, u_{tt}+u_{xxxx}+u\, u_{xx}+u_x^2\, =\, 0 $$
which admits the vector field
$$ X \, = \, {\pa \over \pa x} + {1 \over {t^2}} {\pa \over \pa t} -
\( {2 x \over {t^2} } + {10 t \over 3} \) {\pa \over \pa u} $$
as a partial $\mu$-symmetry (of order $\ell=3$) with $\d \mu = (2/t) \d
t$ (i.e. with $\ga=t^2$). One easily obtains indeed
$$ \begin{array}{rl}
\De^{(1)}:= & Y(\De) \ = \ (1/\ga) \, ( -10 t - 3 u_x - 2 t u_{xt} -
{5\over 3} t^3 u_{x x} - x u_{x x}) \\
\De^{(2)}:= & Y(\De^{(1)})|_{\De^{(1)}=0} \ = \ \rho \ ( 2 + u_{x t} +
t^2 u_{x x}) \end{array} $$ where $\rho=\rho(t)$.
The most general solution of the equation
$\De^{(2)} = 0$ is
$$ u\, = \, F(t) + G \( x - {t^3\over 3} \) - 2tx \ , $$
where $F,G$ are arbitrary functions of the indicated arguments.
Substituting this in the Boussinesq equation gives, after some
calculations, the general solutions
$$ u(x,t)= -{t^4\over 3} - 2t x +2 c_1 t - {12\over (x - t^3/3-c_1)^2}
\ {\rm and} \ u(x,t)=- {t^4\over 3} + c_2 t - 2t x + c_3 \eqno(6.1)$$
where $c_1,c_2,c_3$ are arbitrary constants.
Looking now for the solutions of the Boussinesq equation which are
invariant under the $\mu$-prolongation of $X$, one obtains from the
invariance equation $Q=0$,
$$u(x,t) \, = \, w(z) -2zt -t^4 \ ; \ z := x- (t^3/ 3) \ , $$
and the Boussinesq equation becomes
$$ (w_z^2 +w w_{z z} + w_{zzzz}) - 2t(3 w_z + z w_{z z}) \ = \ 0 \
, $$
which is of the form (3.5), as expected.
Solving the system of the two ODEs appearing in the parentheses in the
above equation gives the same solutions obtained above (6.1) but with
the restrictions $c_1=c_2=0$.
\subsubsection*{Example 6.}
In this and the following example we will consider two-dimensional
systems of PDEs; in these we write $u^1=u(x,y)$ and $u^2=v(x,y)$.
It is not difficult to verify that the system
$$ \cases{
u_y + x^2 v_x - y v_y \ = \ 0 \ , & \cr
x u_x - y u_y + x^2 v_x^2 + y^2 \ = \ 0 \ . & \cr} \eqno(6.2) $$
or more in general, any system of equations of the form $\zeta_\a (y,
v_y, x v_x, u_y + x^2 v_x, x u_x + x^2 y v_x) = 0$, admits the vector
field $$X\, =\, x{\pa\over \pa x}$$
as a $\mu$-symmetry with $\mu = \Lambda_x \d x + \Lambda_y \d y$ and
$$ \Lambda_x = \pmatrix{0 & y \cr 0 & 0 \cr} \ ; \ \Lambda_y =
\pmatrix{0 & x \cr 0 & 0 \cr} \ . $$
In this example, the matrix $\ga$ is given by
$$ \ga \ = \ \pmatrix{1 & xy \cr 0 & 1 \cr}$$
and it is easy to verify that all systems $\zeta_\a=0$ are indeed
symmetric under the standard symmetry in evolutionary form
$$ \XT := \ \ga \cdot X \ =\ \( x u_x + x^2 y v_x \) {\pa \over {\pa
u} } + x v_x {\pa \over \pa v} \ . $$
The reduction, imposing $Q^a=0$, i.e., $u_x=v_x=0$, gives $u=u(y),\,
v=v(y)$, and the above system becomes a system of equations $\zeta_\a
(y, v_y, 0, u_y, 0) = 0$ involving $u,v$ with their derivatives with
respect to $y$ only.
In particular, the system (6.2) is reduced in this way to
$$ \cases{ u_y - y v_y \ = \ 0 \ , & \cr - y u_y + y^2 \ =\ 0 \ , &
\cr} $$
with solution $u=y^2/2+c_1$, $v=y+c_2$.
\subsubsection*{Example 7.}
Consider the system
$$ \cases{
u_x \ = \-v_x \log (|v_x|) + v \ , & \cr
v_x \ = \ 2 v_y - y^2 + u_y + (v_x - v_y)^2 \ , & \cr} $$
and the vector field
$$ X \ = \ {\pa \over \pa x} + v {\pa \over \pa u} $$
with $\mu=\Lambda_x\d x+\Lambda_y\d y$ and
$$ \Lambda_x = \pmatrix{1 & 0\cr 0 & 1 \cr} \ ; \
\Lambda_y = \pmatrix{0 & 0\cr 0 & 1 \cr} \ . $$
The matrix $\gamma$ is then
$$ \gamma \ = \ \pmatrix{\exp x & 0\cr 0 & \exp(x+y) \cr} \ . $$
Direct computation shows that $X$ is a partial $\mu$-symmetry of this
system: indeed, according to definition 5$'$, the first application of
the $\mu$-prolongation does not give zero but produces the new system
$$ 0 \, = \, 0 \ ; \ v_x \, = \, v_y $$
and one needs another application of the $\mu$-prolongation of $X$.
Imposing $Q^a=0$ we get $u_x=v$, $v_x=0$; i.e. $v=v(y)$ $u=w(y)+
xv(y)$. Thus the reduction yields from the second equation of the
system (the first one is satisfied)
$$ w_y + x v_y + v^2_y + 2 v_y - y^2 \ = \ 0 \ , $$
which has the form (3.5), as expected in the case of partial
$\mu$-symmetries; this forces $v_y=0$ and leads to the solution $u =
y^3/3 + c x$, $v=c$, with $c$ an arbitrary constant.
\subsubsection*{Example 8.}
Let us consider the Euler equation
$$ u_t \ + \ u \, u_x \ = \ 0 \ . $$
It is shown in \cite{GaMo} that this admits as $\mu$-symmetry, with
$$ \mu \ := \ u \, \d x \ - \ (u^2 / 2 ) \, \d t \ = \ \a\ \d x + \b\
\d t \ , $$
the vector field
$$ X \ = \ \exp[- (u^2/2) t ] \ \( [B(u) - A(u) t / u] \,
{\pa \over \pa t} \ + \ A(u) \, {\pa \over \pa u} \) $$
where $A(u), \, B(u)$ are arbitrary functions. Note that for this $\mu$
the compatibility condition $D_t \a = D_x \b$ is satisfied only on the
solution manifold $S_\De$.
The $\mu$-symmetry $X$ corresponds to a nonlocal ordinary symmetry
$\XT$ of exponential type,
$$ \XT \ := \ e^{\int (u \d x - (u^2 /2 ) \d t ) } \ X \ . $$
\subsubsection*{Example 9.}
Let us consider the Calogero-Degasperis-Ibragimov-Shabat equation
$$ u_y \ = \ u_{xxx} \ + \ 3 \, u^2 \, u_{xx} \ + \ 9 \, u \, u_x^2 \
+ \ 3 \, u^4 \, u_x ; $$
motivated by \cite{SeS} we will choose the horizontal form
$$ \begin{array}{l}
\mu \ = \ \a \, \d x \ + \ \b \, \d y \\
\a \ := \ u^2 \ \ ; \ \ \b \ := \ 2 u u_{xx} + 6 u^3 u_x + u^6 - u_x^2
\ . \end{array} $$
This does not satisfy (1.7) in the full jet space $J^{(3)} M$; on the
other hand, (1.7) is satisfied on the solution manifold $S_\De$.
Indeed, on $S_\De$ we have formally $\mu = D P$ with $P = \int u^2 \d x
$, as readily checked by explicit computation. Obviously $D_x P = \a$.
As for $\b$, we have on $S_\De$
$$ D_y P \ = \ \int 2 u u_y \d x \ = \ \int \[ 2 u u_{xxx} + 6 u^3
u_{xx} + 18 u u_x^2 + 6 u^5 u_x \] \ \d x \ ; $$
integrating by parts this is just $\b$ given above.
The vector fields
$$ X_1 := \pa_x + u_x \pa_u \ \ {\rm and} \ \ X_2 := \pa_y + u_y \pa_u
$$
turn out to be $\mu$-symmetries for the CDIS equation with the $\mu$
given above\footnote{Actually, any $\mu$-symmetry (with the $\mu$
considered here) vector field $X = \xi (x,u^{(1)}) \pa_x + \eta
(x,u^{(1)}) \pa_y + \phi (x,u^{(1)}) \pa_u$ for the CDIS equation is
written as $X = h_1(x,y,u,u_x) X_1 + h_2(x,y,u,u_x) X_2$.}.
The nonlocal symmetries associated to these are $\XT_k = \ga \cdot
X_k$, and the function $\ga$ is (see section 4) $\ga = e^P = \exp [
\int u^2 \d x ]$. This is general a formal expression only, but a
proper function when restricted to $S_\De$.
Note that both $X_1$ and $X_2$ have trivial characteristics; it is
quite remarkable that nevertheless they originate a nontrivial nonlocal
symmetry.
\subsubsection*{Example 10}
The Burgers equation $$u_{xx}-uu_x-u_t=0 $$ admits the vector field $$
X = (2 w_x + u \, w) {\partial\over{\partial u}} $$
where $w=w(x,t)$ is any solution of the heat equation $w_t=w_{xx}$, as
a $\mu$-symmetry with
$$ \mu \ = \ {1\over 2}u \, \d x + \( {1\over 2} u_x - {1\over 4} u^2
\) \d t \ := \ \a \, \d x \ + \ \b \, \d t \ . $$
Clearly $D_t \a = D_x \b$ is satisfied only on the solution
manifold of the Burgers equation; therefore, this $\mu$-symmetry is
equivalent to a nonlocal exponential symmetry. Actually, this nonlocal
symmetry is also a potential symmetry \cite{BK}: this follows from the
property of the Burgers equation of being a conservative divergence
equation.
\vfill\eject
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\end{document}