% This paper appeared in "Acta Applicandae Mathematicae",
% vol. 28 (1992), p. 43
\magnification=1200
%\input EBL.macro
\def\a{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\e{\varepsilon}
\def\eps{\varepsilon}
\def\g{\gamma}
\def\l{\lambda}
\def\la{\lambda}
\def\s{\sigma}
\def\th{\theta}
\def\z{\zeta}
\def\phi{\varphi}
\def\D{\Delta}
\def\Ga{\Gamma}
\def\La{\Lambda}
\def\S{\Sigma}
\def\Th{\Theta}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\H{{\cal H}}
\def\K{{\cal K}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\N{{\cal N}}
\def\U{{\cal U}}
\def\W{{\cal W}}
\def\SD{S_\Delta}
\def\c{\cdot}
\def\x{\times}
\def\pa{\partial}
\def\grad{\nabla}
\def\ss{\subset}
\def\sse{\subseteq}
\def\all{\forall}
\def\1{{(1)}}
\def\n{{(n)}}
\def\LRA{\Leftrightarrow}
\def\({\left( }
\def\){\right) }
\def\[{\left[ }
\def\]{\right] }
\def\^#1{{\widehat #1}}
\def\~#1{{\widetilde #1}}
\def\.#1{{\dot #1}}
\def\Fix{{\rm Fix}}
\def\Ker{{\rm Ker}}
\def\Corollary{{\it Corollary: }}
\def\Remark{{\it Remark: }}
\def\Example{{\it Example: }}
\def\Proof{{\it Proof: }}
\def\Theorem{{\it Theorem: }}
%\baselineskip=20pt
\parindent=0pt
\parskip=10pt
{\nopagenumbers
\parskip=10pt
%\footnote{}{ {\tt version 5.02 - 6/8/91 } }
\vfill
\centerline{\bf Reduction and Equivariant Branching Lemma:}
\centerline{\bf Dynamical Systems, Evolution PDEs, and Gauge Theories}
\bigskip \bigskip \vfill
\centerline{\it Giuseppe Gaeta}
\centerline{\it Centre de Physique Theorique}
\centerline{\it Ecole Polytechnique, F-91128 Palaiseau (France)}
\vfill
{\parindent=50pt \parskip=3pt
\item{1.} Introduction \hfill 1
\item{2.} Symmetry of differential equations \hfill 2
\item{3.} The reduction lemma \hfill 4
\item{4.} The equivariant branching lemma \hfill 6
\item{5.} Symmetries of PDEs \hfill 10
\item{6.} Gauge symmetries and Lie point vector fields \hfill 12
\item{7.} Reduction lemma for gauge theories \hfill 14
\item{8.} Symmetric critical sections of gauge functionals \hfill 18
\item{9.} Equivariant branching lemma for gauge functionals \hfill 19
\item{10.} Evolution PDEs \hfill 21
\item{11.} Symmetries of evolution PDEs \hfill 23
\item{12.} Reduction lemma for evolution PDEs \hfill 27
\item{ } References \hfill 30
}
\vfill
\eject}
\pageno=1
\parskip=20pt
{\bf 1. Introduction}
\bigskip
One of the simplest yet most useful tools in equivariant bifurcation theory is
the so called
"Equivariant Branching Lemma" (EBL in the following).
This was first proved by Cicogna [1] and Vanderbauwhede [2] in the context of
bifurcation of
stationary solutions; Golubitsky and Stewart [3] provided then an extension to
the case of Hopf
bifurcation (of periodic solutions). The EBL is also effective in the context
of the so called
"quaternionic bifurcation" [4,5,6], but this latter case seems to be of no use
in applications.
All the above mentioned results deal with bifurcation problem which are
equivariant under the action
of a linear representation of a (compact) Lie group [7-12], and have
proved to be of great use in application, see e.g. [10].
The EBL was recently extended by Cicogna [13] to the case of bifurcation
problems symmetric under
general - i.e. not necessarily linear - groups of Lie-point transformations.
For a treatment of
these transformations in the context of differential equations, we refer to
[14-18].
The purpose of the present note is to extend the EBL to gauge symmetries and
gauge-symmetric
problems; this will be done by looking at gauge symmetries as a specific class
of Lie-point
symmetries in an appropriate space. We will also obtain a weaker result
(reduction lemma) for
evolution PDEs.
In the preparation of this work, after obtaining our results, we became aware
that they - as well as
the previous similar ones - can be also seen as deriving from the "Symmetric
Criticality Principle"
of Palais [19,20] (we thank prof. Bourguignon for pointing out this fact and
the references). This
principle seems to have passed unnoticed in the context of bifurcation theory
(probably due to the
fact it was originally meant to deal just with gauge theories), but it is
actually a very powerful tool and worth being widely known.
On the other side, the powerful results obtained by equivariant bifurcation
theory over the last few
years are probably not well known to gauge theorists; one good reason for this
is
that equivariant bifurcation theory deals usually with finite-dimensional
linear
group actions, but recently it has been shown that the basic results of the
theory extend to Lie-point, and therefore gauge, symmetries as well [21,22]. A
related discussion, focused on gauge theories, is presented in [23].
%\vfill \eject
\bigskip
\bigskip
{\bf 2. Symmetry of differential equations}
\bigskip
Despite the simplicity of the EBL, we will best understand it by looking at it
as composed of two
parts, a "reduction lemma" and a "branching lemma"; the reduction part is in
facts completely
general and does not depend on any bifurcation phenomena or assumption.
Let us consider a time-evolution ordinary differential equation (ODE), which
could also be the
bifurcation equation relative to some bifurcation problem,
$$ \.u = F(u) \kern 2cm F(0)=0 \eqno(1) $$
where $u$ belongs to a smooth manifold $\U$ which we will consider as embedded
in $R^N$ with
coordinates $\{ u^1 ,... , u^N \}$; $F(u)$ will then be a $C^\infty$ tangent
vector field on $\U$,
$$F : \U \to T \U \eqno(2) $$
Next, assume that eq. (1), which will also be written as
$$ \D (u, \.u ) \equiv \.u - F(u) =0 \eqno(3) $$
has a Lie-point time-independent group of (evolutionary) symmetries $G_\D$,
with Lie algebra
$\G_\D$, whose elements will be denoted by $\g$.
This means the following (for details we refer to e.g. [14] or to the other
texts quoted above):
a generic vector field (VF) $\eta$ on $\U$ will be written as
$$\eta = \phi^i (u) {\pa \over \pa u^i} \eqno(4) $$
It determines a VF $\eta^\1$ (its first prolongation) in the jet space $J_1 \U
$ (actually, in this
case $J_1 \U = T \U$), which can be thought with coordinates $\{ \.u^1 , ... ,
\.u^N ; u^1 , ... , u^N
; t )\}$. The equation $\D = 0$ determines a manifold $S_\D$ in $J_1 \U$,
$$ S_\D = \{ ( \.u , u ;t ) ~/~ u \in \U ~,~ \.u^i = F^i (u,t) \} \eqno(5) $$
The VF $\g \in \G_\D$ are those such that their first prolongation leaves
invariant $S_\D$, or
$$ \g^\1 : S_\D \to T S_\D \eqno(6) $$
\Remark As the notation of eq. (1) suggests, we want actually to deal with
autonomous ODEs. In this
case, $S_\D$ will actually be of the form
$$ S_\D = S_\D^0 \x R $$
with $S_\D^0$ belonging to the reduced jet space $J_1^0 \U = J_1 \U / R $ which
can be thought as
the subspace of $J_1 \U$ with coordinates $\{ ( \.u^1 , ... , \.u^N ; u^1 , ...
, u^N ) \}$.
\Remark Obviously, any nonautonomous ODE can be seen as autonomous by adding a
new variable $u^{N+1}
= t$; in this way one can consider VFs on $R \x \U$, i.e. of the general form
$\eta = \phi^i (u,t)
\pa / \pa u^i + \tau (u,t) \pa / \pa t$; we will keep to (1), (4) for
notational simplicity.
Now, to any subgroup $G_i \ss G_\D$ we can associate a pointwise invariant
manifold, $\Fix (G_i)
\sse \U$:
$$ \Fix (G_i ) = \{ u \in \U ~/~ gu = u ~~ \all g \in G_i \} \sse \U \eqno(7)
$$
(this could be empty); at the Lie algebra level, if $\G_i$ is the Lie algebra
of $G_i$, we have
$$ \Fix (G_i ) \sse W_i \equiv \Ker ( \G_i ) = \{ u \in \U ~/~J\g \cdot u = 0
~~ \all \g \in \G_i \}
\sse \U \eqno (8) $$
The equality sign between $\Fix (G_i )$ and $ W_i$ holds for $G_i$ a connected
Lie group; if this
is not the case, $\Fix (G_i )$ can in general be a proper subspace of $W_i$.
\Example Let us consider a linear example: let $\U = R^3$, and $G_1$ be given
by $SO(2) \x Z_2$,
where $SO(2)$ represents rotations around the axis $u^3$, and $Z_2$ reflections
across the
$(u^1 , u^2 )$ plane. Then $W_1$ is the whole $u^3$ axis, while $\Fix (G_1 ) $
is just the origin.
Remark that for $G_2 = SO(2) \x Z_2$ with $SO(2)$ as before and $Z_2$ a
reflection in the plane
$(u^1 ,u^3 )$ (or any plane including the axis $u^3$) one has $\Fix (G_2 ) =
W_2$.
To any point $u \in \U$ are associated an isotropy subgroup $G_u \sse G_\D$ and
an isotropy
subalgebra $\G_u \sse \G_\D$ (which is the Lie algebra of $G_u$):
$$ G_u = \{ g \in G_\D :~ gu =u \} \sse G_\D \eqno(9) $$
$$ \G_u = \{ \g \in \G_\D :~ \g u =0 \} \sse \G_\D \eqno(10) $$
%\vfill \eject
\bigskip
\bigskip
{\bf 3. The reduction lemma}
\bigskip
We can now state the reduction lemma:
{\it Reduction Lemma:} A solution $u(t)$ of (1) with initial datum $u(t_0
)=u_0$
will satisfy $u(t) \in W_0 \equiv \Ker (\G_{u_0} ) ~ \all t \ge t_0$.
This can also be stated as
{\it Reduction Lemma:} The manifolds $W_i \equiv \Ker ( \G_i )$ are invariant
under the flow of $\D$
for any $\G_i \sse \G_\D $.
\Remark Clearly, if $\G_\a \ss \G_\b$, then $W_\b \sse W_\a$; in other words an
order relation among
subalgebras of $\G_\D$ implies an inclusion relation among the invariant
manifolds $W_i$; the
invariant manifolds corresponding to maximal isotropy subalgebras will not
contain any other
invariant manifold.
\Remark If, as before, $\G_\a \ss \G_\b$ and $W_\b \ss W_\a$, a solution with
$u_0 \in W_\a
\backslash W_\b$ can flow into the invariant submanifold $W_\b$. We stress that
this can be a limit
process, but it can also be a process which takes place in a finite time. As an
example (again,
linear), let $\U = R^2$ and consider the equation $\.u_i = - k^2 u_i^\a$, $k
\not= 0$, $u\in R^2$, so that $G_1 =
SO(2) \in G_\D$, with $G_2 = \{ e \} \ss G_1$ and $W_1 = \{ 0 \} $, $W_2 =
R^2$. Then for
$u(t_0 ) \not= 0$, we have that if $\a =1$ the solution $u(t)$ tends to $W_1$
as
a limit point, while for $0 < \a < 1$ this is reached in a finite time. Anyway,
it should be remarked that in this ODE case $F(u)$ fails to be $C^\infty$ (or
even $C^1$) just on $W_1$, which indeed could be not invariant, as e.g. for $k
=3/2$, $\a = 1/2$.
\Proof Let us now prove the reduction lemma. First of all, we notice that,
writing the evolution VF
$\e : \U \to T \U$, i.e. the VF such that $\.u = \eps \c u$,
$$ \e = F^i (u) {\pa \over \pa u^i} \eqno(11) $$
the condition for $\eta \in \G_\D$ is precisely (see [22])
$$[\eta , \e ] = 0 \eqno(15) $$
>From this the lemma follows at once: in facts, let $\g \in \G_0 \ss \G_\D$.
Then by definition $\g
\cdot u =0 ~ \all u \in W_0$, and $\e \cdot 0 = 0$ since $\e = F(u) \pa_u$ and
$F(0) = 0$. But $\g
\in \G_\D \Longrightarrow [\e , \g ] =0$, so that $\g \e u - \e \g u =0$, i.e.
$\e : W_0 \to T W_0$
\hfill $\bullet$
The reduction lemma implies a useful consequence: let $F_\a (u)$ be the
restriction of $F(u)$ to
$W_\a \ss \U$; then $F_\a : W_\a \to T W_\a$, and one has the
\Corollary Let $u(t)$ be a solution of $\.u = F(u)$ and $v(t)$ a solution of
$\.v = F_\a (v)$; let
$u(t_0 ) =v(t_0 ) \in W_\a$. Then $u(t)=v(t) ~~ \all t \ge t_0$.
In other words, in order to study solutions of (1) one can consider the
simplest (or not more
difficult) equation $\.x = F_0 (x)$, where $F_0 (x) = F(u) \vert_{W_0}$.
\Remark If one has to find critical points of a potential $V(x)$ defined on a
manifold $\M \sse
R^N$, i.e. solutions of the equation $\grad V (x) =0$, these can be seen as
stationary solutions of
the equation $ \.x^i = F^i (x) = - \grad^i V(x)$
In this way, we see that the reduction lemma can be recast in the following
form in variational case:
{\it Reduction Lemma (variational case):} If $V: \M \to R$ and $V(x)$ is
invariant under the action
of the Lie group $G: \M \to \M$, and $G_x =G_\a \sse G$ is the isotropy group
of the point $x \in
\M$, then $\grad V(x) \in T_x \Fix (G_x ) \sse T_x W_\a$. Therefore $\grad V
(x_0 )=0$ if and only
if $\( y , \grad V(x) \) =0$, $\all y \in T_x W_\a$, where $(.,.)$ is the
standard scalar product
induced in $T_x W_\a$ by the scalar product in $R^N$.
\Remark This is actually at the basis of Michel's theorem [21], which inspired
the first version of
the EBL. I.e., we have followed a path which is just opposite to the historical
one.
\Remark It should be noticed that the argument used to prove the reduction
lemma are purely
geometrical; in particular, nothing changes if the manifold $\U$ happens to be
of countably infinite
dimension, or if $\M$ is an Hilbert space or an infinite dimensional Riemannian
manifold (so that a
scalar product is well defined).
\vfill \eject
\bigskip
\bigskip
{\bf 4. The equivariant branching lemma}
\bigskip
We can now introduce the bifurcation setting. We consider the case in which one
of the $u^i$'s in
(1) can be regarded as a parameter, also denoted $\la$. For the sake of
simplicity, we will assume
$\U = \U_e \x \La$, where $\La = ( - \mu , \mu ) \ss R$ is the parameter space,
and $\U_e \sse
R^{N-1}$ with coordinates $(u^1 , ..., u^{N-1} )$.
It was shown in [22] that in considering symmetries of the form (4) it suffices
then to consider those with $\phi^N = 0$, i.e. those which do not act on $\la$;
in this case the invariant manifolds $W_\a$ will be foliated as $W_\a = \La \x
W_\a^{(\la)}$. If moreover $$ {\pa \phi^i (u) \over \pa u^N } \equiv {\pa
\phi^i \over \pa \la } = 0 \eqno(16) $$ we have a trivial fibration $$W_\a =
\La \x W_\a^0 \eqno(18) $$
We can restate the bifurcation lemma in the bifurcation setting as follows:
{\it Reduction Lemma (bifurcation setting):} Let us consider the equation
$ \.u = F (\la , u)$, where $ \la \in \La ,~ u \in \U ;~ F(\la , u_0 ) = 0 ~
\all \la $
,with $F$ a $C^\infty$ vector field, $F: \La \x \U \to T \U$, and let $G_\D^0$
be its
$\la$-independent symmetry group which stabilizes $u_0$ (i.e. $g u_0 =0$), with
Lie algebra
$\G_\D^0$. Then for any subalgebra $\G_\a^0 \ss \G_\D^0$, the manifold $W_\a^0
= \{ u \in \U ~/~ \g
u = 0 ~ \all \g \in \G_\D^0 \}$ is invariant under the flow of the equation,
and so is therefore
$\La \x W_\a^0$.
>From this the EBL follows at once. Let us first introduce {\it standard
bifurcation assumptions}: $i)$
$F(\la , u_0 )=0 ~ \all \la \in \La$; $ii)$ if $L(\la ) = F_u ( \la , u_0 )$,
$\Ker L(\la ) = 0$ for
$\la \not= 0$, and $L(0)$ is a Fredholm operator of index zero; $iii)$ if $\s_i
(\la )$ are the
critical eigenvalues of $L(\la )$, $d \s_i (\la ) / d \la > 0$ for $\la = 0$.
Let us moreover assume
a {\it stability condition}: there is an open compact set $\K \ss \U$, of the
same dimension $m$ as
$\U$ and whose border $\pa \K$ is $m-1$ dimensional (a disk in $\U$) such that
$F( \la ,u)$ points
inward on $\pa \K$ and $u_0 \in \K$. Then we have the
{\it Equivariant Branching Lemma (stationary case): } If $\G_\D^0$ admits a
subalgebra $\G_\a^0 \ss
\G_\D^0$ such that $W_\a^0$ is one-dimensional (with $T_0 W_\a^0 \ss T_0 \Ker
L(0)$ ), then there is a branch of stationary solutions $u_\a (\la)$
bifurcating
from $u_0$ and such that $u_\a (\la ) \in W_\a^0 ~ \all \la \in \La$.
\Remark Clearly, $T_0 W_\a^0$ is the tangent space to $W_\a^0$ in $u=u_0$.
Also, $\Ker L(0)$ is
obviously a linear space.
\Proof By the reduction lemma, we are authorized to consider the reduction
$F_\a (\la , u)$ of
$F (\la , u)$ to the manifold $W_\a^0$; this manifold contains $u_0$ by
definition of $\G_\D^0$. Let
$\K_\a = \K \cap W_\a^0$; this is an open interval containing $u_0$, so we can
give an orientation to
it, and $\pa \K = K_- \cup K_+$. By the stability condition $F_\a (\la , K_- )
>0$, $F_\a (\la ,
K_+ ) <0$. It follows at once from the bifurcation assumptions that $\.u =
F_\a (\la , u)$ undergoes
a bifurcation at $\la = \la_0$, and there is a branch of stationary solutions;
the reduction lemma
ensures that if $F_\a (\la , u_\a (\la ) ) =0$, then also $F ( \la , u_\a (\la
))=0$. \hfill
$\bullet$
In the Hopf case, one adopts Hopf bifurcation assumptions, i.e. in $ii)$ above
we ask for the
spectrum of $L(\la )$ not to touch the line Re$\s (\la )=0$ for $\la \not=
\la_0$, and that only a
finite number of eigenvalues cross the imaginary axis of the complex plane for
$\la = \la_0$ (the
linear space spanned by the corresponding eigenvectors will be called $\N$),
with $d {\rm Re} \s (\la ) / d \la > 0$ and the remaining of the spectrum at
finite distance from the imaginary axis.
{\it Equivariant Branching Lemma (periodic case): } If $\G_\D^0$ admits a
subalgebra $\G_\a^0 \ss \G_\D^0$ such that
$W_\a^0$ is two-dimensional (with $T_0 W_\a^0 \ss T_0 \N$), then there is a
branch of
periodic solutions $u_\a (\la)$ bifurcating from $u_0$ and such that $u_\a (\la
) \in W_\a^0 ~
\all \la \in \La$.
\Proof As before, we can reduce to $W_\a^0$ by the reduction lemma, and
consider in it the invariant
compact set $\K_\a = \K \cap W_\a^0$. It suffices then to invoke
Poincare'-Bendixson theorem
[26] to ensure the existence of periodic solutions. \hfill $\bullet$
\Remark The above case is nongeneric, but can occurr, for $\G_\a^0$ a maximal
isotropy subalgebra of
$\G_\D^0$; notice that it can also occurr if $\exists \G_\b^0$ such that
$\G_\a^0 \ss \G_\b^0 \sse
\G_\D^0$, and which meets the conditions for the stationary case EBL.
\Remark The stability assumption could be substituted, in both stationary and
periodic case, by a
weaker one: i.e., it suffices to ask $F_\a (\la , u )$ points inward of $\K_\a$
on $\pa \K_\a$.
For higher dimensional $W_\a$, we have a result due to Cicogna [27] (his proof
for the linear case is purely topological and so applies to the Lie-point case
as well):
{\it Equivariant Branching Lemma (stationary case, bis): } If $\G_\D^0$ admits
a subalgebra $\G_\a^0 \ss \G_\D^0$ such that
$W_\a^0$ is $(2m +1)$-dimensional (with $T_0 W_\a^0 \ss T_0 \Ker L(0)$), then
there is a branch of
stationary solutions $u_\a (\la)$ bifurcating from $u_0$ and such that $u_\a
(\la ) \in W_\a^0 ~
\all \la \in \La$.
\Proof It is well known that any VF on $D^{2m+1}$ has at least a zero [28,29].
The reduction lemma and the stability assumptions allows actually to reduce to
a
disk $\K_\a = D^{2m+1} \ss W_\a^0$, and hence the lemma. \hfill $\bullet$
\Remark Such an extension is not possible for the case of stationary solutions:
there is no
equivalent of the fixed point theorem for invariant circles, so that we can not
go beyond the
classical Poincare'-Bendixson theorem.
{\it Equivariant Branching Lemma (variational case, bis):} Consider the case
$\D$ is issued from a
variational problem, i.e. $\D \equiv {\dot u_i } - \pa V(u) / \pa u_i$, so that
stable stationary solutions of $\D$ corresponds to minima of the potential. If
$\G_\D^0$ admits a subalgebra $\G_\a^0 \ss \G_\D^0$ such that $W_\a^0$ is of
any
finite dimension (with $T_0 W_\a^0 \ss T_0 \Ker L(0)$ ), then there is a branch
of minima $u_\a (\la)$ of $V(u)$ bifurcating from $u_0$ and such that $u_\a
(\la
) \in W_\a^0 ~ \all \la \in \La$.
\Remark If in the previous cases $\K_\a$ is not $D^1 , D^2$ but $S^1 , S^2$,
the lemma continues to
hold in a weaker form. This is due to the assumptions on $L(\la )$: infacts,
$S^1 \backslash u_0
\simeq D^1$, and now $u_0$ corresponds to $\pa D^1$; in the same way, $S^2
\backslash u_0 \simeq
D^2$, and $u_0$ corresponds to $\pa D^2$ (e.g. by stereographic projection, see
fig.1). In this way
one can also deal with the case $\K_\a = S^{2m+1}$: since $S^n \backslash u_0
\simeq D^n$, we are
reconducted to the above mentioned case (EBL, stationary case bis). We still
have stationary or
periodic solutions in the appropriate subspaces, but these will not in general
form a smooth branch
bifurcating from $u_0$, see the example.
\Example Let $\U$ be the torus $T^2 = S^1 \x S^1$, with coordinates $\th , \psi
\in I \x I$, $I = [-
\pi , \pi ]$. Consider the equation
$$ \eqalign{ \.\th =& \la \sin \th - \sin^3 \th \cr \.\psi =& - \sin \psi \cr}
\eqno(19) $$
Clearly, $(\th , \psi ) = (0,0)$ is a stationary solution for any $\la$, and at
$\la = 0$ this
undergoes a bifurcation ($L(\la ) = {\rm diag} (\la , -1)$). Also, $\sin \psi
\pa_\psi \in \G_\D$,
and it leaves pointwise invariant the circle $\psi =0$ (and the one $\psi = \pm
\pi$), so $WP/a^0 =
S^1$; it is clear also that we have a bifurcation of stationary solutions.
Notice that in the
absence of the term $\sin^3 \th$ one would have stationary solutions in
$W_\a^0$ as well, but now
they would not bifurcate from $(0,0)$; instead, one would have a discontinuous
transition. In
physical terms, one would have a first order transition instead than a second
order one.
We have then a weaker form of the EBL, which we will state as
{\it Equivariant "Branching" Lemma (discontinuous case): } Let us consider the
equation
$ \.u = F ( \la , u)$, in which $ F (\la , u_0 ) =0 $, with $\la , u , F$ as
before, and $G_\D$ its
$\la$-independent symmetry group; let $G_\D^0 = G_\D \cap G_{u_0}$, with Lie
algebra $\G_\D^0$. Let
the bifurcation assumptions hold for $L(\la ) = F_u (\la , u)$ and let there
be a compact set $\K
\ss \U ,~u_0 \in \K$, invariant under the flow of $F$. Then, let $\G_\D^0$
admit an isotropy
subalgebra $\G_\a$ such that $\K_\a = \K \cap W_\a = S^n$ and such that $T_0
W_\a \ss \N$. If
$n=2m+1$, for $\la > \la_0$ there exists in $\K_\a$ a new stable stationary
solution $\~u_0 (\la )$;
if $n=2$, for $\la > \la_0$ there exists in $\K_\a$ a periodic solution $\~u
(t)$ distinct from $u_0$.
\Proof Immediate from the reduction lemma and the above remarks.
\Remark If no additional assumption is made, we do not know if $\lim_{\la \to
0} \~u_0 (\la) = u_0$
or not.
\Remark Again, one can consider stationary solutions of a problem $\.x = \grad
V( \la , x)$ and
reach the case of $G$-invariant potentials. In this context, in the language
of Landau theory [30] the above lemma guarantees the existence of a (first or
second order) phase transition.
\Remark All the above results hold true if we consider $G_\D^0 , G_\a$ instead
of $\G_\D^0 ,
\G_\a$, and $\Fix (G_\a )$ instead of $W_\a$, etc. We have preferred the Lie
algebra setting because
one has algorithmic ways to compute the Lie algebra of Lie-point symmetries of
differential
equations [31], while no systematic procedure exists for discrete symmetries.
In
the context of first order transitions, anyway, it becomes more relevant to
consider groups than algebras, in view of physical applications [30].
\bigskip
\bigskip
{\bf 5. Symmetries of PDEs}
\bigskip
We want now to make contact with gauge symmetries; to this purpose, let us see
how the above discussion extends to PDEs and gauge theories.
In the case of PDEs, we will denote as $x,t$ the independent variables,
and as $u$ the dependent ones; the independent spatial variables $x$
will belong to a manifold $X$ which we will see as a $d$-dimensional
manifold embedded in $R^m$, with coordinates $\{ u^1 , ... , u^m \}$; (the
time $t$, when appearing, will belong to a manifold $\Theta \sse R$,
i.e. $\Th = R$ or $R_+$ or still $S^1$ if we look for periodic
solutions). The dependent variables will belong to a manifold $\U$,
which again we will consider as embedded in $R^N$ with coordinates ${u^1
, ..., u^N }$ as in the ODE case.
We have therefore a natural setting in terms of a fiber bundle $\pi : E
\to \B$ , where $\B = X \x \Theta$ or $\B=X$ as appropriate (i.e., $\B$ is the
base space of independent coordinates; from now on we consider $\B = X$ unless
discussing
evolution equations), with fiber $\U$ and projection $\pi (u,x) = x$
A generic Lie point VF would then be written as
$$ \eta = \phi \pa_u + \xi \pa_x \equiv \phi^\a (x,u) {\pa \over \pa
u^\a } + \xi^i (x,u) {\pa \over \pa x^i} \eqno(1) $$
We will consider only projectable VFs, i.e. those of the form
$$\eta = \phi^\a (x,u) {\pa \over \pa u^\a } + \xi^i (x) {\pa \over \pa
x^i} \eqno(2) $$
In physical terms, eq. (2) means that the transformations of space-time
do not depend on the fields $u(x)$, and is therefore a natural
assumption.
In the class of projectable VFs, we will single out the evolutionary VFs, i.e.
those of the form
$$\eta = \phi^\a (x,u) {\pa \over \pa u^\a} \eqno(3) $$
which are those which do not affect space-time, but only the fields $u$.
These are in facts local gauge symmetry generators, as we will point out
in a moment; the case $\phi = \phi (u)$ would correspond to global gauge
symmetries.
{\it Remark on terminology:} We will denote local gauge symmetries
simply as gauge ones, and reserve for global gauge symmetries the term
"rigid gauge symmetries".
A PDE of $n$-th order will be written as
$$\D (x, u^\n ) =0 \eqno(4) $$
or also, when in the following we want to point out its evolutionary character
(and writing
then $t$ for the time coordinate)
$$ \D (x,t; u^\n ) \equiv u_t - F(x,t;u^{[n]} ) = 0 \eqno(5) $$
In this case it will be understood that in $F$ do not appear time
derivatives of the $u$'s.
%$F$ can also be seen as a VF on $\B \x J^n \U$.
If the $x,t$ do not appear explicitely in $F$, the equation is
autonomous.
As before, the equation $\D =0$ determines a manifold $\SD$ in $J^n \U$,
$$ \SD = \{ (x,t;u^n ) ~/~J u^{(k)} \in J^k \U ,~k \le n ~;~ \D (x,t; u^n
) =0 \} \eqno(6) $$
and if the equation is autonomous, this is naturally fibered as
$$\SD = \SD^0 \x \B \eqno(7) $$
with $\SD^0$ belonging to the reduced jet space $J^n \U / \B \equiv \U^n$.
Now, as it is well known, [14-18], a solution $u(x,t)$ to (4) is a section
of $\pi : E \to B$ such that the $n$-lift of its graph (which is the
same as the graph of its $n$-th prolongation) $\Ga_{u^n } \equiv \Ga^n_u
\ss J^n \U$ belongs entirely to $\SD$, and the lift of a Lie-point
symmetry of (4) is a differentiable transformation of $ J^n \U$ which
leaves invariant the manifold $\SD \ss J^n \U$; the Lie point algebra of
symmetries of $\D =0$, $\G_\D$, is made up of those VF $\g$ as in (1)
which satisfy
$$\g^n : \SD \to T \SD \eqno(8) $$
We will denote by $\G_\D^{(g)} \ss \G_\D$ the algebra of evolutionary,
or gauge, symmetries (see (3)), by $\G_\D^{(r)} \ss \G_\D$ that of
rigid gauge symmetries (4), and by $\G_\D^{(p)} \ss \G_\D$ that of
projectable symmetries (2). Clearly, one has
$$ \G_\D^{(r)} \ss \G_\D^{(g)} \ss \G_\D^{(p)} \ss \G_\D \eqno(9) $$
\bigskip
\bigskip
%\vfill \eject
{\bf 6. Gauge symmetries and Lie point vector fields}
\bigskip
We want now to point out that, as claimed above, gauge symmetries are a
special case of evolutionary Lie-point symmetries. We will now denote all
the independent variables as $x \in X$, i.e. we suppose space and time
play equivalent roles and do not distinguish between them.
In a gauge theory [32,33,34] the physical fields are sections of a fiber
bundle $\pi : E \to X$ with typical fiber $F= \pi^{-1} (x)$ ($F$ is a
linear space for matter fields; the Lie gauge group itself for gauge
fields); there is an action of the Lie group $G$, the gauge group,
defined in $F$ by the representation $T$, with $T_g : F \to F$ the
operator corresponding to the group element $g \in G$.
One is specially interested, for physical reasons [32], in the case of
$G$ a compact Lie group and $T$ a unitary linear representation.
A gauge transformation is then a function $g : X \to G$, and this acts
on a section $\s (x)$, $\s : X \to F$, of the bundle as
$$ (g \cdot \s ) (x) = T_{g (x)} \s (x) \eqno(10) $$
Obviously, one can also look at gauge transformations at the Lie algebra
level: let $\G$ be the Lie algebra of the group $G$, and $L$ the
representation of $\G$ corresponding to the representation $T$ of $G$
(i.e., $L$ gives the infinitesimal generators for $T$). Then, an
(infinitesimal) gauge transformation will be a function $\g : X \to \G$,
and this will act on a section $\s (x)$, $\s : X \to F$ of the bundle as
$$ (e + \eps \g ) \cdot \s = \s + \eps \delta \s \eqno(11) $$
$$ (\delta \s ) (x) \equiv (\g \cdot \s ) (x) = L_{\g (x) } \s (x)
\eqno(12) $$
Now, given the Lie algebra $\G$, we can choose a basis in it, $\{ \ell_1
, ... , \ell_s \}$, such that
$$ [J\ell_i , \ell_j ] = c_{ij}^k \ell_k \eqno(13) $$
and that $\G$ is the linear span of $\{ \ell_1 , ... , \ell_s \}$, i.e.
$\all \ell_0 \in \G$, $\exists a=(a_1 , ... , a_s ) \in R^s$ such that
$$ \ell_0 = \sum_{i=1}^s a_i \ell_i \eqno(14) $$
Clearly, to such $\ell_i$'s correspond $L_i$'s with the same properties,
i.e.
$$ [L_i , L_j ] = c_{ij}^k L_k \eqno(15) $$
and $\all L_0 \in L$, $\exists a \in R^s$ such that
$$ L_0 = \sum_{i=1}^s a_i L_i \eqno(16) $$
Now, any function $\g : X \to \G$ induces a function $\a : X \to R^s$ by
$$ \g (x) = \sum_{i=1}^s \a_i (x) L_i \eqno(17) $$
where $\a (x) = ( \a_1 (x) , ... , \a_s (x) )$, $\a_i (x) \in R$.
\Remark In algebraic language, the set $\Ga$ of infinitesimal gauge
transformations $\g : X \to \G$ is a module over $\G$ [35].
To connect with the formalism used to discuss Lie-point symmetries, let
us just look at the action of $\G$ on $F$: if we choose in $F$ a basis
$\{ u_1 , ..., u_n \}$, and $\ell_0 \in \G$ is represented in $L$ by the
matrix $L_0 = L_{ik}$, then it is also possible to consider a
representation $\^L$, equivalent to $L$, in terms of differential
operators or, which is the same, of tangent vector fields on $F$; now to
the matrix $L_0 =L_{ik} $ corresponds the operator
$$ \^L_0 = ( L_{ik}^+ u_k ) { \pa \over \pa u_i } \eqno(18) $$
so that $L_{ik} u_k \equiv - \^L_0 u$ (using $L^+ = - L$, since we have a
unitary representation of a compact Lie group, as recalled above).
Therefore, to the infinitesimal gauge transformation $\g : X \to \G$ is
naturally associated a Lie point VF
$$ \^\g = - \sum_{i=1}^s a_i (x) \^L_i \equiv \a_i (x) [L_i ]_{mn} u_n {\pa
\over \pa u_m } \equiv \phi^m (x,u) {\pa \over \pa u_m } \equiv \phi
(x,u) \pa_u \eqno(19) $$
\Remark The linearity of the representation $L$ is reflected in the fact
that $\phi$ is linear in the $u$'s,
$$ {\pa^2 \phi^m \over \pa u_i \pa u_j } = 0 \eqno(20) $$
Also, $L^+ = - L$ ensures $\pa \phi^m / \pa u_m =0$.
\Example It can be worth considering shortly a very simple example, just
to fix ideas. Let $\U = R^2$, $G = SO(2) \simeq U(1)$ and $T_\th =
\pmatrix{ \cos \th & - \sin \th \cr \sin \th & \cos \th \cr} $, so that
$s=1$ and the Lie algebra $\G$ is generated by
$$ L_1 = \pmatrix{0&-1 \cr 1 & 0 \cr} = - L_1^+ \eqno(21) $$
To this corresponds the VF
$$ \^L_1 = - u_1 {\pa \over \pa u_2 } + u_2 {\pa \over \pa u_1 }
\eqno(22) $$
In facts, $(I + \eps L_1 ) u = u + \eps \delta u$; $\delta u = \pmatrix{
- u_2 \cr u_1 \cr} = - \^L_1 u$. To any $\g : X \to \G$ corresponds
then, via the $\a$ introduced above, a VF
$$ \^\g = - \a (x) \^L_1 = \( \a (x) u_1 \) {\pa \over \pa u_2 } -
\( \a (x) u_2 \) {\pa \over \pa u_1 } \eqno(23) $$
\Remark In (18) one has $L^+$ instead than $L$ in order to ensure that
$ [ \^L_i , \^L_j ] = c_{ij}^k \^L_k $ with the same $c_{ij}^k$ as in (15);
if using $L$, one would have a minus sign in the $c_{ij}^k$.
\bigskip
\bigskip
{\bf 7. Reduction Lemma for gauge theories}
\bigskip
We do now aim at extending the EBL to critical section of gauge
invariant functionals, or gauge functionals for short.
Let $u=u(x)$ with $x\in X \sse R^d $ and $u \in U \sse R^n$, with $X$ a
smooth manifold of dimension $q$ embedded in $R^d$ (with coordinates $\{
x_1 , ..., x_d \}$) and $\U$ a smooth manifold of dimension $p$ embedded
in $R^n$ (with coordinates $\{ u_1 , ..., u_n \}$), and let $L [u]$ be a
smooth functional defined by a smooth density $\L : U \x X \to R$, i.e.
$$ L [u ] = \int_M \L (u(x) , x ) dx \kern 2cm M \sse X \eqno(1) $$
Let us now consider a compact Lie group $G$ (with Lie algebra $\G$)
acting on $U$ by the representation $\La$. If for any smooth function $g
: X \to G$ one has $L [ \La_g u ] = L [u]$ then we say that $L$ is
gauge-invariant for $G$. In
facts, this is equivalent to
$$ \L ( \La_{g(x)} u(x) , x ) = \L (u(x),x) \eqno(2) $$
due the arbitrarity of $g: X \to G$.
At the Lie algebra level, we have that under a function $\g : X \to \G$,
the functional (1) becomes
$$ \eqalign{ (I + \eps \g ) L [u] =& \int_M \left[ \L ( u(x) + \eps (\g \cdot
u) (x ) , x ) \right] dx = \cr ~ & = \int_M \left[ \L (u(x),x) + \eps
\delta \L (u(x),x) \right] dx = L [u] + \eps ( \delta L ) [u] \cr } \eqno(3)
$$
where the variation $\d L$ is given by
$$ \d L [u] \equiv \int_M \left[ {\pa \L \over \pa u } (u(x),x) \cdot
(\g \cdot u) (x) \right] dx \eqno(4) $$
Writing $\g$ as a vector field, $\g = \phi (u,x) \pa_u$, this becomes
$$ \d L [u] = \int_M \[ {\pa \L \over \pa u_i } (u(x),x) \cdot \phi^i
(u(x),x) \] dx \eqno(5) $$
\Remark The above notation suggests that $\L$ depends on $u$ but not on
its derivatives, as in facts we will assume in the following. Anyway,
the case in which $\L$ depends e.g. on first derivatives of $u$ is
readily seen to be equivalent to the above, enlarging the space $U$ to
$U \x R^{pq}$, where the new variables are $\pa_i u_j \equiv \pa u_j /
\pa x_i$, $i=1,...,d$, $j=1 , ... , n$, with constraints to take into
account the relations existing between the $u$'s and the $\pa u$'s (i.e.
we assign a {\it contact structure} [36,37]), and others to ensure that in
facts
$x\in X$, $u \in U$. Also, in this case $\G$ will act not by $\g$ but rather
by its first prolongation $\g^\1$, see [14] for details; see also [23].
For the sake of semplicity, in the following we will also assume that
$\pa \L / \pa x =0$, i.e. the density $\L$ is autonomous (or
homogeneous, as all the points on $M$ are equivalent).
The group of gauge transformations has a Lie algebra $\Ga$,
$$ \Ga = \{ \eta = \phi (x,u) \pa_u ~/~ \phi (x,u) = \a_i (x) L_i \}
\eqno(6) $$
where $\a , L_i$ are as in the previous section; given a section $\s$ of
the bundle $\pi : E \to X$, $E=X \x U$, its symmetry (isotropy) algebra
is
$$ \Ga_\s = \{ \eta \in \Ga ~/~ \eta \c \s = 0 \} \eqno(7) $$
Given a subalgebra $\Ga_0$ of $\Ga$, we can define the space of
$\Ga_0$-invariant sections , $\W_0$, by
$$ \W_0 = \{ \s \in \Sigma (E,X) ~/~ \eta \c \s =0 ~~J\all \eta \in \Ga_0 \}
\eqno(8) $$
where $\Sigma (E,X)$ is the set of sections of $\pi : E \to X$.
\Remark If the action of $\G$ on the fiber $U$ is linear (we have seen
that this amounts to have $\phi$ linear in $u$), then $\W_0$ is a linear
space [19].
\Remark In general, it can be proven that if the action of $\G$ is
smooth with compact orbits, $\W_0$ is a smooth submanifold of $\Sigma$.
The variation of $\d L$ under $\s \to \s + \eps \tau$, with $\tau \in
\Sigma$, can be written in the notation used for (5) as
$$ \d L [\s ] = \eps \int_M \[ \tau^i (x) {\pa \over \pa u^i } \L (\s
(x) ) \] dx \equiv \eps \int_M \[J(\tau \c \grad ) \L \] dx \eqno(9) $$
This shows that for the variation of $L [\s ]$ to vanish for every
$\tau$, it is needed to have
$$ \grad \L ( \s (x) ) = 0 \kern 2cm \all x \in M \eqno(10) $$
A section $\s (x) $ is said to be critical for $L$ if $\d L [\s ] =0$
under $\s \to \s + \tau$, for every $\tau \in \Sigma$. The above
discussion means that we have the following {\it criticality criterion:} $\s $
is
critical for $L$ as in (1) if and only if (10) is satisfied.
We can now state the RL for the case of gauge functionals
\medskip
{\it Reduction Lemma (gauge functionals):} If $\L : U \to R$ and $\L$ is
invariant under the action of the group of gauge diffeomorphisms generated by
$\Ga$, and $\Ga_\s = \Ga_0 \sse \Ga$ is the isotropy algebra of the
section $\s \in \Sigma$, then $\grad \L (\s (x)) \in T_{\s } \W_0$.
Therefore, $\d L [\s ] =0$ if and only if $\d L =0$ under $\s \to \s +
\eps \tau$ for all $\tau \in \W_0$.
\medskip
\Proof The gradient appearing in (10) is, by definition (see (9)), in $T
U_x$, where $U_x \equiv \pi^{-1} (x)$ is the fiber through $x$. More
precisely, $( \grad \L ) (\s (x)) \in T_{ \s (x)} U_x$. It suffices
therefore to consider the finite dimensional restriction of our setting
to $U_x$. The algebra $\Ga_\s$ can also be characterized as
$$ \Ga_\s = \{ \g : X \to \G ~/~J\g (x) \in \G_{\s (x)} ~~ \all x \in X
\} \eqno(11) $$
where $\G_{\s (x) } \sse \G$ is the isotropy subalgebra of $\s (x) \in
U$. Analogously, $\W_\s$ can be characterized as
$$ \W_\s = \{ \th \in \Sigma (E,X) ~/~ \th (x) \in \W_{\s (x) } ~~ \all x \in
X \} \eqno(12) $$
Now, we know by the discussion of the RL in variational case and for
finite dimension that, if $\L : U \to R$ is invariant under $\G_\a$,
then $\grad \L \in T W_\a$; in the present case $\G_\a \equiv \G_{\s
(x)}$ (in facts by definition, or by (11), $\g (x) \in \G_{\s (x)} ~
\all \g \in \Ga_\s$). Therefore, for any $x$ we have $\grad \L (\s (x))
\in T_{\s (x)} W_{\s (x)} \sse T_{\s (x)} U_x$. By the definition (12)
we have that $\grad \L [\s ] \in T_{\s (x)} \W_0$: indeed, for any
$\tau \in \Sigma (E,X)$, $(\tau (x) \c \grad ) \L (\s (x) ) \in \W_\s$, since
$\grad \L (\s (x)) \in T W_{\s (x)}$.
>From the above discussion it also follows that given $\tau (x) = \tau_0
(x) + \tau_1 (x)$, with $\tau_0 \in \W_0$ and $\tau_1 \in W_0^c \equiv \Sigma
(E,X) \backslash
\W_0$, the variation of $L[\s ]$, for $\s \in \W_0$, under $\s \to \s + \eps
\tau$ is the same as the one under $\s \to \s + \eps \tau_0$, i.e. what
is affirmed by the lemma. \hfill $\bullet$
\Remark the above lemma can be restated as follows: the sections which
are critical among sections with prescribed symmetry are also critical
{\it tout court}. For this reason, an analogue of this lemma [19,20] was
also called "Principle of Symmetric Criticality".
As remarked in the introduction, the result of Palais just mentioned
seems to be not so widely known - especially in the bifurcation
community - as it would deserve to be. Therefore, we quote it here
(adapting the notation to that of the present paper) from [19]:
\medskip
{\it Symmetric Criticality Theorem (Palais):} Let $G$ be a compact Lie
group, $X$ a smooth manifold, $\pi : E \to X$ a smooth $G$-fiber bundle
over $X$, and $\Sigma$ a Banach manifold of sections of $E$. Let $G$ act
on $\Sigma$ by $(g \s ) (x) = g (\s (x) )$ and let $L : \Sigma \to R$ be
a smooth $G$-invariant function on $\Sigma$. Then the set $\W$ of $G$-
equivariant sections in $\Sigma$ is a smooth submanifold of $\Sigma$,
and if $\s \in \W$ is a critical point of $L \vert_\W $ then $\s $ is in
fact a critical point of $L$.
\medskip
\Remark Palais does actually also consider the case of $G$ acting
nontrivially on the base manifold $X$; we have excluded this case from
his statement as well as from our discussion just for simplicity, but it
does not present new difficulties.
\Remark For the proof of the Symmetric Criticality Theorem, see [19], as
well as for extensions of it, examples, and counterexamples to naive
generalizations. A number of applications to mathematical physics are
given in [20].
\Remark Strictly speaking, the Symmetric Criticality Theorem concerns
only sections $\s$ such that $\Ga_\s = \{ \g : X \to \G_0 \}$ for some
subalgebra $\G_0 \sse \G$, but the generalization to $\Ga_\s = \Ga_0
\sse \Ga$ is straightforward.
\bigskip
\bigskip
{\bf 8. Symmetric critical sections of gauge functionals}
\bigskip
We can now use the RL in order to get informations about critical
sections of gauge functionals. From now on, an isotropy subalgebra will be a
subalgebra which is the isotropy algebra of a point different than the origin.
We will need, as in the finite
dimensional case, a {\it stability assumption}: there is an open compact set
$K
\ss U$, topologically a disk, containing the origin, such that $\grad \L (u)$
points outward of $K$ on $\pa K$. We have then that:
{\it Symmetric existence lemma:} Given a maximal isotropy subalgebra
$\Ga_0 \sse \Ga$, there is a critical section $\s \in \W_0$ for $L$.
\Proof Let us consider $L_0$, the restriction of $L$ to $\W_0$. If we
consider $W_0 = \{ u \in U ~/~ \G_0 \sse \G_u \}$, then $K_0 = K \cap
W_0$ is a compact set, and $\grad \L$ points outward of $K_0 $ on $\pa
K_0$. It is easy to see that there are critical sections $\s_c \in \W_0$
of $L_0$, which satisfy $\s_c (x) \in K_0$. The RL ensures these are
also critical for $L$. \hfill $\bullet$
\Remark Notice that this holds for any finite dimension of $W_0$.
\Remark For a linear action of $\G$ on $U$, the trivial section $\s_0
(x) =0$ has always full symmetry, $\Ga_{\s_0} = \Ga$; the critical
section whose existence is granted by the lemma could just be $\s_0$.
This is also the reason for considering $\Ga_0$ a maximal isotropy
subalgebra: for nonmaximal ones $\Ga_1 \ss \Ga_0$, the lemma continues
to hold, but it could happen that for every critical section $\s_1$ in
$\W_1$, $\Ga_{\s_1} = \Ga_0$, i.e. actually $\s_1 \in \W_0$; i.e.,
repeatedly applying the lemma over a chain of subgroups does not give
any new information.
\Remark Obviously, the lemma continues to hold if $\grad \L$ points
inward of $K_0$ on $\pa K_0$; one prefers to look for minima of $L$ due to
clear physical reasons.
The above remarks suggest to consider the case of $\s_0$ a local (non
degenerate) maximum for $L$. This guarantees the existence of a small
disk $B \ss U$ such that $\grad \L$ points outward of $B$ on $\pa B$.
Our stability assumption will then also require that ${\bar B} \ss K$.
We have then:
{\it Symmetric existence lemma (bis):} Given the above assumptions and a
maximal isotropy subalgebra $\Ga_0 \sse \Ga$, there is a critical
section $\s \in \W_0$ for $L$, with $\s \not= \s_0$.
\Proof Just as before, considering the compact set $K \backslash B$ in
the place of $K$. Now, the critical section will satisfy $\s_c (x) \in
K_0 \backslash B_0$, where $K_0 = X \cap W_0$ as before, and $B_0 = B
\cap W_0$. \hfill $\bullet$
\Remark The property $\s_c (x) \in K_0 \backslash B_0$ depends crucially
on the triviality of the bundle $E = X \x U$. For nontrivial $E$, one
can just say $\s_c (x) \in K_0$. In other words, for nontrivial $E$, the
theorem holds only locally.
\Remark We stress that the above results guarantee criticality of $\s \in \W_0$
also "in directions transverse to $\W_0$", but no statement can be made about
stability: a section which is a minimum for $L_0$ could be a saddle point for
$L$.
\Remark This is also an appropriate point to stress that in our discussion we
avoid topological matters, which are of a different nature than those discussed
here. In other words, our discussion deals with (and holds only for) trivial
bundles or, equivalently, local sections.
\vfill \eject
\bigskip
\bigskip
{\bf 9. Equivariant Branching Lemma for gauge functionals}
\bigskip
Finally, let us introduce in $\L$, and therefore in $L$, a $\C^\infty$
dependence on a real parameter $\l \in \La = [-a , a ] \ss R$, so to
have
$$ L_\l [\s ] = \int_M \L (\l , \s (x) ) dx \eqno(13) $$
Define the $n \x n$ real symmetric matrix $H(\l )$ as
$$ H_{ij} (\l ) = { \pa^2 \L ( \l , 0 ) \over \pa u_i \pa u_j
}J\eqno(14) $$
and denote its eigenvalues as $h_i (\l )$, $i= 1 , ... , n$.
We want to consider the situation of $\s_0 (x) \equiv 0$ a critical
section $\all \l \in \La$, stable for $\l < \l_0 =0$ and unstable for
$\l > \l_0$. This means that $u=0$ is a minimum of $\L (\l , u)$ for $\l
< 0$, and a saddle point or a maximum for $\l \ge 0$. For simplicity of
notation, we will just consider the case in which $u=0$ loses stability
in all directions at the same time, the degeneracy being entirely due to
the symmetry; i.e. $G(u)$ is not contained in any proper linear subspace
of $U$ for $u \not= 0$.
We have therefore, for what concerns the eigenvalues $h_i (\l)$, the
following {\it bifurcation assumptions}:
$$ h_i (\l_0 ) =0 \kern 1cm ; \kern 1cm { \pa h_i (\l_0 ) \over \pa \l
}J> 0 \eqno(15) $$
(the matrix $H (\l )$ is playing here the role played by $L_0$ in the
finite dimensional ODE case, see sect.4)
We have then the
\medskip
{\it Equivariant Branching Lemma (gauge):} If the above bifurcation and
stability assumptions
are verified, and $\G_0 \sse \G$ is a maximal isotropy subalgebra of
$\G$ for the action of $\G$ on $U$, then there is a branch of critical
sections $\s_\l (x)$ for $L_\l [\s ]$, bifurcating from $\s_0$ at $\l =
\l_0$ and such that $\s_\l \in \W_0 ~~ \all \l \ge \l_0 $.
\medskip
\Remark As already pointed out, $\s_\l \in \W_0$ is equivalent to $\s_\l
(x) \in W_0 ~~J \all x \in X$.
\Proof We do actually use both the fact that $\L$ does not depend on
derivatives of $\s$ neither on $x$, and the trivial structure of the
bundle $E = X \x U$.
In facts, if we just consider a fiber $\pi^{-1} (x)$, and $\L : \La \x
\pi^{-1} (x) \to R$, we recover the finite dimensional situation studied
above, and we are granted there is a branch of subsets $U_\l \ss U$
bifurcating from the origin $U_0 = \{ 0 \}$ at $\l = \l_0$ and such that
$u \in U_\l$ is a (degenerate) minimum of $\L (\l ,u)$ (in general,
$U_\l = G (u)$ $\all u \in U_\l$, i.e. the degeneracy is entirely due to
the symmetry and $U_\l$ is just a group orbit; we also recall that $G(u)
\simeq G / G_u $). Moreover, the finite dimensional EBL ensures that
there is such a branch $U_\l^0$ satisfying $U_\l^0 \in W_0 ~~J \all \l$.
Now, if we consider the bundle
$$ C_\l^0 = M \x U_\l^0 \ss E $$
with $\pi : C_\l^0 \to U_\l^0 \ss W_0 \ss U$, sections $\tau$ of this
bundle are transversally critical sections of $E$, i.e. for any
variation $\tau \to \tau + \eps \d \tau$ with $\d \tau (x) \in N_{\tau
(x)} U_\l^0$ (where $ N U_\l^0$ is the normal bundle to $U_\l^0$), we
have $L_\l [\tau + \eps \d \tau ] = L_\l [ \tau ] + O (\eps^2 )$.
In other words, not only critical sections for $L_\l$ restricted to
$\W_0$ are critical sections for unrestricted $L_\l$, but in turn
critical sections for $L_\l$ restricted to $U_\l^0$ are also critical
for $L_\l $ on $\W_0$, and therefore for unrestricted $L_\l$.
We can now just consider that by definition of $U_\l$, $L_\l [ \tau +
\eps \d \tau ] = L_\l [\tau ] + O( \eps^2 )$ for $\d \tau (x) \in
T_{\tau (x)} U_\l$, so that every section $\tau : M \to U_\l^0$ is
critical for $L$. Such a degeneracy should not be surprising: as
recalled above, $U_\l^0 \simeq G(u)$, $u \in U_\l^0$, and $G(u) \simeq G
/ G_0$ (indeed, $G_u = G_0$ for $u \in W_0$ and $G_0$ a maximal isotropy
subgroup), so that the degeneracy does actually just correspond to the
action of the gauge symmetry. \hfill $\bullet$
\Remark This seems an appropriate point to stress that our proofs are
generalizable to $x$-dependent $\L$, or to higher order $\L$ (i.e. $\L$
depending on higher derivatives of $u$), but they rely essentially on
the fibered structure of $E$ and the fact the gauge symmetry preserves
this structure. I.e., they are not generalizable to general Lie-point
(i.e. not projectable) symmetries. See also [13].
\bigskip \bigskip {\bf 10. Evolution PDEs} \bigskip
Let us now consider evolution PDEs. Among the independent variables, we
single out the time $t \in \Th$, where usually $\Th = R $ or $R_+$ (but
if we want to consider only time-periodic solutions, we should set $\Th
= S^1$), and denote by $x \in X \sse R^d $ the spatial ones. In the
following discussion, we will usually set $X = R^1$ for ease of
notation, but it will be quite clear that our results continue to hold
for any smooth finite dimensional manifold $X$. As for the dependent
variable $u=u(x,t)$, we set $u \in \U \sse R^n$, where $\U$ is a smooth
finite dimensional manifold, which we will think as embedded in $R^n$,
i.e. with coordinates $\{ u_1 , ... , u_n \}$. Therefore, a function
$u(x,t)$ can also be thought as a section $\s_u$ of the trivial fiber
bundle $E = \U \x \( X \x \Th \) $, with projection $\pi : E \to \U \x
\( X \x \Th \)$, $\pi \s (x,t) = (x,t)$, and fiber $\U$. The base space
$\( X \x \Th \)$ will also be denoted by $\B$; with this, $E = \U
\x \B$, $\pi : E \to \B$.
We will write an evolution PDE in the form
$$ \D (u^{(N)} ) \equiv u_t - F [u] =0 \eqno(1) $$
where $F$ depends smoothly on $u$ and its $x$-derivatives up to a finite
order $N$. The space spanned by $u$ and its $x$-derivatives of order up
to $N$ will be denoted in the following as $\M$. Therefore, in (1)
$$ F : \M \to T \U \eqno(2) $$
We will also write (1) in a slightly different way, introducing the
evolutionary VF
$$ f = F[u] \pa_u \equiv F^i [u] {\pa \over \pa u^i }J\eqno(3) $$
Notice that $f$ is not a vector field on $\U$, since $F$ depends on
derivatives of $u$ and not only on $u$ itself. It can instead be seen as
a vector field on $\M$ (such that the components in the directions
corresponding to $x$-derivatives of $u$ are zero), i.e.
$$ f : \M \to T \M \eqno(4) $$
It should be quite clear that this $f$ is {\it not} the prolongation of
any VF on $\U$.
In the following, we will find it useful to denote the set of smooth VFs
on a manifold $M$ by $V(M)$. Therefore,
$$ f \in V(\M ) \eqno(4') $$
With this notation, the equation $\D$ (by this we will mean $\D =0$)
reads also
$$ u_t = f \c u \eqno(5) $$
Now, the solution manifold $\SD = \{ u^{(N)} ~/~ \D ( u^{(N)} ) = 0 \}
\sse J^N \U$ is actually a fibered one: let us denote by $\~\U$ the
space of first time derivatives of $u$ (so that $\~\U \simeq T_x \U$),
and by ${\cal R}$ that of higher $t$-derivatives and of mixed (i.e.,
involving both $x$ and $t$) derivatives. Then, it is clear that actually
one has
$$ S_\D = \SD^e \x {\cal R} \kern 1cm ; \kern 1cm \SD^e \ss \~\U \x \M
\x \B \equiv \^E \eqno(6) $$
and it will be convenient to consider only $\SD^e$, from now on denoted
$\SD$ tout court, which is a smooth submanifold of the bundle $\^\pi :
\^E \to \B$. With this notation,
$$ \SD \sse \^E \eqno(7) $$
We have remarked before that a function $u(x,t)$ can be naturally seen
as a section $\s_u$ of the bundle $E$. This is naturally lifted to a
section $\^\s_u$ of the bundle $\^E$; this corresponds to the function
$\^u (x,t)$, $\^u : \B \to \~\U \x \M$ which associates to a point
$(x,t) \in \B$ the values of $u_t (x,t)$, of $u(x,t)$, and of
$x$-derivatives of $u$ of order up to $N$ (i.e., of $D_J u(x,t)$, where
$J$ is a multiindex in $x$, $\vert J \vert \le N$). The space $\^\U \x
\M$, which is the fiber of $\^E$, will also be denoted by $\~E$, so that
$\^u : \B \to \~E$.
With this notation, a function $u(x,t)$ is a solution to (1) if and only
if the corresponding section $\^\s_u$ lies entirely in $\SD$:
$$ \D [u] =0 \LRA \^\s_u \sse \SD \sse \^E \eqno(8) $$
\Remark As the notation used in (1), (4) suggests, we are actually
interested in considering autonomous (in both $x$ and $t$) equations,
i.e. $\pa_t F = \pa_x F =0$. In this case, we have
$$ \SD = \~\SD \x \B \kern 1cm ; \kern 1cm \~\SD \sse \~\U \x \M \equiv
\~E \eqno(9) $$
\Remark We can still use the particular form of eq.(1): in facts, this is
solved for $u_t$, so that $\SD$ can be seen as a section of the bundle
$\^E'$ (with total space ${\^E}' = \^E$) over $\M \x \B$; for autonomous
equations, also $\~\SD $ can be seen as a section of the bundle $\~E$
over $\M$. In both cases, the fiber of the bundle is $\~\U$.
\bigskip \bigskip {\bf 11. Symmetries of evolution PDEs} \bigskip
Let us now consider a VF on $\U$, $\eta \in V (\U )$, which we write
as
$$ \eta = \phi (u) \pa_u \equiv \phi^i {\pa \over \pa u^i } \eqno(1) $$
\Remark In general, we could take $\eta = \phi (x,u) \pa_u + \xi (x)
\pa_x$, which is a projectable VF on $\U \x X$, and the discussion to
follow would give similar results (we stick to (1) in order to keep the
notation as simple as possible). It is instead essential that $\phi_t = 0$,
$\xi_t =0$, and even more that $\eta$ has zero component in the $t$
direction. We will shortly discuss this later on.
Now, $\eta \in V (\U ) $ induces a VF $\^\eta \in V( \M )$ by
$$ \^\eta = \^\phi \c \^\grad \eqno(2) $$
where $\^\phi ,~ \^\grad$ are vectors of components
$$ \eqalign{ \^\phi =& \( \phi , D_x \phi , ... , D_x^N \phi \) \cr
\^\grad =& \( {\pa \over \pa u} , {\pa \over \pa (D_x u)} , ... ,
{\pa \over \pa (D_x^N u)} \) \cr } \eqno(3) $$
with $D_x$ the total $x$ derivative [14], so that $\^\eta$ is simply
$$ \^\eta = \sum_{m=0}^N \( D_x^m \phi \) {\pa \over \pa (D_x^m u)}
\eqno(4) $$
(i.e. the Lie derivative along $\eta$, or $\phi$, acting on functions
defined on $\M$).
In the case of multidimensional $X$, this is better rewritten in terms
of the multiindex $J$ [14] as
$$ \^\eta = \sum_{\vert J \vert \le N} \( D_J \phi \) \c \grad_J \kern
1cm; \kern 1cm \grad_J = {\pa \over \pa (D_J u)} \kern 1cm \(
\grad \equiv \grad_0 \) \eqno(5) $$
\Remark In geometrical terms, $\^\eta$ is the projection along $T \M$ of
$\eta^N$, the $N$-th prolongation of $\eta$, which is in $V (J^N \U )$.
As for the action of $\eta$ on $\~\U$, by standard prolongation formula
again, we get
$$ \eta^{(t)} = \( D_t \phi \) {\pa \over \pa u_t } \equiv \phi_u u_t
{\pa \over \pa u_t } \eqno(6) $$
(in the second equality we have used $\pa_t \phi =0$).
We have therefore
$$ \eta^N \c \D \equiv \( \eta^{(t)} + \^\eta \) \c \D \eqno(7) $$
and the condition [14] for $\eta$ as in (1) to be a symmetry of $\D$,
i.e. for $\eta \in \G_\D$, is
$$ \[ \phi_u u_t - \^\eta F \]_{\SD} \equiv \[ \phi_u u_t - ( \^\phi \c
\^\grad ) F \]_{\SD} = 0 \eqno(8) $$
and since $u_t \vert_{\SD} \equiv F[u]$, this also reads
$$ (F \c \grad ) \phi - (\^\phi \c \^\grad ) F = 0 \eqno(9) $$
By recalling the $f$ introduced above, one sees that (9) is nothing
else than the commutation relation
$$ \[ f , \^\eta \] =0 \eqno(10) $$
among vector fields in $V ( \M )$.
\Remark Condition (10) is formally analogous to the one obtained for
symmetries of ODEs (this latter involving elements of $V (\U)$ alone).
In facts, the idea behind our treatment is that of looking at an
evolution PDE as an infinite dimensional dynamical system. Condition
(19) will indeed permit to parallel the discussion conducted in the ODE
case, with of course some new features and difficulties.
The formula (19) suggests we can also define a Poisson bracket $\{ . , .
\} : V(\M ) \x V ( \M ) \to V( \M )$. Given $A,B \in V(\M )$,
$$ A = \a^J \grad_J \kern 1cm ; \kern 1cm B = \b^J \grad_J \eqno(11) $$
their Poisson bracket will be
$$ \{ \a , \b \} = (\a \c \^\grad ) \b - (\b \c \^\grad ) \a \eqno(12)
$$
or, in component notation,
$$ \{ \a , \b \}^j = (\a^k \c \^\grad_k ) \b^j - (\b^k \c \^\grad_k )
\a^j \eqno(13) $$
The obvious relation among $[.,.]$ and $\{ .,. \}$ is given by
$$ \[ A,B \] = \{J\a , \b \}^J \grad_J \eqno(14) $$
With this, (19) reads also
$$ \{ \~F , \~\phi \} = 0 \eqno(15) $$
where $\~F = (F , 0, ... ,0)$ in the notation (3).
\Remark One can check that $\{ .,. \}$ is actually a Poisson bracket by
verifying the Jacobi identity holds. All the usual properties of Poisson
brackets are then granted; in particular, given two VFs $\eta_1 = \phi
(u) \pa_u$ and $\eta_2 = \psi (u) \pa_u$ in $\G_\D$, i.e. such that
$\{J\~F , \~\phi \} = \{J\~F , \~\psi \} =0$, their bracket $\~\z = \{
\~\phi , \~\psi \}$ does also satisfy $\{J\~F , \~\z \} =0$. This is no
surprise, since $\^\eta$ is the (projection of the) $N$-prolongation of
$\eta$, and the above relation then follows from [14] $\[ pr^{(N)}
\eta_1 , pr^{(N)} \eta_2 \] = pr^{(N)} \[ \eta_1 , \eta_2 \]$, so that
actually $\~\z = \( \z , D_x \z , ... \)$ where $\z = \{ \phi , \psi \}$.
To summarize our discussion, we have the
{\it Symmetry criterion (evolution PDEs):} Let $\D = u_t - F[u]$ be an
evolution PDE, and $\G_\D$ its Lie point symmetry algebra. Then, with
the above notations,
$$ \eta = \phi (u) \pa_u \in \G_\D \LRA \[ f , \^\eta \] =0 \LRA \{ \~F
, \^\phi \} = 0 $$
{\it Aside: }
Let us briefly discuss what happens for $\eta$ of more general
form than in (1). We consider indeed
$$ \eta = \phi (u,x,t) \pa_u + \xi(x) \pa_x + \tau (t) \pa_t $$
i.e. a projectable VF, which in turn keeps separate the space and time
variables (this guarantees that $\eta^{(N)}$ transforms evolution
equations into evolution equations).
Then (7) continues to hold, with
$$ \eta^{(t)} = \Phi^{(t)} {\pa \over \pa u_t} \kern 1cm ; \kern 1cm
\Phi^t = D_t ( \phi - \tau u_t ) + \tau u_{tt} $$
and in (3),(4) we have
$$ \^\eta = \^\Phi \c \^\grad \kern 1cm ; \kern 1cm \^\Phi = \( \phi ,
\Phi^{(1)} , ... , \Phi^{(N)} \) $$
where
$$ \Phi^{(m)} = D_x^m ( \phi - \xi u_x ) + \xi D_x^m u_x $$
or, in multiindex notation,
$$ \^\eta = \Phi^{(J)} \c \grad_J $$
These follows from the general prolongation formula [14] and from
$\xi_u = \tau_u = \xi_t = \tau_x =0$. Now, (7) reads
$$ \[ \Phi^{(t)} - ( \Phi^{(J)} \grad_J ) F \]_{\SD} \equiv
\[ \Phi^{(t)} - ( \^\Phi \c \^\grad ) F \]_{\SD} = 0 $$
which gives
$$\( \phi_u u_t + \phi_t - F \tau_t \)_{\SD} - ( \^\Phi \c \^\grad )
F =0 $$
or, in the notation (9),
$$ \phi_t + \[ (F \c \grad ) \phi - ( \^\Phi \c \^\grad ) F \] = F
\tau_t $$
We see that for $\tau_t =0$ (notice that for $F$ autonomous, $\pa_t$ is
always a symmetry of $\D$, so we can take it out from $\eta$, and assume
directly $\tau=0$) this is analogous to (9), i.e. we get $\phi_t (u)
\pa_u = \[ \~\eta , f \]$ and in particular
$$ \phi_t = \{ \~\phi , \~F \} \eqno\bullet $$
We do now return to consider $\eta$ as in (1).
Now, let $\G_0$ be a subalgebra of $\G_\D$. To it, we associate the
spaces
$$ \eqalign{ W_0 =& \{ u \in \U ~/~ \eta u =0 ~~ \all \eta \in \G_0 \}
\sse \U \cr \W_0 =& \{ m \in \M ~/~ \^\eta m =0 ~~ \all \eta \in \G_0 \}
\sse \M \cr } \eqno(16) $$
For completeness of notation, we also consider the isotropy
subalgebra $\G_u$ of $u \in \U$ and $\G_\s$ of the section $\s $ of
$\M \x \B$, and the fixed space conjugated to $u$ and to $\s$.
For $u \in \U$, we have
$$ \eqalign{ \G_u =& \{ \eta \in \G_\D ~/~ \eta u = 0 \} \sse \G_\D
\cr
W_u =& \{ v \in \U ~/~ \eta v = 0 ~ \all \eta \in \G_u \} = \{ v \in
\U ~/~ \G_u \sse \G_v \} \sse \U \cr } \eqno(17) $$
while for $\s \in \Sigma ( E)$, after noticing that $\eta$ as in
(10) acts on $\s$ in such a way that $\eta : \s \to T \s \LRA \eta
\c \s = 0$ (notice this is not true if $\eta$ has a component in the
$\pa_x$ direction) these are
$$ \eqalign{ \G_\s =& \{ \eta \in \G_\D ~/~ \eta \c \s = 0 \} \cr
W_\s =& \{ \s' \in \Sigma (E) ~/~ \eta \c \s ' = 0 ~~ \all \eta \in
\G_\s \} \cr} \eqno(18) $$
\bigskip \bigskip {\bf 12. Reduction Lemma for evolution PDEs} \bigskip
With the notation $\^u (x,t) = \( u(x,t) , D_x u(x,t) , ... , D_x^N u(x,t)
\)$,
we can now state the RL for evolution PDEs as follows:
{\it Reduction Lemma (evolution PDEs):} Let $\G_0$ be a subalgebra of
$\G_\D$, the symmetry algebra of the evolution PDE $\D [u] = u_t -
F[u]$, and let $\W_0 \sse \M$ be the corresponding fixed space. Then,
for an initial datum $u(x,0) = u_0 (x)$ such that $\^u (x,0) \in \W_0
other words, if $\^u (x,0) : X \to \W_0$, then $\^u (x,t) : \B \to
\W_0$.
\Proof Just repeat the proof of RL in ODE case. Indeed, that was
purely geometrical and did not depend on the dimension of $\U$. We
have that by definition the flow of $\^\eta $ on $\W_0$ is trivial
for $\eta \in \G_0$, so that if $\Psi (m;t)$ is the evolute of $m \in
\M$ under the flow induced by $F$ after time $t$, by $[f, \^\eta ]
=0$ it follows $\^\eta \Psi (m;t) = \Psi (m;t) ~ \all m \in \W_0 ,~
\all t \ge 0$; this yields $\Psi (m;t) \in \W_0$, and therefore the
lemma. \hfill $\bullet$
\Corollary $F : \W_0 \to T W_0 $
\Proof $F: \M \to T \M$ and leaves invariant $\W_0$, i.e. $F: \W_0
\to T \W_0$. But actually $F : \M \to T \U \ss T \M$, and $T \W_0
\cap T \U = T W_0$. Hence $F : \W_0 \to T W_0$. \hfill $\bullet$
One would obviously like to have a statement simply in terms of
$u(x,t)$ and not of $\^u (x,t)$; to this aim we notice that in the
same way as $\M$ is fibered as $\pi : \M \to \U$, also $\W_0$ is
fibered, by the same projection $\pi$, as $\pi : \W_0 \to W_0$. In
other words, we have the
{\it Lemma:} \kern 2cm $ \^u (x,t) \in \W_0 ~~ \all x \LRA u(x,t) \in W_0 ~~
\all
x$
\Proof First of all, notice that the dependence on $t$ is inessential
here, so that we can just consider $u_0 (x) \equiv u(x,t_0 )$ and $\^u_0
(x) = \^u (x,t_0 )$. Then, by definition of $\W_0$,
$$ \^u_0 (x) \in \W_0 \Rightarrow D_J \phi (u) \equiv {\^\phi}^J (\^u
) =0 \kern 1cm , ~ \vert J \vert \le N \eqno(1) $$
so that in particular we must have $\phi (u) =0 $, i.e.
$$ \^u_0 (x) \in \W_0 \Rightarrow u_0 (x) \in W_0 \eqno(2) $$
To see that the converse is also true, let $u_0 (x) \in W_0 ~ \all
x$; by definition $\phi (u)$ vanishes along $W_0 $. On the other
side, if $u(x)$ is in $W_0 ~ \all x$, then $\pa_x u_0 (x) \in T W_0 ~
\all x$. Now just notice that $D_x \phi = \phi_u u_x = (u_x \c \grad
) \phi$, i.e. is the gradient of $\phi$ in a direction lying in $T
W_0$, and by the above discussion $D_x \phi = 0$. This argument is
readily generalized to higher orders, yielding that $\[ \pa^k \phi /
\pa u^k \]_{W_0} = 0 ~ \all k >0$. Therefore,
$$ u_0 (x) \in W_0 ~~ \all x \Rightarrow \^u_0 (x) \in \W_0 \eqno(3) $$
and the lemma is proved. \hfill $\bullet$
The above discussion proves that we can restate our RL as
{\it Reduction Lemma (evolution PDE):} Let $\G_0$ be a subalgebra of
$\G_\D$, the symmetry algebra of the evolution PDE $\D [u] = u_t -
F[u]$, and let $\W_0 \sse \M$ be the corresponding fixed space. Then,
for an initial datum $u(x,0) = u_0 (x)$ such that $u_0 (x) \in W_0 ~
\all x$, we have $u (x,t) \in W_0 ~ \all x ,~ \all t \ge 0$.
\Remark One could have obtained the RL directly in this form by
noticing earlier the relation among $W_0 $ and $\W_0$. We have
preferred this two-steps path in order to stress the analogy with the ODE
case.
As always with RL, we will therefore consider the restriction $F_0$
of $F$ to $\W_0 \sse \M$. this gives the restriction $\D_0 $ of $\D$,
$$ \D_0 [v] \equiv v_t - F[v] \eqno(4) $$
$$ v \in W_0 \sse \U \kern 1cm ; \kern 1cm F_0 : \W_0 \to T W_0 $$
The RL has then the usual
\Corollary Let $u(x,t)$ be a solution of $\D$, and $v(x,t)$ a solution
of $\D_0$, with $u(x,0) = v(x,0) \in W_0 ~ \all x$. Then $u(x,t) =
v(x,t) \in W_0 ~ \all t \ge 0$.
As usual, given $u_0 (x) \in W_0 ~ \all x$, we can study the simpler
equation $\D_0$ instead than $\D$.
\Remark It is a trivial observation that, given an arbitrary smooth
$u_0 (x)$, $u_0 : X \to \U$, we can consider $\G_0^{(x)}$ as defined
by
$$ \G_0^{(x)} = \{ \eta \in \G_\D ~/~ \eta u_0 (x) = 0 \} =
\G_{u_0 (x) } $$
and $\G_0 = \cap_{x \in X} \G_0^{(x)}$ is such that, by definition,
$u_0 (x)$ lies entirely in the corresponding $W_0$. This tells how to
use the RL for arbitrary $u_0 (x)$; clearly, it can very well happen
in this case that $\G_0 = \{ e \}$, $W_0 = \U$, $\W_0 = \M$, $\D_0 =
\D$, i.e. no reduction arises.
\Remark We notice that, as a consequence of the corollary, given an
evolution equation $\D$ with symmetry algebra $\G_\D$, the existence
of solutions to $\D$ invariant under $\G_0 \sse \G$ can be studied by
means of (is equivalent to the existence of solutions to) $\D_0$.
\Remark The above remark does also suggest a possible way to attack
the problem of which kind of conditional symmetries [38,39,40] can appear
in the solutions of $\D$ (these correspond in short to invariance of
solutions under Lie-point transformations which are not in $\G_\D$).
We will not pursue this point here.
\vfill \eject
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\baselineskip=10pt
\parindent=10pt
\parskip=10pt
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