% 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 $W_0$. 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