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\centerline{\bf \smc A Solution of Ovsiannikov's Reduction 
Problem$^*$\footnotetext"$^*$"{Presented at the Eleventh International 
Colloquium on
 Differential Equations held August 18-23, 2000 in Plovdiv, Bulgaria.}}
 
  
\vskip.25in

\centerline{\smc Terrance   Quinn}
\centerline{\it Department of Mathematics, Ohio University Southern Campus}
\centerline{\it Ironton, Ohio, 45638, U.S.A.}
\centerline{\it e-mail:    quinnt\@ohiou.edu}
\bigskip
\centerline{\smc Sanjay  Rai}
\centerline{\it Department of Mathematics, Jacksonville University}
\centerline{\it Jacksonville, FL  32211, U.S.A.}
\centerline{\it e-mail: srai\@ju.edu} 
\medskip
\centerline{\smc and}
\medskip
\centerline{\smc C.C.A. Sastri}
\centerline{\it Department of Mathematics and Statistics, 
Dalhousie University }
\centerline{\it Halifax, N.S. B3H 3J5 Canada}
\centerline{\it e-mail: sastri\@mscs.dal.ca}
 
\vskip 1in


\centerline{\bf Abstract}
{\narrower\eightpoint\noindent
In his book ``Group Analysis of Differential
Equations'', Ovsiannikov poses a problem  called the 
problem of reduction.  Roughly speaking, it asks for 
necessary and sufficient conditions under which symmetry
reduction, in a certain sense, is possible.  When it {\it is},
the construction of non-invariant solutions --- as well as of 
invariant ones --- of partial differential equations (PDEs)
becomes easier.  Thus a solution of the reduction problem
would be useful.  A solution, under certain assumptions that
appear to be natural, is presented in this paper. 
The method used
indicates not only {\it when} the reduction is 
possible but also {\it how} it can be carried out.}
\bigbreak
 
%\vskip.2in
\subhead{1. Introduction}\endsubhead

The purpose of this paper is to present 
a solution of the reduction problem [12] of Ovsiannikov
under certain conditions that appear to be natural.  To
state the problem, let $\Delta =0$ be a system of partial 
differential equations (PDEs) for $q$ unknowns $u_1, u_2, \ldots, u_q$
in $p$ independent variables 
$x_1, x_2, \ldots, x_p$. We shall
write $x = (x_1, x_2, \ldots, x_p), u = (u_1, u_2, \ldots, u_q)$.
Let $G$ 
be the full local Lie group of point symmetries of the system.
Thus $G$ acts on the manifold $M = {\Bbb R}^p \times {\Bbb R}^q$.
Throughout 
this paper, all objects --- manifolds, mappings, vector
fields --- will be assumed smooth.  Also, all group actions
considered will be local and semiregular, i.e., for all
points $z$ in $M$ on which $G$ acts, the orbits
$G z = \{g z | g \in G\}$ are submanifolds of 
$M$ of locally constant dimension.  Let $\Gamma$ be a solution
manifold, defined implicitly by $F(x, u) =0$,
where $F: {\Bbb R}^p \times {\Bbb R}^q \to {\Bbb R}^q$, 
$F(x, u) = (F^1(x, u), \ldots, 
F^q(x, u))$,
is assumed to be of full rank.  Let $H \subset G$ be a Lie
subgroup.  Let ${\Cal U} \subset \Gamma$ be an open
set small enough that the submanifolds $Hz, z \in {\Cal U}$, are all of the 
same dimension.  Since the group actions are local,
 we shall rename ${\Cal U}$ as $\Gamma$.  
Ovsiannikov
makes the assumption that $H\Gamma$ (the union of all the 
orbits $Hz$, as $z$ varies over $\Gamma$) is itself a manifold
and calls it the orbit manifold. By definition, then,
$H\Gamma$ is invariant under $H$, but in general $\Gamma$ is not
invariant under $H$. When $\Gamma$ {\it is} invariant 
under $H$,
the corresponding solution is called an invariant
(or $H$-invariant) solution and $\Gamma$ an invariant manifold.
Given $H$ and $\Gamma$, Ovsiannikov defines two quantities,
namely the defect $\delta$ and the rank $\rho$ as follows:
$$ \delta  \equiv \delta (\Gamma, H) = \dim (H \Gamma) -\dim \Gamma$$
and 
$$ \rho \equiv \rho (\Gamma, H) = \dim (H \Gamma)- \dim Hz, \q z \in \Gamma.$$
The defect $\delta$ measures the extent to which $\Gamma$ fails to be 
invariant under $H$.  The rank $\rho$ is the local codimension in $H\Gamma$
of the orbit of a single point in $\Gamma$, and 
this is equal to the maximal
number [8, 12] of functionally independent $H$-invariant
functions on $H\Gamma$. (Any smooth function defined
on $H\Gamma$ and invariant under $H$ can be expressed
as a function of $\rho$ functionally independent invariant
functions.) 
Ovsiannikov has shown [12] that if $H^\p \subset H$
is a Lie subgroup and if $\delta^\p= \delta(\Gamma, H^\p)$ and
$\rho^\p = \rho(\Gamma, H^\p)$
then $\delta^\p \leq \delta$ and $\rho^\p \geq \rho$.  From a practical point of
view, invariant solutions --- corresponding to $\delta =0$ --- are the easiest
ones to construct: one can use Lie's algorithm, as refined
by Ovsiannikov [12] and Olver [8].  If a solution is not invariant,
the best one can hope for is reducing the defect while 
keeping the rank the same.  This is what is meant by
reduction in the sense of Ovsiannikov.  We can now
state his

{\bf Reduction Problem:}  Given $H$ and $\Gamma$ as defined above,
what are the necessary and sufficient conditions 
for the existence of a Lie subgroup $H^\p \subset H$ s.t.
$\delta^\p <  \delta$ and $\rho^\p = \rho$?

Before we state the assumptions under which the problem
can be solved, a couple of remarks are in order:

\roster
\item"1." Reduction in the sense of Ovsiannikov is different 
from reduction or reducibility as defined by one of 
us in earlier work [16, 18] and as commonly understood in the literature.
There, a solution is called irreducible
if there is no non-trivial subgroup of $G$ under
which it is invariant.  The difference is that in 
Ovsiannikov's case, control over {\it two} parameters --- the
rank and the defect --- is sought, whereas in our 
earlier definition, only one parameter, the defect, 
matters.  Therefore, to make the terminology compatible, 
we let ``rank-invariant 
reduction'' mean reduction in the sense of Ovsiannikov.
Note further that while Ovsiannikov's definition
is useful in determining whether a given solution is reducible, 
our earlier 
definition is related to the question of whether  a given 
system of PDEs has irreducible solutions with 
respect to $G$. For an example, see [14]. 

\vskip .15in

\item"2."  Ovsiannikov introduces the notion of a partially
invariant solution [12]: $\Gamma$ is partially 
invariant if its defect is 
strictly positive and if $H \Gamma$ 
is not the whole of $M$.  For our purposes, what is of interest  
 is a measure of the 
lack of invariance of a solution, namely the 
defect, whatever it happens to be.

 

\endroster

 

There has been considerable work on partial 
invariance [1, 3-7, 9, 11-19], but as far as we   know, Ovsiannikov's
reduction problem has not been addressed.  Now, in 
order to state the conditions under which we solve
the problem, some notation is needed.  Let $\frak g, \frak h$
be the Lie algebras of $G$
and $H$, respectively. If $\bold w$ is a vector
field in $\frak h$, let ${\frak h}(z) = \{{\bold w} (z) | {\bold w} \in 
{\frak h}\}$, $z 
\in \Gamma$.
Then ${\frak h}(z)$ is a vector space; even if ${\frak h}$ is infinite
dimensional, ${\frak h}(z)$ is finite dimensional, since
it is a subspace of $TM|_z$, the tangent space of $M$ at $z$, and 
$\dim(TM|_z) = p+q$.  Let $H(\Gamma)$ be the maximal
Lie subgroup of $H$ which leaves $\Gamma$ invariant.  Clearly,
such a subgroup exists.  For if $\Gamma$ is invariant under
$H$, $H(\Gamma) = H$, and if the only subgroup of $H$ under
which $\Gamma$ is invariant is the trivial subgroup, then 
$H(\Gamma)$ is the trivial subgroup.  Otherwise, $H(\Gamma)$
is a proper subgroup of $H$. That it is  a Lie group
follows [8, 12] from the fact that if a manifold is invariant
under the action of two vector fields, then it is also invariant
under the action of their Lie bracket.  In what follows, we shall
adopt the notation ${\frak a} \la {\bold w}_1, {\bold w}_2, \ldots \ra$
to denote the Lie algebra 
with basis $\{{\bold w}_1, {\bold w}_2, \ldots\}$ and 
$\la {\bold w}_1, {\bold w}_2, \ldots \ra$ to denote 
the corresponding vector space.

We are now ready to state the assumptions:
\roster
\item"(i)"  $H\Gamma$ is a manifold.
\vskip .15in

\item"(ii)"  For $z \in \Gamma$, $\dim (T(H\Gamma)|_z) = 
\dim \la {\frak h}(z), T\Gamma|_z \ra$, 
where $T\Gamma|_z$ and $T(H\Gamma)|_z$ are the tangent spaces,
respectively, of $\Gamma$ and $H\Gamma$, at $z$, and $\la {\frak h}(z), 
T\Gamma|_z\ra$
is the span of ${\frak h}(z)$ and $T\Gamma|_z$.
\vskip .15in

\item"(iii)"  The action of $H$ ``splits along $\Gamma$'', i.e.,
$${\frak h}(\Gamma)(z) = {\frak h}(z) \cap T\Gamma|_z, \qquad z \in \Gamma, $$
where ${\frak h}(\Gamma)$ is the Lie algebra of $H(\Gamma)$.
\endroster

We note that Ovsiannikov himself makes assumption (i);
the other two are new.  To see that these assumptions
are not always satisfied, let's consider the following
examples:

\roster
\item"1."  Let $\Gamma$ be the $x$-axis in ${\Bbb R}^2(x,y)$,
 ${\bold w} = \partial_x + 2x \partial_y$,
and ${\frak h} = {\frak a} \la {\bold w} \ra$. 
Near $(0,0)$, (i) $H\Gamma$ is not a manifold, (ii) $\dim 
\la {\frak h}(z), T\Gamma|_z \ra$
is not locally constant and (iii) ${\frak h}(\Gamma)$ is the zero algebra,
so that ${\frak h}(\Gamma)(0,0)$ is the zero vector space, but
${\frak h}(0, 0)\cap T\Gamma|_{(0,0)} = \la (1, 0) \ra$, so the action of $H$
does not split along $\Gamma$.
\vskip .15in

\item"2." Let $\Gamma$ be the $x$-axis in ${\Bbb R}^2(x,y)$ and 
${\frak h} = {\frak a}
\la {\bold w}_1, {\bold w}_2 \ra$, where  
${\bold w}_1 =  \partial_x$,  ${\bold w}_2 =   
\partial_y.$\newline
All three assumptions are satisfied.
\vskip .15in

\item"3." Let  $\Gamma$ be the $x$-axis in ${\Bbb R}^2(x,y)$
and ${\frak h}= {\frak a}
\la {\bold w}_1, {\bold w}_2 \ra$, where ${\bold w}_1 
= \partial_y$ and ${\bold w}_2 = \partial_x + 2x \partial_y$.
 Assumptions (i) and (ii) are satisfied, but 
not (iii).
For ${\frak h}(\Gamma)$ is the zero algebra, so that
$$ {\frak h}(0, 0) \cap T\Gamma|_{(0,0)} = \la (1,0) \ra \neq {\frak h}
(\Gamma) (0, 0). $$
\endroster

\subhead{2. The Reduction Problem}\endsubhead

In what follows, given $z \in \Gamma$, we mean by proj the 
projection onto the orthogonal complement of $T \Gamma|_z$ in 
$TM|_z$.  We now need a result of Ovsiannikov (see Proposition 2.2
below), which we shall state with a new proof.  Our proof has
the advantage of suggesting a solution to the reduction 
problem.  We start with

\proclaim{Lemma 2.1}  Let ${\bold w}_1, {\bold w}_2, \ldots 
{\bold w}_r$  be elements of ${\frak h}$ s.t. $\forall z \in 
\Gamma$, 
$\{{\bold w}_1(z),
\ldots, {\bold w}_r(z)\}$
is a basis for ${\frak h}(z)$.  Let $Q=(\{{\bold w}_j (F^i)\})$.  Then
rank $Q(z) = \dim \text{ proj } {\frak h}(z)$.
\endproclaim

\demo{Proof}  Let $\lambda_1, \lambda_2, \ldots, \lambda_r \in {\Bbb R}$
be arbitrary, and 
let ${\bold  w} = \lambda_1 {\bold w}_1 + \cdots + \lambda_r {\bold w}_r$.  
Then, at $z$ 
$$\leqalignno{ Q(\{{\bold w}_j (z)\} \left(\matrix
\lambda_1\\
\vdots\\ \lambda_r
\endmatrix  \right) &=
\left( \matrix
({\bold w}_1 F^1) (z) \cdots ({\bold w}_r F^1) (z)\\
\hdotsfor1\\
({\bold w}_1 F^q)(z) \cdots ({\bold w}_r F^q)(z)
\endmatrix \right) \enskip \left(
\matrix
\lambda_1\\
\vdots\\
\lambda_r
\endmatrix
\right)\cr
&= \left(\matrix
({\bold w}  \cdot \nabla F^1)(z)\\
\hdotsfor1\\
\hdotsfor1\\
({\bold w}\cdot \nabla F^q)(z)
\endmatrix  \right)\cr} $$
so that $Q(\{{\bold w}_j(z)\}) \left( \matrix
\lambda_1 \\ \vdots \\ \lambda_r
\endmatrix \right)=0$
iff ${\bold w}(z)$ is orthogonal to $\nabla F^i(z), \q i = 1, \ldots, q$,
i.e., iff ${\bold w}(z) \in T\Gamma|_z$,
since $F$ is of full rank.

Hence $\dim \ker Q(z) = \dim \ker \text{ proj}|_{{\frak h}(z)}$.
But $\dim\left(\left\{\left( \matrix
\lambda_1 \\ \vdots\\ \lambda_r
\endmatrix \right)  \right\}, \lambda_i \in {\Bbb R}  \right) = 
\dim {\frak h}(z) = r$
by hypothesis.

Hence, by the rank-nullity theorem, 
$$\leqalignno{\dim \text{ range } (Q(z)) &= \dim \text{ range } \text{proj}
|_{{\frak h}(z)}\cr
&= \dim \text{ proj } {\frak h}(z). \hfill  \qquad \qed\cr}$$
 
\enddemo

\proclaim{Proposition  2.2 (Ovsiannikov)}  $\delta = \text{ rank } Q(z)$.
\endproclaim

\demo{Proof}  From the  definition of defect, we have
$$\leqalignno{\delta &= \dim H\Gamma - \dim \Gamma\cr
&= \dim T(H\Gamma)|_z - \dim T\Gamma|_z \cr
&=\dim ({\frak h}(z)) + \dim T \Gamma|_z - \dim ({\frak h}(z) \cap 
T\Gamma|_z)-
\dim T\Gamma|_z\cr
&= \dim ({\frak h}(z)) -\dim ({\frak h}(z) \cap T\Gamma|_z)\cr
&= \dim \text{ proj } {\frak h}(z)\cr
&= \text{ rank } Q(z).\hfill  \qquad \qed \cr}   $$
\enddemo

\proclaim{Corollary 2.3} $\delta = \dim {\frak h}(z) - \dim {\frak h}
(\Gamma)(z)$.
\endproclaim
This follows directly from assumption (iii) and the proof of Prop. 2.2.
\hfill \qquad \qed


We recall [8, 12] that by the existence and uniqueness theorem, a vector 
field ${\bold v}$ leaves a
submanifold $\Gamma$ invariant iff ${\bold v}(z) \in T \Gamma |_z$ for 
all $z \in \Gamma$.  So given $H$ and $\Gamma$ it is 
straightforward to find $H(\Gamma)$.  Suppose that $\{{\bold w}_\alpha\}$ is a 
basis  for the 
possibly infinite dimensional Lie algebra $\frak h$.  Then
${\bold w} = \sum^k_{j=1} \lambda_{\alpha_j} {\bold w}_{\alpha_j} 
\in {\frak h}(\Gamma) \Longleftrightarrow {\bold w} (F^i) =
\sum^k_{j=1} \lambda_{\alpha_j} {\bold w}_{\alpha_j} (F^i) = 0$ along
$\Gamma$, \enskip
$i = 1, \ldots, q \Longleftrightarrow (\lambda_{\alpha_1}, 
\ldots, \lambda_{\alpha_k})$ is a constant vector 
in $\ker({\bold w}_{\alpha_j}(F^i))(z)$ for 
all $z \in \Gamma$.



\example{Proposition  2.4}  $\rho(\Gamma, H) = \rho(\Gamma, H(\Gamma))$.
 \endexample
 
\demo{Proof}  We have, by definition 
$$ \leqalignno{\rho(\Gamma, H) & = \dim (H\Gamma) - \dim Hz
\q (z \in \Gamma)\cr
& =\dim T(H\Gamma)|_z - \dim Hz\cr
&= \dim T (H \Gamma)|_z - \dim {\frak h}(z)\cr
&= \dim T\Gamma|_z + \delta - \dim {\frak h}(z)\cr
&= \dim T\Gamma|_z - \dim {\frak h}(\Gamma) (z) + \delta - 
(\dim {\frak h}(z) - \dim {\frak h}(\Gamma)(z))\cr
&= \dim T\Gamma|_z - \dim {\frak h}(\Gamma)(z)\cr
&=\rho (\Gamma, H(\Gamma))   \cr}$$
by semiregularity. \qquad \qquad \qquad  \qquad \qed
\enddemo 


We next state a result of Ovsiannikov's with a new, and 
simpler, proof.  


\proclaim{Lemma 2.5} Let $H^\p \subset H$ be a Lie subgroup.  Then 
(i)  $\delta^\p \leq \delta$ and (ii) $\rho^\p \geq \rho$. 
\endproclaim

\demo{Proof}  $$(i) \enskip \delta^\p = \dim \text{ proj } {\frak h}^\p(z) \leq \dim 
\text{ proj } {\frak h}(z) = \delta$$ and 
$$\leqalignno{(ii) \enskip \rho^\p &= \dim T \Gamma|_z - \dim {\frak h}^\p 
(\Gamma)(z)\cr
&\geq \dim T\Gamma|_z - \dim {\frak h}(\Gamma)(z)\cr
&= \rho. \hfill \qquad \qed\cr}$$
The solution to Ovsiannikov's reduction problem follows easily now.
\enddemo


\proclaim{Corollary 2.6}
$H^\p \subset H$ reduces $(\Gamma, H)$ in the sense of Ovsiannikov iff 
(i) $\delta^\p = \dim \text{ proj } {\frak h}^\p (z) < \dim \text{ proj } 
{\frak h}(z) = \delta$,
and (ii) $ {\frak h}^\p(\Gamma) (z) =  {\frak h}(\Gamma) (z)$. 
\endproclaim

We observe that since ${\frak h}^\p(\Gamma) (z) \subset {\frak h}(\Gamma)(z)$,
$\dim{\frak h}^\p(\Gamma)(z) = \dim {\frak h}(\Gamma)(z)$ iff
${\frak h}^\p(\Gamma)(z) = {\frak h}(\Gamma)(z)$.  \qquad\qquad 
\qed

For completeness, we include a reformulation of the 
Ovsiannikov (rank-invariant) reduction problem.  In an
immediate sense, this will only modestly extend the results so far,
but will also correspond to an analogous problem raised in 
[16, 18].  The basic observation is that the defect $\delta$
and the rank $\rho$ are numerical quantities.  In Ovsiannikov's 
work, minimizing $\delta$ and $\rho$ corresponds to minimizing the
complexity of the algorithm and maximizing the invariance of
possible solutions.  Strictly speaking then, it is not necessary
to restrict ourselves to subgroups of a given $H$.  The objective 
can simply be to minimize $\delta$ and $\rho$.  So, as in  [16, 18]
we extend  the context to the full Lie symmetry group $G$.
The extended problem is then to determine
the existence of a rank-invariant reduction in $G$, that is, to
find $H^\p \subset  G$ ($H^\p$ not necessarily a subgroup of $H$) such
that $\delta^\p < \delta$ and $\rho^\p = \rho$. Again, assuming that
actions split along $\Gamma$ we obtain the following:

\proclaim{Corollary 2.7}
$H^\p \subset  G$ gives a rank-invariant reduction of $(\Gamma, H)$
iff 
\roster
\item"(i)"  $\dim$ proj ${\frak h}^\p(z) < \dim$ proj ${\frak h}(z)$ and 

\item"(ii)" $\dim {\frak h}^\p(\Gamma)(z) = \dim {\frak h}(\Gamma)(z)$.
\qquad \qquad \qed
\endroster 
\endproclaim

We observe that it is possible to have a chain of reductions
$H^{(k)} \subset \cdots H^{(2)} \subset H^{(1)} \subset H$,
but the best possible reduction is obtained by 
choosing $H^\p = H(\Gamma)$. Then, since $\Gamma$ is invariant 
under
$H(\Gamma)$, $\delta^\p =0$ and, by Prop. 2.4, $\rho^\p =\rho$.
In other words, if ${\frak h}(\Gamma) \neq \{0\}$, then there is a 
rank-invariant reduction.  The 
converse is not true in general.  (See Examples 3 and 4 below.)

 

\noin
{\bf Examples}

\roster
\item"" We remark that all three of our assumptions
are satisfied by the examples 
considered by Ovsiannikov.

\vskip .25in


\item"1." This example is  due to Ovsiannikov ([12], p. \!286
and p. \!293), although  the analysis given here is different
from his. 

Consider the equations of transonic flow:
$$ u_y = v_x, \qquad v_y = -u u_x, \qquad x, y \in {\Bbb R}.$$
This is a nonlinear system whose full Lie symmetry group is 
infinite dimensional    and acts on $M = {\Bbb R}^4 (x, y, u, v)$.
Let $H$ be the two-dimensional
subgroup of this group with generators ${\bold w}_1 = x \partial_x + 
y \partial_y$
and ${\bold w}_2 = \partial_v$. If $z = (x, y, u, v)$ and $\lambda=
\frac{x}{y}$, then 
$F(z) = (F_1(z), F_2(z)) = (u - f(\lambda), v - g(\lambda) + c\;ln\;y)$,
where $f$ and $g$ are some functions
and $c$ is a constant.  $\Gamma$ is defined as the zero-set 
of $F$, i.e., it is given by $u = f(\lambda),
v= g(\lambda) - c\;ln\;y$.
So $\dim \Gamma =2$.   
Now we have
$Q(z) = \left(  
\matrix
0 & 0\\
c & 1
\endmatrix\right) $, so 
that $\delta = \text{ rank } Q(z) =1$.
The kernel of $Q(z)$ consists of vectors of the form 
$\left( \matrix
\alpha_1\\
\alpha_2
\endmatrix \right)$
where $\alpha_2 = -c \alpha_1$.  Choosing $\alpha_1 =1$, 
we find that 
${\frak h}(\Gamma)$ is the algebra generated by ${\bold w} = x
 \partial_x + y \partial_y - c \partial_v$ and so $\rho = \dim 
 \Gamma - \dim {\frak h}(\Gamma) =1$.
 Ovsiannikov chooses ${\frak h}^\p$ to be the algebra generated by 
 $x \partial_x + y \partial_y - c \partial_v$, i.e., s.t. ${\frak h}^\p
 ={\frak h}(\Gamma)$.  The 
 conditions of Corollary 2.6 are satisfied, and rank-invariant
  reduction takes place.
 
\vskip .25in 
 
\item"2." . 
This example is also due to Ovsiannikov ([12], p. \! 279),
although, again, the analysis given here is different from his. The ambient space is
$M = {\Bbb R}^4 (x, y, z, v)$; if ${\frak z} = (x, y, z, v)$,
$F({\frak z}) = (F_1 ({\frak z}), F_2 ({\frak z})) = (y - \lambda xz, v)$, 
and $\Gamma$ is defined
as the zero-set of $F$.  Thus $\Gamma$ is given by $y-\lambda x z =0$,
$v = 0$.  Clearly, $\dim \Gamma =2$.  Let $\frak h$ be the algebra generated by 
${\bold w}_1 = x \partial_y$ and ${\bold w}_2 = \partial_z$.
It is easy to see that
$\dim(H\Gamma) =3$ and $\dim (H {\frak z}) = 2$, so that $\rho =1$.
We have 
$ Q({\frak z}) = \left( \matrix
x & - \lambda x\\
0 & 0
\endmatrix \right)$, so that if $x \neq 0$,
rank $Q({\frak z}) =1 = \delta$, the defect.

The kernel of $Q({\frak z})$ consists of all vectors $\left(\matrix
\alpha_1\\ \alpha_2
\endmatrix  \right)$ of the 
form $\alpha_1 = \lambda \alpha_2$, i.e., $\alpha_2 \left(
\matrix
\lambda\\ 1
\endmatrix  \right)$, $\alpha_2 \neq 0$.

Setting $\alpha_2 =1$, we find that ${\bold w}({\frak z}) = (\lambda x 
\partial_y + \partial_z)
({\frak z})$ is in $T \Gamma|_{\frak z}$,
i.e., $\bold w$ is the generator of ${\frak h}(\Gamma)$. 
Ovsiannikov chooses
${\frak h}^\p$ to be the algebra generated by $\lambda x \partial_y + \partial
_v$,
i.e., s.t. ${\frak h}^\p = {\frak h}(\Gamma)$.  
The conditions of 
Corollary (2.6) are satisfied, since
$$\leqalignno{ \delta^\p &= \dim 
\text{ proj } {\frak h}^\p ({\frak z}) 
= \dim \text{ proj } {\frak h} (\Gamma) ({\frak z}) 
=0 < \delta =1 \cr
\text{ and } & {\frak h}^\p (\Gamma) ({\frak z}) = 
 {\frak h}(\Gamma)({\frak z}), \cr}$$
and hence rank-invariant reduction occurs.





 
 \vskip .25in


\item"3."  Suppose that $\Gamma$ is the $x$-axis in ${\Bbb R}^3 (x, y, z)$ 
and let ${\bold e}_2 = (0, 1, 0), {\bold e}_3 = (0, 0, 1)$ and let ${\frak h}
= {\frak a} \la {\bold e}_2, {\bold e}_3 \ra$.
Then $\delta (\Gamma, H) = 2$ and $\rho(\Gamma, H) = 1$.
Let ${\bold w}$ be any non-zero
vector field in ${\frak h}$ and let ${\frak h}^\p = 
{\frak a} \la  {\bold w}\ra$.  Then $\delta (\Gamma, H^\p) =1$
while the rank remains $\rho (\Gamma, H^\p) = 
\rho (\Gamma, H) =1$.  Note that
${\frak h}(\Gamma) = {\frak h}^\p(\Gamma) = \{0\}$.

In our next example, we show that the basic situation given 
in Example 3 occurs in the system of partial differential
equations governing   transonic flow.

\vskip .25in


\item"4." As in Example 1, with ${\bold w} = x \partial_x + y \partial_y
-c \partial_v$,
${\frak h}^\p = {\frak a} \la  {\bold w} \ra$ gives a rank-invariant 
reduction. But 
Ovsiannikov's algorithm can be used again, this time to obtain
solutions partially invariant with respect to ${\frak h}^\p = 
{\frak a} \la {\bold w} \ra$. Let
$z = (x, y, u, v)$.  For
arbitrary $c$, an extended set of solutions [17] partially invariant 
with respect to $H^\p$ includes
functions which satisfy
$$\leqalignno{ u_y &= \mp (-u)^{\frac{1}{2}} u_x\cr
v &= k (\lambda, u) - c \; ln \; y, \q \lambda = \frac{x}{y}, \q y \neq 0,\cr} $$
where $k$ satisfies a derived equation. In particular, 
we may suppose $k_\lambda =0$.  Now with ${\frak h} = {\frak a} \la
{\bold w}_1, {\bold w}_2 \ra$ and 
$\Gamma$ one of these further solutions, we obtain
$$ Q(z) = \left[\matrix
-(xu_x + y u_y) & 0\\
c & 1
\endmatrix  \right] $$
As in [17], there exist solutions for 
which $xu_x + yu_y \neq 0$, so that $rankQ(z) = \delta =2$. And
because ${\frak h}(\Gamma) = 
\{0\}$, we also get $\rho =2$.  Hence, much 
as in Example 3 above, if ${\bold v}$ is any non-zero vector
field in ${\frak h}$, let ${\frak h}^{\p\p} = {\frak a}
\la {\bold v} \ra$.  Then ${\frak h}^{\p\p}(\Gamma) =
{\frak h}(\Gamma) = \{0\}$,
$\delta^{\p\p} = 1 < \delta =2$ and $\rho^{\p\p} = \rho =2$.
Therefore, there exist
partially invariant solutions with respect to $H^\p$ which 
are rank-invariant reducible with respect to $H$, but which are
not invariant with respect to any subgroup of $H$.

\vskip .25in


\item"5."  If $\delta >1$ and $z \in \Gamma$, then there exists 
${\bold w} \in {\frak h}$
such that proj ${\bold w}(z) \ne 0$.  So the dimension of the 
linear space
$\dim \la {\bold w}, z \ra , {\frak h}(\Gamma)   (z) \ra =
\dim {\frak h}(\Gamma)(z) +1 < \dim {\frak h}(\Gamma) + \delta$.
In general however, this may not yield a reduction.
For since ${\bold w}$ and ${\frak h}(\Gamma)$
may not be in involution, the real linear dimension of 
the Lie algebra ${\frak a} \la {\bold w}, {\frak h}(\Gamma) \ra (z)$
may also be larger than $\dim {\frak h}(\Gamma)(z) + 1$.

\vskip .25in

\endroster

 

\subhead{3. Concluding Remarks}\endsubhead

Local to a point $z \in \Gamma$, a given group of 
symmetries may not ``split'' along $\Gamma$.  In such cases,
$H\Gamma$ may fail to be a manifold.  We may nevertheless
enquire into the orbit structure.  One way to do this would be 
to investigate general
group actions on sets of submanifolds.  This is independent
of differential equations as such and becomes a geometric
question that could apply, for example, to confoliations [2] 
and moduli spaces.  
Other applications could be in ``spontaneous
symmetry 
breaking (SSB)'' in  particle
physics.  An SSB can be characterized by the orbit 
(under an ``internal'' group 
action) of a function which minimizes a potential [10].  
One feature of a generalized theory for orbits of 
manifolds could
be an appropriately generalized definition 
of dimension of a subset of a manifold. 
For instance, one could take the supremum, over a neighborhood, of dimension
in the usual sense.  In 
that case we obtain $\delta \geq \text {rank } Q(z)$, where strict
inequality may occur.  Preliminary results indicate,
however, that a more refined analysis is needed.
As is well known, ``dimension'' can be defined in 
many ways, depending on the setting.  In the context
of orbits $H\Gamma$, certain topological definitions promise
to be of use. But this would take us  beyond the 
scope of the present paper; we shall pursue these matters
elsewhere.

\subhead{Acknowledgements}\endsubhead

We are indebted to Maria Fe Elder for all her 
help in preparing this manuscript.  In addition, 
one of us (C.C.A.S) is pleased to take this
opportunity to thank Mr. T. Chelluri and 
Professors L. Gr\"{u}nenfelder, H.J. Sussmann, and S.R.S. Varadhan 
for helpful discussions.
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\ref \no  18  
\by C.C.A. Sastri, K.A. Dunn and D.R.K.S. Rao           
\paper    Ovsiannikov's Method and the Construction of
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\endref


 





\ref \no  19  
\by W. Strampp           
\paper        Partially Invariant Solutions of Systems of Differential
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\jour     J. Math. Anal. App.
\vol     97
\yr     1983
\pages      509-528 
\endref
















 

\endRefs




\enddocument

\ref \no  1  
\by           
\paper        
\jour     
\vol     
\yr     
\pages       
\endref

\ref \no  2  
\by   
\book         
\publ 
\publaddr  
\yr     
\endref


\subhead{1. Introduction}\endsubhead

\proclaim{Theorem}
\endproclaim

\definition{Definition 2.1}
\enddefinition

\remark{Remark }
\endremark

\example{}


\proclaim{Proposition  }
\endproclaim

\proclaim{Corollary  }
\endproclaim

\proclaim{Lemma  }
\endproclaim


\demo{Proof}
\enddemo



\endexample

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