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%  LaTeX 2.09
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\documentstyle[12pt,bezier]{article}
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\title{Infinitesimal Classification of First Order Systems of
Two Partial Differential Equations in Two Variables}
\author{S.~V.~Duzhin
\thanks{This work was partially supported by the following grants:
RFBR 98-01-00555a, RFBR 00-15-96084, NWO-RFBR 047-008-005.}}
%
\def\t{\theta}
\def\s{\sigma}
\def\l{\lambda}
\def\p{\partial}
\def\GL{\mathop{\rm GL}\nolimits}
\def\IG{\mathop{\rm IG}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\E{{\cal E}}
\def\R{{\bf R}}
\newtheorem{theorem}{Theorem}
%
\begin{document}
\maketitle

\section{Introduction}
We give a classification of symbols of partial differential systems
that consist of two equations imposed on two functions in two independent
variables, with respect to the natural action of the group of linear
changes of dependent and independent variables.

The classification turns out to be finite, consisting of 5 types in the
real case and of 4 types in the complex case.
Apart from the elliptic, parabolic and hyperbolic types [1]
constituting the well-known classification of second order equations
for a function $u(x,y)$, in the present case we also have two
non-equivalent `degenerate' types.

Bifurcation diagrams for the singular orbits in the space of symbols
imply certain properties of the singularities of linear
and quasilinear systems of the considered type.
I am indebted to S.~V.~Chmutov who explained me the meaning of versal
deformations and bifurcation diagrams in the general context of an
arbitrary Lie group action on a manifold.

In the last section of the paper we find, for every type of the symbol,
the topological type of the corresponding integral Grassmanian
--- an object important in the study of singularities of solutions
of partial differential equations [5].

All theorems stated in this text are quite straightforward,
so we confine ourselves just to short comments, omitting
the complete proofs.

\section{Classification of Symbols}

A system of $k$-th order partial differential equations imposed on
$m$ functions of $n$ independent variables determines a submanifold
$\E$ in the jet space $J_k(X,Y)$ where $\dim X=n$, $\dim Y=m$.
The {\it symbol\/} of the system at a point $\t\in\E$ is defined
as the intersection of the tangent space to $\E$ and the tangent
space to the fiber $F_\t$ of the natural projection $J_k\to J_{k-1}$ 
containing $\t$.
If $a\in X$ and $b\in Y$ are the source and the target of the jet
$\t$, respectively, then the tangent space $T_\t(F_\t)$
is naturally isomorphic to $S^kV^*\otimes W$ where $V=T_a(X)$,
$W=T_b(Y)$ and $S^k$ denotes the $k$-th symmetric power (see [2]).
In this sense, a symbol can be thought of as a linear subspace of the
space $S^k(\R^n)^*\otimes \R^m$.

Denote by $G_\t$ be the group of those local diffeomorphisms of
the pair $\bigl(X\times Y,\,(a,b)\bigr)$ which, after being lifted
to the jet space $J_k(X,Y)$, do not move the point $\t$.

\begin{theorem}
Let $\t,\t'\in \pi_{k,0}^{-1}(a,b)$ be two points lying in the same fiber of
the natural projection $\pi_{k,0}:J_k(X,Y)\to J_0(X,Y)=X\times Y$. Then:
\begin{enumerate}
%
\item There exists a local diffeomorphism $\phi$ in a neighbourhood of $(a,b)$
such that $\phi^{(k)}(\t)=\t'$.
%
\item The groups $G_\t$ and $G_{\t'}$ are conjugate via $\phi$.
%
\item The action of $G_\t$ in $T_\t(F_\t)$ agrees with the
action of $G_{\t'}$ in $T_{\t'}(F_{\t'})$ under the identification
$T_\t(F_\t)\cong T_{\t'}(F_{\t'})\cong S^kV^*\otimes W$.
%
\item In the case $X=\R^n$, $Y=\R^m$ and $\t=(0,0,\dots,0)$ in the
standard jet coordinates the group $G_\t$ consists of those
local diffeomorphisms $(x,y)\mapsto(\tilde{x},\tilde{y})$ which satisfy
${\p\tilde{y}\over\p x}\vert_{(0,0)}=0$.
%
\item The action of this group in $T_\t(F_\t)$ depends only on the linear
part of the diffeomorphism and decomposes as
$G\to\GL(V)\times\GL(W)\to\GL(T_\t(F_\t))$, where
the last arrow is nothing but the natural left-right action of the
direct product $\GL(\R^n)\times\GL(\R^m)$ in $S^k(\R^n)^*\otimes \R^m$.
%
\end{enumerate}
\end{theorem}
Thus, in a purely algebraic setting, the problem of symbol classification
can be stated as follows. {\it Fix a quadruple of natural numbers
$n$, $m$, $k$, $r$, and let $V$ and $W$ be linear spaces 
of dimensions $n$ and $m$ over a certain field.
Describe the orbits  of the group $\GL(V)\times\GL(W)$ in the Grassmanian
of $r$-dimensional planes in $S^kV^*\otimes W$ with respect to the action
that comes from the natural left-right action of this group in the underlying
space $S^kV^*\otimes W$}. The problem makes sense over an arbitrary field;
we will be only concerned with the reals and the complexes.

In the special case $m=1$, $k=2$, $r=1$ (corresponding to one second order
equation) we obtain the problem of classification of (real or complex)
quadratic forms in $n$ variables. Its well-known solution gives rise to
the usual notion of type (hyperbolic, parabolic, elliptic) for the second
order equations.

In this note we will solve, and derive the consequences of, the symbol
classification problem in the case $n=2$, $m=2$, $k=1$, $r=2$, which,
simple as it is, has never been studied in the literature, to the best
of our knowledge. The algebraic problem to study is that of classifying
two-dimensional linear spaces of linear operators in the plane with
respect to the independent linear changes both in the source and in the
target.

A complete list of all quadruples $(n,m,k,r)$ for which the classification
of symbols is finite, will be given in a separate paper [4].

\section{Planes of linear operators in the plane}

The direct product $\GL(\R^2)\times\GL(\R^2)$ acts in $\End(\R^2)$
according to the rule
$$
  (X,Y)\cdot A = XAY.
$$
We are to find the orbits of the induced action on the set of 2-planes
in $\End(\R^2)$.

In the four-dimensional space $\End(\R^2)$ we single out a three-dimensional
cone $K$ that consists of all degenerate operators.
The trace left by this cone on a two-plane $L$ in $\End(\R^2)$ and
considered up to linear transformations of the plane, is an important
invariant of $L$ which is sufficient to distinguish all the orbits
but two. The latter are made up of planes that are entirely contained
in $K$ and differ in that, for one of them, all the operators of the
plane possess a common 1-dimensional kernel, while for the other, they
have one and the same 1-dimensional image.
\begin{theorem}
Two-dimensional planes of linear operators in $\R^2$ under the linear
changes of variables in the source and in the target split into five
orbits. Therefore, the symbol of any system of partial differential
equations with $n=m=r=2$ belongs to one of the five possible types.

The orbits are listed in this table:

\begin{center}
\begin{tabular}{|l|c|l|l|l|}
\hline
Type  & $\dim({\cal O})$ & $L\cap K$ & Normal form of $L$ & System of d.e.\\
\hline
Hyperb. & 4 & two lines  & $\langle\pmatrix{1&0\\0&0},\pmatrix{0&0\\0&1}\rangle$
    & $\left\{\begin{array}{l}
                u_y=0\\
                v_x=0
             \end{array}\right.$ \\
\hline
Ellipt. & 4 & one point  & $\langle\pmatrix{1&0\\0&1},\pmatrix{0&1\\-1&0}\rangle$
    & $\left\{\begin{array}{l}
                v_x=-u_y\\
                v_y=u_x
              \end{array}\right.$\\
\hline
Parab.  & 3 & one line   & $\langle\pmatrix{1&0\\0&1},\pmatrix{0&1\\0&0}\rangle$
    & $\left\{\begin{array}{l}
                v_x=0\\
                v_y=u_x
              \end{array}\right.$\\
\hline
Deg. I  & 1 & all plane  & $\langle\pmatrix{1&0\\0&0},\pmatrix{0&0\\1&0}\rangle$
    & $\left\{\begin{array}{l}
                u_y=0\\
                v_y=0
              \end{array}\right.$\\
\hline
Deg. II & 1 & all plane  & $\langle\pmatrix{1&0\\0&0},\pmatrix{0&1\\0&0}\rangle$
    & $\left\{\begin{array}{l}
                v_x=0\\
                v_y=0
              \end{array}\right.$\\
\hline
\end{tabular}
\end{center}
\end{theorem}

These five orbits can be conveniently vizualized in terms of elementary
projective geometry.
The Grassmanian of two-planes in the four-dimensional space $\End(\R^2)$
is isomorphic to the variety of projective lines in the three-dimensional
projective space $P(\End\R^2))\cong\R P^3$.
The cone of degenerate operators shows up as a non-degenerate quadric
$Q\subset\R P^3$.
The action of the group of left and right linear changes carries over into
the action of the group of projective transformations of $\R P^3$ that
preserve the quadric $Q$.
A line $l$ belongs to the ellptic, parabolic or hyperbolic orbit if
it intersects $Q$ in 0, 1 or 2 points, respectively.
The two families of straightline generators of the quadric correspond
to the two degenerate types.

\section{Versal Deformations}
%
Let $G$ be a Lie group smoothly acting on a manifold $M$.
A {\it versal deformation\/} [3] of a point $p\in M$ is a local
mapping $f: (\Lambda,0) \to (M,p)$ of a certain parameter space
$\Lambda\cong\R^s$ whose image is transverse to the orbit $G(p)$
and has dimension equal to the codimension of $G(p)$.

The partition of the space of parameters $\Lambda$ into subsets
corresponding to different orbits is called {\it bifurcation diagram\/}.

If the orbit of the point $p$ has dimension equal to the dimension of
$M$, then its bifurcation diagram is trivial and it has no versal
deformations with $s>0$.
The type of such point $p$ is said to be {\it stable\/}.

\begin{theorem}
The versal deformations of the normal forms listed in Theorem 2
can be chosen in the following way:
\begin{enumerate}
\item The hyperbolic type is stable.
\item The elliptic type is stable.
\item For the parabolic type $\{v_x=\l u_y,\,v_y=u_x\}$.
\item For the first degenerate type
      $\{u_y=\l_1 v_x,\,v_y=\l_2 u_x +\l_3 v_x\}$.
\item For the second degenerate type
      $\{v_x=\l_1 u_y,\,v_y=\l_2 u_x +\l_3 u_y\}$.
\end{enumerate}
\end{theorem}

The bifurcation diagram for a parabolic point is one-dimensional
and consists of an elliptic region on one side of the point considered
and of a hyperbolic region on the other side.

Bifurcation diagrams for both degenerate types are similar; they
can be represented as a neighbourhood of zero in a three-space
split by the cone $\l_3^2+4\l_1\l_2=0$ into an elliptic region (inside
the cone) and a hyperbolic region (outside the cone); the points of the
cone itself belong to the parabolic type.

Thus, the diagram of orbit adjacency looks like this:

\begin{center}
\unitlength10pt
\begin{picture}(34,8)(-1,0)
 \put(-1,6){\framebox(8,2){Degenerate I}}
 \put(-1,0){\framebox(8,2){Degenerate II}}
 \put(14,3){\framebox(6,2){Parabolic}}
 \put(27,6){\framebox(7,2){Elliptic}}
 \put(27,0){\framebox(7,2){Hyperbolic}}
 \put(8,1){\vector(2,1){5}}
 \put(8,7){\vector(2,-1){5}}
 \put(21,4.5){\vector(2,1){5}}
 \put(21,3.5){\vector(2,-1){5}}
\end{picture}
\end{center}

In the complex case all these results remain valid with one exception:
the elliptic and hyperbolic orbits merge into one.

\section{Degeneration of Symbols for Linear and Quasilinear Systems}

So far, we have been concerned with the symbol of the system
$\E\subset J_1(\R^2,\R^2)$ at one fixed point $\t\in\E$.
Let us now see what can be said about the dependence of the symbol
$\s(\E,\t)$ on this point $\t$.

For a general non-linear system  the set $\{\s(\E,\t)\}$
constitutes a six-para\-meter family, because the manifold $\E$ is
six-dimensional.

For a quasilinear system
\begin{equation}
\left\{
\begin{array}{l}
  a_1(x,y,u,v)u_x+b_1(x,y,u,v)u_y+c_1(x,y,u,v)v_x+d_1(x,y,u,v)v_y=0\\
  a_2(x,y,u,v)u_x+b_2(x,y,u,v)u_y+c_2(x,y,u,v)v_x+d_2(x,y,u,v)v_y=0
\end{array}\right.
\end{equation}
this family is constant along the fibers of the projection
$J_1\to J_0$ and thus in fact depends only on four parameters.

For a linear system
\begin{equation}
\left\{
\begin{array}{l}
  a_1(x,y)u_x+b_1(x,y)u_y+c_1(x,y)v_x+d_1(x,y)v_y=0\\
  a_2(x,y)u_x+b_2(x,y)u_y+c_2(x,y)v_x+d_2(x,y)v_y=0
\end{array}\right.
\end{equation}
this family only depends on two parameters.

The results obtained in the previous section lead to the following
conclusions about the behaviour of the regions in the parameter
spaces where the symbol of the system belongs to one or another
type.

\begin{theorem}
For a linear system (2), the degenerations of the symbol of types
I and II can be eliminated by a small movement.
The parabolic points of a typical system form a curve in the plane
$(x,y)$, on one side of which the symbol of the system is elliptic
while on the other side it is hyperbolic.
This picture is stable with respect to small deformations.

For a quasilinear systems (1) all degenerations of the symbol are stable.
In a neighbourhood of every point $(x,y,u,v)$ that belongs to either of the
two degenerate types I or II, there is an entire curve made up of points
of the same type; in a transversal to this curve one can always find
parabolic, elliptic and hyperbolic points which, up to a diffeomorphism,
fill up a quadratic cone, its inner part and its outer part.
\end{theorem}

\section{Integral Grassmanians}
According to [5], with every point $\t$ of a submanifold
$\E\subset J_k(X,Y)$, one can associate an algebraic variety
$\IG(\E,\t)$ which  consists of all $n$-planes tangent to $\E$ and
integral for the metasymplectic structure naturally defined on the 
jet bundle.
This variety is called the {\it integral Grassmanian\/} of $\E$ at $\t$;
its topology yields important information about the singularities 
that may appear in the solutions of the given system.

It is readily verified that the algebraic (and hence the topological)
type of the integral Grassmanian $\IG(\E,\t)$ is completely determined
by the type of the symbol $\s(\E,\t)$ in the sense of the previous
classification and thus Theorem 2 allows to describe all possible
types of integral Grassmanians for the first order systems $2\times2$.

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\begin{theorem}
The integral Grassmanians for the first order systems with two dependent
and two independent variables may belong to one of the following five
topological types defined by the type of the symbol:
\medskip

{\rm
\begin{center}
\begin{tabular}{|l|cl|}
\hline
Symbol type    & \multicolumn{2}{c|} {Integral Grassmanian}   \\
\hline
\begin{picture}(3.5,2.5)
\makebox(3.5,2.5){Elliptic}
\end{picture}  &  \usebox{\sphere}   &
\begin{picture}(7.5,2.5)
\makebox(7.5,2.5){Sphere $S^2$}
\end{picture}          \\
\hline
\begin{picture}(3.5,1.5)
\makebox(3.5,1.5){Hyperbolic}
\end{picture}  &  \usebox{\torus}    &
\begin{picture}(7.5,1.5)
\makebox(7.5,1.5){Torus $S^1\times S^1$}
\end{picture}           \\
\hline
\begin{picture}(3.5,2)
\makebox(3.5,2){Parabolic}
\end{picture}    &  \usebox{\ptorus}   &
\begin{picture}(7.5,2)
\put(0,0.9){\makebox(7.5,1)[c]{Torus with a meridian}}
\put(0,0.1){\makebox(7.5,1)[c]{collapsed to a point}}
\end{picture} \\
\hline
\begin{picture}(3.5,2)
\makebox(3.5,2){Degenerate I}
\end{picture}    &  \usebox{\twoplns}  &
\begin{picture}(7.5,2)
\put(0,0.9){\makebox(7.5,1)[c]{Two intersecting}}
\put(0,0.1){\makebox(7.5,1)[c]{planes in $\R P^3$}}
\end{picture} \\
\hline
\begin{picture}(3.5,2)
\makebox(3.5,2){Degenerate II}
\end{picture}  &  \usebox{\twosphr}  &
\begin{picture}(7.5,2)
\put(0,1.2){\makebox(7.5,1)[c]{Three-manifold}}
\put(0,0.2){\makebox(7.5,1)[c]{$S^2\times[0,1]/(x,0)\sim(-x,1)$}}
\end{picture} \\
\hline
\end{tabular}
\end{center}
}
\end{theorem}

\section*{References}
%
\begin{enumerate}
\item I.~G.~Petrovsky. Lectures on Partial Differential Equations
(in Russian). Fizmatgiz, Moscow, 1961.
\item H.~Goldshmidt. Integrability criteria for systems of nonlinear
partial differential equations. J. Differential Geometry {\bf 1} (1967),
p.~269--307.
\item V.~I.~Arnold, A.~N.~Varchenko, S.~Gussein-Zade. Singularities of
differentiable mappings (in Russian). Vol.~1. Nauka, Moscow, 1982.
\item S.~V.~Chmutov, S.~V.~Duzhin. Simple and unimodular cases in the
classification of linear spaces of vector-valued homogeneous forms.
In preparation).
\item V.~V.~Lychagin. Topological aspects of geometrical theory of
differential equations. Lecture Notes in Math. {\bf 1214}, Springer,
1985.
\end{enumerate}
\end{document}
