\section{Introduction}
Roughly speaking, a \BT/ is a method for generating new solutions for a given partial
differential equation by starting with a `seed' solution to the same (or a different)
PDE and solving an auxiliary system of ODEs.  B\"acklund's original example was a
transformation that produced new pseudospherical surfaces from old, and it is equivalent
to the following system:
\begin{equation} \label{SGsingleBT}
\begin{aligned}
v_x -u_x &= \tfrac12\sin((u+v)/2), \\
v_y +u_y&= -\tfrac12\sin((u-v)/2).
\end{aligned}
\end{equation}
Given an arbitrary smooth function $u(x,y)$, this overdetermined system for $v$ is inconsistent.  However,
if $u$ satisfies the {\em sine-Gordon equation} $u_{xy}=\sin u$
then the system is consistent, and
the function $v(x,y)$, determined up to a constant of integration,
will also satisfy the sine-Gordon equation.  The transformation works in reverse, too:
given a solution $v(x,y)$ for sine-Gordon, the system determines a 1-parameter family of solutions $u(x,y)$ for the same PDE.

It is this type of B\"acklund transformation---connecting solutions of two
second-order \MA/ PDE in the plane, not necessarily the same equation---which is the general subject of
this paper.  (A second-order \MA/
equation is a PDE where the highest-order derivatives may appear nonlinearly but
only in the form of the determinant of the Hessian.)
Another important example of this type is the system
\begin{equation}\label{Liouvillewave}
\begin{aligned}
z_x -u_x&=  - 2\exp((u+z)/2), \\
z_y +u_y&= \exp((u-z)/2).
\end{aligned}
\end{equation}
In this example, if $z(x,y)$ satisfies the wave equation (in characteristic coordinates, $z_{xy}=0$),
then  the system determines a 1-parameter family of solutions of
{\em Liouville's equation} $u_{xy}=e^u$, and conversely.   \BT/s of this subtype---where one of the two
PDE involved is the standard wave equation---are the specific concern of this paper.

Liouville's equation also has the rare property that it is {\em Darboux-integrable}---in other
words, it can be solved by the method of Darboux.  (This will be defined below.)
The main point of this paper is that this is not a coincidence; more precisely, we will prove

\begin{thm}\label{ourtheorem}
Let  $(\scM^5,\I)$ be a hyperbolic \MA/ system.  If there is a {\em normal} \BT/ with 1-dimensional fibers
linking this system with the wave equation, then the first prolongation of $\I$ is
Darboux-integrable.  Conversely,  if the first prolongation of $\I$ is Darboux-integrable,
then near any point $p \in \scM$ there is an open set $U\subset\scM$ around $p$ such that
the restriction of $\I$ to $U$ is linked to the wave equation by a normal \BT/.
\end{thm}

The technical terms in this theorem must be explained.  Any single PDE or system of PDE
may be re-cast as an {\em exterior differential system (EDS)}
or differential ideal (i.e., an ideal, with respect to wedge product,
 in the ring of differential forms on a manifold, that is also closed
 under the exterior derivative),
in a way that solutions are in one-to-one correspondence
with submanifolds to which all the differential forms in the EDS pull back to be zero.  
(These submanifolds, which  must usually also satisfy a nondegeneracy condition,
are known as {\em integral manifolds} of the EDS.)
In particular, a \MA/ equation in the plane can be re-cast as the following type of EDS:

\begin{defn*}A {\em \MA/ exterior differential system} is a differential ideal $\I$
on a 5-dimensional manifold $\scM$, such that near any point of $\scM$,
$\I$ is generated algebraically by one 1-form $\theta$
 and two 2-forms $\Omega_1,\Omega_2$.
(Hence, $d\theta$ must equal a linear combination of the $\Omega$'s, plus possibly
a wedge product with $\theta$ as factor.)
The 1-form $\theta$ is required to be a {\em contact form}, i.e.,
$\theta \& d\theta \& d\theta \ne 0$.
A \MA/ system is {\em hyperbolic} if the $\Omega$'s may be chosen so that both are decomposable.
\end{defn*}

For example, for Liouville's equation we may take $\scM$ to
be $\R^5$ with coordinates $x,y,u,p,q$, and let
\begin{equation}\label{Liouvilleforms}
\theta=du-p\,dx-q\,dy,\qquad
\Omega_1=(dp - e^u dy) \& dx,\qquad
\Omega_2=(dq - e^u dx) \& dy.
\end{equation}
(Note that $d\theta = -\Omega_1 - \Omega_2$.)
Given a solution $u=f(x,y)$ of the PDE, we can construct
a surface $\Sigma \subset \R^5$ such that $i^*\theta=0$, $i^*\Omega_1 = i^*\Omega_2 = 0$
(where $i:\Sigma \hookrightarrow \R^5$ is the inclusion map)
by setting $u=f(x,y)$, $p=f_x(x,y)$ and $q=f_y(x,y)$.  Conversely,
any surface $\Sigma$ satisfying $i^*\theta=0$, $i^*\Omega_1 =i^*\Omega_2=0$ and
the nondegeneracy condition $i^* dx \& dy \ne 0$
is the graph of a solution constructed in this way.

In the body of the paper, we will also use another type of EDS:
\begin{defn*} A {\em Pfaffian exterior differential system} is a differential
ideal $\I$ on an arbitrary manifold $\scM$, defined by a vector bundle $I \subset T^*\scM$,
such that a differential form belongs to $\I$ if and only if it is a linear combination
of wedge products involving sections of $I$ or their exterior derivatives.
(In practice, our Pfaffian systems will be specified by giving a list of
1-forms that span the fiber of $I$ at each point.)  The {\em rank} of a Pfaffian
system is the rank of the vector bundle.
\end{defn*}

A Pfaffian system satisfies the {\em Frobenius condition} or is said to
be {\em integrable} if the exterior derivative
of any section of $I$ is in the {\em algebraic} ideal generated
by $I$.
Any Frobenius system is locally equivalent to a (possibly underdetermined)
system of ordinary differential equations; see Chapter 1 in \cite{CFB}.


\bigskip
Theorem \ref{ourtheorem} is about relations between exterior differential systems; in particular,
we have the following definition from \cite{CFB}:
\begin{defn}\label{general-Backlund-defn}
A {\em \BT/ between two exterior differential systems} $(\scM,\I)$ and
$(\Mbar,\Ibar)$ is a manifold $\B$ equipped with submersions $\pi:\B\to \scM$
and $\pibar:\B \to \Mbar$ (see diagram below) and vector bundles $J,\Jbar \subset T^*\B$
such that
\newline
(i) the fibers of $\pi$ and $\pibar$ are transverse in $\B$;
\newline
(ii) the rank of $J$ equals the dimension of the fibers of $\pi$, and
sections of $J$ pull back to the fibers of $\pi$ to span the
cotangent space of each fiber;
\newline
(iii) $\Jbar$ is similarly related to the fibers of $\pibar$;
\newline
(iv) the algebraic ideal $\J$ generated by $\pi^* \I$ and sections of $J$ is
the same as the algebraic ideal generated by $\pibar^* \Ibar$ and sections of $\Jbar$,
and $\J$ is a differential ideal.
\end{defn}

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\put(20,25){\makebox(0,0){$\B$}}
\put(16,21){\vector(-3,-4){9}}
\put(24,21){\vector(3,-4){9}}
\put(7,16){$\pi$}
\put(30,16){$\pibar$}
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The impact of the last condition is that if $\scN \subset \scM$
is an integral manifold of $\I$, then sections of $J$ pull back to
$\pi^{-1}(\scN)$ to satisfy the Frobenius condition, so that
integral manifolds of $\J$ inside $\pi^{-1}(\scN)$ may be constructed by
solving ODE; moreover, the image under $\pibar$ of each of 
these integral manifolds is an integral manifold
of $\Ibar$.  Because the definition is symmetric, this also works in the other direction: 
given an integral manifold of $\Ibar$, 
we can solve a Frobenius system on the inverse image in $\B$
to obtain a family of integral manifolds of $\I$.
For example, given a solution $z(x,y)$ of the wave equation,
substitution in \eqref{Liouvillewave} gives an overdetermined system of ODE for
a solution $u(x,y)$ of Liouville's equation, and in this context the
Frobenius condition is exactly the compatibility condition for the ODE system.

The condition of {\em normality} for \BT/s, assumed in Theorem \ref{ourtheorem},
will be explained in \S2.

\bigskip

A hyperbolic \MA/ system is a special case of {\em hyperbolic EDS}:
\begin{defn*}A {\em hyperbolic EDS of class $k$} is a differential ideal defined on
a manifold of dimension $k+4$ that,\ near any point of manifold,
is generated algebraically by $k$ 1-forms and two decomposable 2-forms.
\end{defn*}

Associated to a given hyperbolic EDS $\I$ of class $k$ are two characteristic distributions,
one corresponding to each decomposable 2-form generator.
At each point, the distribution is given by the 2-dimensional subspace of the
tangent space annihilated by the $k$ 1-forms of the system
and the factors of the chosen decomposable 2-form.  (These annihilators
form a rank $k+2$ Pfaffian system, known as a {\em characteristic system} of $\I$.)
A hyperbolic EDS $\I$ is {\em integrable by the method of Darboux},
or {\em Darboux-integrable} for short, if both characteristic distributions have two
independent first integrals, i.e., functions which are constant
along all curves tangent to the distribution, and whose differentials are
pointwise linearly independent from the 1-forms of $\I$.
Such functions are also known as {\em characteristic invariants}, since they are constant
along integral curves of the distribution.

The Darboux-integrability condition has the virtue that it is easy to check, using
only differentiation and linear algebra, by calculating the successive derived
systems of each characteristic system.  An extensive discussion
of hyperbolic EDS and Darboux-integrability, with worked-out examples, is
available in Chapter 6 of \cite{CFB}.
For the purposes of this paper, we will need a few more facts about the method of Darboux:
\begin{itemize}
\item It is known that any \MA/ system which is Darboux-integrable (i.e.,
has two characteristic invariants for each distribution) is equivalent
to the standard wave equation under a contact transformation (see, e.g., Thm. 2.1
in \cite{BGG}).
%So, Darboux-integrable \MA/ systems are well-understood.
\item If a hyperbolic \MA/ system $\I$ has a pair of independent first integrals for exactly
one of its characteristic distributions, then $\I$ is said to be {\em integrable
by the method of Monge} or {\em Monge-integrable} for short.
(The analogous term for hyperbolic EDS of arbitrary class $k$ is {\em Darboux semi-integrable}.)
\item If a hyperbolic EDS of class $k$ fails to be Darboux-integrable, it is possible that
its {\em prolongation}, which is a hyperbolic EDS of class $k+2$, is Darboux-integrable.
Thus, a given \MA/ system may lead to a hyperbolic system that is Darboux-integrable only after
sufficiently many prolongations.
\end{itemize}
Prolongation of an EDS is essentially the process
of adding higher derivatives as new variables and adjoining to the ideal
the differential equations satisfied by the higher derivatives.
For example,
for Liouville's equation we add variables $r$ and $t$ to stand for $u_{xx}$ and $u_{yy}$
respectively, and adjoin the 1-forms $\theta_1 = dp - r\,dx -e^u dy$,
$\theta_2 = dq - e^u dx - t\,dy$.  The new ideal is a Pfaffian system on $\R^7$
(with coordinates $x,y,u,p,q,r,t$) generated by 1-forms $\theta_0,\theta_1,\theta_2$.
(Note that
the 2-forms $\Omega_1$ and $\Omega_2$ given in \eqref{Liouvilleforms} are now
in the ideal generated algebraically by these $\theta_0,\theta_1,\theta_2$.)
The set of algebraic generators of the new ideal is
completed by computing the exterior derivatives of $\theta_1,\theta_2$,
and these are expressible as linear combinations
of the decomposable forms $\Omega'_1 = (dr - p e^u dy) \& dx$, $\Omega'_2 =
(dt - q e^u dx) \& dy$, modulo multiples of $\theta_0,\theta_1,\theta_2$.
(Thus, the new ideal is a hyperbolic EDS of class 3.)
Let $\Delta_1, \Delta_2$ be the corresponding characteristic distributions for
the prolongation (i.e., $\Delta_i$ is annihilated by $\theta_0,\theta_1,\theta_2$
and the factors of $\Omega'_i$).
To see that the system is Darboux-integrable,
note that $x, r - \tfrac12 p^2$ are invariants for $\Delta_1$ and
$y, t -\tfrac12 q^2$ are invariants for $\Delta_2$.

\begin{rem}  Both Darboux-integrability and the transformation \eqref{Liouvillewave} enable
one to express all solutions of Liouville's equation via specifying two arbitrary functions
and integrating systems of ODE, and these two solution methods are
equivalent, although Darboux's method requires one to solve more ODEs.
For, as mentioned above, substituting the wave equation solution $z=f(x)+g(y)$
in \eqref{Liouvillewave} produces two
compatible ODEs for $u(x,y)$.
Given an initial value for $u$, these
may be integrated simultaneously in the $x$- and $y$-directions
to propagate a solution over an open set in the $xy$-plane.
On the other hand, under Darboux's method we obtain ODEs by setting one invariant
in each characteristic system to be an arbitrary function of the other, yielding in this case
the equations
$$p_x-\tfrac12 p^2 = \phi(x), \qquad q_y-\tfrac12 q^2 = \psi(y),$$
which, together with $u_x=p$ and $u_y=q$, may be integrated to obtain the solution.
(In fact, the data for these two methods are related by $\phi=f''$ and $\psi=g''$, but
in other cases we cannot expect the relationship to be this simple.)
\end{rem}

Next, we will put Theorem \ref{ourtheorem} in context with other results both classical and modern.
Much of what was known in the 19th century about solving second-order PDE for
one function of two variables was summarized in Goursat's treatise \cite{GLecons}.
In Volume 2, \S181 of that work, we find the following result:
\begin{thm}[Darboux-Goursat]
Suppose that a second-order PDE for $z$ as a function of $x,y$ has the property
that there exists a Pfaffian system
$$dF_i = \Phi_i\,d\alpha + \Psi_i d\beta, \qquad 1\le i \le \ell,$$
and formulas
\begin{equation}\label{dgxv}
x = V_1, \quad y = V_2, \quad z=V_3,
\end{equation}
where $\Phi_i$, $\Psi_i$, $V_1, V_2, V_3$ are functions of $F_1,\ldots, F_\ell,$ $\alpha,\beta$,
$f(\alpha)$, $g(\beta)$ and finitely many of the derivatives of $f$ and $g$,
such that \eqref{dgxv} satisfies the Frobenius condition
for arbitrary choices of functions $f$ and $g$, and gives an implicit solution of the PDE for
arbitrary choices of initial data for the Frobenius system.
Then the PDE is Darboux-integrable after finitely many prolongations.
\end{thm}
The hypotheses of the Darboux-Goursat theorem are fulfilled if the given PDE is
linked to the standard wave equation by a \BT/.  (For, the d'Alembert formula
gives solutions of the wave equation $u_{\alpha\beta}=0$ in the form
$u=f(\alpha)+g(\beta)$ for arbitrary $f$ and $g$, and the Pfaffian
system in the theorem is given by the equations of the \BT/.)
Compared with the Darboux-Goursat theorem, one direction of our theorem has a stronger hypothesis
(essentially, that $\ell=2$ and only first derivatives of $f$ and $g$ are involved)
and a stronger conclusion (that at most one prolongation is required to get Darboux-integrability).

In Theorem 6.5.14 in \cite{CFB} it is shown, by an elementary argument, that
Darboux-integrability of the prolongation implies that there is a \BT/ between the
prolongation (not the original \MA/ system, but one defined by an EDS on a 7-dimensional
manifold) and the wave equation (defined by an EDS on a 5-dimensional manifold).
However, this asymmetric transformation--relating the 2-jets of solutions of
one PDE to the 1-jets of another---is less than satisfying, compared to
more symmetrical transformations like \eqref{Liouvillewave}.  Our analysis in \S4 shows
that it is a much more delicate matter to show that there exists a \BT/ between
two \MA/ systems.  We should also note that
the argument given in \cite{CFB} for the other direction (\BB/-equivalence
to the wave equation implies Darboux-integrability) is unfortunately incorrect,
and the proof we give in \S3 of this paper is along completely different lines.

\bigskip

We now briefly outline the rest of the paper.  In \S2 we set up the basic machinery required for
the first half of the proof, namely, the $G$-structure for \BT/s originally introduced by the first author
in \cite{C1}.  In \S3 we prove the forward direction in our theorem by
following the implications (for the invariants of the $G$-structure) of the existence of a \BT/ to the wave equation.
In \S4 we prove the converse direction by constructing, for any given Darboux-integrable Monge-Amp\`ere equation,
an involutive exterior differential system whose solutions are such transformations.
In \S5, we discuss our results in the context of earlier classifications
of Darboux-integrable equations and of \BT/s to the wave equation; 
we also outline an alternate
proof technique for the converse direction, which can in some examples be used
to establish global existence of the transformation.
In \S6 we discuss further steps in our research program.

We are grateful to the referee who read the first version of this paper, and gave us many 
useful comments and suggestions.