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The Monge-Ampere equationOriginally Posted by theta
 = V)
was studied extensively in the 1800's most notably by Lie. He
showed that it was possible to find first integrals to this
equation. If one solves
for r and t and substitutes into our PDE and separate wrt s then
we obtain the Monge equations
Considering a linear combination of these two shows that they can
be factored ifare two real distinct solutions of
and further, the factors are
In your case wherethen we have
which has factors
. Thus, we have
which gives rise to
or
leading to the first integrals
and
Sophus Lie (1877) showed that via a contact transformation that it
possible to transform PDE like yours to the wave equation. In
your case the transformation is
(U_X+U_Y) -<br />
\frac{1}{2}(U_X+U_Y)^2-\frac{1}{2}(X-Y)^2-U<br />
)
)
givingWith the general solution as
, gives the general solution of your PDE via (1).
