GENERALIZED LAPLACE INVARIANTS AND CLASSICAL INTEGRATION METHODS
FOR SECOND ORDER SCALAR HYPERBOLIC PARTIAL DIFFERENTIAL
EQUATIONS IN THE PLANE

Martin Jur\'{a}\v{s}

Key words: Darboux integrable, exactly integrable,
        Laplace invariants, intermediate integrals.

1991 Math. Subject Classification: 58G16.

Abstract: In this paper we announce several new results concerning
classical integration methods for second order scalar hyperbolic
partial differential equations in the plane.
We find that the vanishing of the generalized Laplace
invariants  is  both necessary and sufficient  for the equation
to be Darboux integrable. We invariantly characterize
the various general cases of Darboux integrability due to Goursat.
We also find necessary and sufficient conditions
for an equation to admit a general or a complete intermediate integral.
Our methods are based upon the introduction of an adapted coframe for
the prolonged equation manifold. This coframe is constructed
by generalizing the classical Laplace transform used to integrate
certain linear equations in the plane.